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url stringlengths 15 1.13k	text stringlengths 100 1.04M	metadata stringlengths 1.06k 1.1k
http://mathoverflow.net/users/4428/sasha	# Sasha less info reputation 11527 bio website location age member for 5 years, 3 months seen Jun 9 at 8:32 profile views 6,644 29 Sheaves without global sections 19 Composing left and right derived functors 17 Extending vector bundles on a given open subscheme, reprise 16 Understanding Adjointness of Sheaves in Algebraic Geometry 15 Automorphisms of a weighted projective space # 13,775 Reputation +10 Holomorphic Poisson brackets on Fano manifolds +10 Atiyah class for non-locally free sheaf +10 Branch locus of a 6:1 cover of the grassmannian G(1,3) +10 When does the direct image functor commute with tensor products? # 4 Questions 9 Highest weights of the restriction of an irreducible representation of a simple group to a Levi subgroup 7 Littlewood-Richardson rule and commutativity morphism 6 Flatly compactifiable morphisms 5 Universal nondegenrate map over the space of complete linear maps # 135 Tags 874 ag.algebraic-geometry × 233 75 ct.category-theory × 15 131 ac.commutative-algebra × 26 67 complex-geometry × 17 126 homological-algebra × 28 58 vector-bundles × 16 124 derived-category × 24 51 sheaf-theory × 14 83 reference-request × 19 49 rt.representation-theory × 16 # 1 Account MathOverflow 13,775 rep 11527	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9114129543304443, "perplexity": 3794.3760718503877}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-27/segments/1435375095184.64/warc/CC-MAIN-20150627031815-00279-ip-10-179-60-89.ec2.internal.warc.gz"}
http://mathhelpforum.com/number-theory/162148-euler-s-criterion.html	# Math Help - Euler's criterion 1. ## Euler's criterion "Let p = a prime. Show $x^2$ ≡ a (mod $p^2$) has 0 solutions if $x^2$ ≡ a (mod p) has 0 solutions, or 2 solutions if $x^2$ ≡ a (mod p) has 2." I thought I should use Euler's criterion for this(?), which says $x^2$ $a (mod p^2)$ if and only if $a^{(p-1)/2}$ ≡ 1 (mod p), but I don't know where to go from there. Also, I know that a (mod p) can have only either exactly two solutions or zero solutions (by Lagrange), so the question remains how to map a (mod p) to a (mod $p^2$) Thanks !	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 8, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.952474057674408, "perplexity": 368.9717018935902}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-41/segments/1412037663007.13/warc/CC-MAIN-20140930004103-00099-ip-10-234-18-248.ec2.internal.warc.gz"}
https://blog.jpolak.org/?p=1464	# Pretty Plots with Pygame and Python The other day I learned a little about the pygame python module that provides access to sdl functions on Linux along with some higher level functions. Basically, this means that you can use pygame to access some pretty low level graphics to draw things. I thought it might be fun to use it to make pretty pictures. Here’s an example where points are plotted, with the colour of each point a function of the coordinates. These simple examples should indicate how this type of low-level drawing can be used, perhaps to visualise complicated data or make interesting animations. In sdl as with many computer graphics systems, coordinates start from the top left with $(0,0)$, and proceed positively right and down. The equations presented below give the red-green-blue (RGB) values of the color of the point at $(x,y)$. Each value is taken modulo $255$ (actually, I should have taken the value modulo $256$, since each color goes from $0$ to $255$, but I didn’t think carefully about this when I wrote the program!). \begin{align} r &= x\\ g &= x-y\\ b &= x + y \end{align} \begin{align} r &= x^2\\ g &= x\\ b &= x+y \end{align} \begin{align} r &= 255\cos(x)\\ g &= 255\sin(x)\\ b &= xy \end{align} \begin{align} r &= 100 + y\\ g &= x/(10 + y)\\ b &= x – y^2 \end{align} \begin{align} r &= x + y + xy\\ g &= x – y + 240\\ b &= x + y^4 \end{align} \begin{align} r &= x^2\\ g &= 4y – x^4\\ b &= y^3 – y^2 + y \end{align} \begin{align} r &= xy\\ g &= x^5-y^5\\ b &= x^{13} \end{align} \begin{align} r &= x^2y^3\\ g &= y + x^2 + 7xy\\ b &= x + y^2 + x^3 + x^4 + y^6 \end{align} \begin{align} r &= 50\sin(x^2 + y^2)\\ g &= 40 + 10\sin(x) + 20\cos(y)\\ b &= 200 \end{align} \begin{align} r &= (1 + x + 100x^3)/(1 + 25y^2)\\ g &= y/4\\ b &= (1 + 200y^3)/(1 + x^4) \end{align*} Here is the code (since it’s so short, consider it public domain):	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 1.000009298324585, "perplexity": 843.7797017969989}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178381803.98/warc/CC-MAIN-20210308021603-20210308051603-00349.warc.gz"}
http://mathhelpforum.com/discrete-math/166516-number-solutions-linear-equation.html	# Math Help - Number of solutions to linear equation 1. ## Number of solutions to linear equation How many solutions are there to the equation x1+x2+x3+x4 = 10, where for every i, 1 <= xi <= 4 ? 2. One way is to find the coefficient of $x^{10}$ in the expansion of $(x^4+x^3+x^2+x)^4$, which can be done in WolframAlpha. I would not say that this restatement makes the counting easier; it just reduces the problem to a well-known question for which automatic tools exist. 3. Can you solve it without that trick? I mean without using automatic tools, let's say its a question on the test, what would you do then?:P 4. You could also try brute force. Consider the following table. Let $a_{i,j}$ be the element at row $i$ and column $j$. We want $a_{i,j}$ be the number of solutions of $x_1+\dots+x_i=j$ where $1\le x_k\le 4$ for all $1\le k\le i$. $ \begin{array}{c\|cccccccccc} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10\\ \hline 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 2 & 0 & 1 & 2 & 3 & 4 & 3 & 2 & 1 & 0 & 0\\ 3 & 0 & 0 & 1 & & & \mathbf{10} & \mathbf{12} & \mathbf{12} & \mathbf{10} & \\ 4 & 0 & 0 & 0 & 1 & & & & & & \mathbf{44} \end{array} $ The non-bold elements of the table are computed directly. Next, for $i>1$ and $j>4$, we have $a_{i,j}=a_{i-1,j-4}+a_{i-1,j-3}+a_{i-1,j-2}+a_{i-1,j-1}$. 5. OK, its nice and all, I have tried to solve it in a different way that seems so correct but doesn't work and I'm confused. Look what I did: because xi >= 1 we can change variable to xi = yi + 1 then we have y1 + 1 + y2 + 1 + y3 + 1 + y4 + 1 = 10 => y1+y2+y3+y4= 6. now we want to find the number of solutions to y1+y2+y3+y4= 6 where for every i, 0<=yi<=4. for getting the number of solutions to the latter equation where yi >=0, we can use the formula for dividing 6 identical objects into 4 different boxes it's \|U\|= nCr(4-1+6,4-1)=84, the size of our universe. now we can use the inclusion & exclusion in order to find number of solutions where y1 >= 5 or y2 >= 5 or y3 >= 5 or y4 >= 5, there are only 16 such solutions. so the final answer is 84-16=68. but the answer is incorrect. where is my mistake? I cant figure it out, just looking for it for hours. 6. oh, just one little detail that makes a big different, it should be yi <=3 instead of 4, but thanks! 7. because xi >= 1 we can change variable to xi = yi + 1 then we have y1 + 1 + y2 + 1 + y3 + 1 + y4 + 1 = 10 => y1+y2+y3+y4= 6. now we want to find the number of solutions to y1+y2+y3+y4= 6 where for every i, 0<=yi<=4. It should be, 0 <= y_i <= 3. Your method is otherwise correct. I was just looking into it at this thread. Let s be the sum of all $y_i$'s, namely, 6, and let m be the maximum of $y_i$'s, namely, 3. What makes it easy to subtract the number of variants where some $y_i>m$ is that $s\le 2m+1$. In this case, when $y_1=m+1$, $y_2+y_3+y_4=s-(m+1)\le m$, so the restriction that $y_i\le m$ does not have to be taken into account and the number of solutions of $y_2+y_3+y_4=s-(m+1)$ can be calculated by the general formula. 8. ## You don't need alpha for this one Here is a way to get the coefficient of x^10 from the generating function. The ordinary power series generating function (as already noted) is $f(x) = (x + x^2 + x^3 + x^4)^4$ $= x^4 (1 + x + x^2 + x^3)^4$ $=x^4 \left( \frac{1-x^4}{1-x} \right) ^4$ $=x^4 (1-x^4)^4 (1-x)^{-4}$ $=x^4 (1 -4x^4 + 6x^8 -4x^{12} + x^{16}) \sum_{i=0}^{\infty} \binom{4+i-1}{i} x^i$ Looking at the last equation, we can see the coefficient of x^10 is $\binom{9}{3} - 4 \binom{5}{2}$ 9. Hello, Ginsburg! A slight variation of emarakov's solution . . . $\text{How many solutions are there to the equation}$ . . $x_1+x_2+x_3+x_4 \:=\: 10\:\text{ where }1 \le x_i \le 4\,?$ . . $\begin{array}{ccc} x_1,x_2,x_3,x_4 & \text{Permutations} \\ \hline \\[-4mm] (1,1,4,4) & \frac{4!}{2!\,2!} \;=\;6 \\ \\[-3mm] (1,2,3,4) & 4! \;\;=\;24 \\ \\[-3mm] (1,3,3,3) & \frac{4!}{1!\,3!} \;=\;4 \\ \\[-3mm] (2,2,2,4) & \frac{4!}{3!\,1!} \;=\;4 \\ \\[-3mm] (2,2,3,3) & \frac{4!}{2!\,2!} \;=\;6 \\ \\[-4mm] \hline \\[-4mm] 10. By the Erdos principle, there is always a simple explanation for a simple answer Ginsburg was quite close in an earlier post. Number of solutions to $x_1 + x_2 + x_3 + x_4 = 10$ with $1 \leq x_i \leq 4$ is the number of solutions to (substitue $y_i = x_i - 1$) $y_1 + y_2 + y_3 + y_4 = 10 - 4 = 6$ with $0 \leq y_i \leq 3$ Number of solutions to $y_1 + y_2 + y_3 + y_4 = 6$ with $0 \leq y_i$ is $\binom{6 + 4 - 1}{4 - 1} = \binom{9}{3}$ We need to throw away solutions where one of $y_i \geq 4$. For the case of $y_1 \geq 4$, let $z = y_1 - 4$, the number of solutions to $z + y_2 + y_3 + y_4 = 6 - 4 = 2$ is $\binom{2 + 4 - 1}{4 - 1} = \binom{5}{3}$ Since exactly one of $y_i$ could be larger than 3 in any solution. We have $4\binom{5}{3}$. Subtracting invalid solutions, the number of possible solutions is: $\binom{9}{3} - 4\binom{5}{3}$ Consistent with awkward's solution. 11. Originally Posted by snowtea By the Erdos principle, there is always a simple explanation for a simple answer Ginsburg was quite close in an earlier post. Number of solutions to $x_1 + x_2 + x_3 + x_4 = 10$ with $1 \leq x_i \leq 4$ is the number of solutions to (substitue $y_i = x_i - 1$) $y_1 + y_2 + y_3 + y_4 = 10 - 4 = 6$ with $0 \leq y_i \leq 3$ Number of solutions to $y_1 + y_2 + y_3 + y_4 = 6$ with $0 \leq y_i$ is $\binom{6 + 4 - 1}{4 - 1} = \binom{9}{3}$ We need to throw away solutions where one of $y_i \geq 4$. For the case of $y_1 \geq 4$, let $z = y_1 - 4$, the number of solutions to $z + y_2 + y_3 + y_4 = 6 - 4 = 2$ is $\binom{2 + 4 - 1}{4 - 1} = \binom{5}{3}$ Since exactly one of $y_i$ could be larger than 3 in any solution. We have $4\binom{5}{3}$. Subtracting invalid solutions, the number of possible solutions is: $\binom{9}{3} - 4\binom{5}{3}$ Consistent with awkward's solution. And is consistent with a direct count! CB 12. Originally Posted by CaptainBlack And not consistent with a direct count! CB CB, I find 44 solutions, listed below. Do you differ with that list? Code: 1 1 1 4 4 2 1 2 3 4 3 1 2 4 3 4 1 3 2 4 5 1 3 3 3 6 1 3 4 2 7 1 4 1 4 8 1 4 2 3 9 1 4 3 2 10 1 4 4 1 11 2 1 3 4 12 2 1 4 3 13 2 2 2 4 14 2 2 3 3 15 2 2 4 2 16 2 3 1 4 17 2 3 2 3 18 2 3 3 2 19 2 3 4 1 20 2 4 1 3 21 2 4 2 2 22 2 4 3 1 23 3 1 2 4 24 3 1 3 3 25 3 1 4 2 26 3 2 1 4 27 3 2 2 3 28 3 2 3 2 29 3 2 4 1 30 3 3 1 3 31 3 3 2 2 32 3 3 3 1 33 3 4 1 2 34 3 4 2 1 35 4 1 1 4 36 4 1 2 3 37 4 1 3 2 38 4 1 4 1 39 4 2 1 3 40 4 2 2 2 41 4 2 3 1 42 4 3 1 2 43 4 3 2 1 44 4 4 1 1 13. Originally Posted by awkward CB, I find 44 solutions, listed below. Do you differ with that list? Code: 1 1 1 4 4 2 1 2 3 4 3 1 2 4 3 4 1 3 2 4 5 1 3 3 3 6 1 3 4 2 7 1 4 1 4 8 1 4 2 3 9 1 4 3 2 10 1 4 4 1 11 2 1 3 4 12 2 1 4 3 13 2 2 2 4 14 2 2 3 3 15 2 2 4 2 16 2 3 1 4 17 2 3 2 3 18 2 3 3 2 19 2 3 4 1 20 2 4 1 3 21 2 4 2 2 22 2 4 3 1 23 3 1 2 4 24 3 1 3 3 25 3 1 4 2 26 3 2 1 4 27 3 2 2 3 28 3 2 3 2 29 3 2 4 1 30 3 3 1 3 31 3 3 2 2 32 3 3 3 1 33 3 4 1 2 34 3 4 2 1 35 4 1 1 4 36 4 1 2 3 37 4 1 3 2 38 4 1 4 1 39 4 2 1 3 40 4 2 2 2 41 4 2 3 1 42 4 3 1 2 43 4 3 2 1 44 4 4 1 1 It is a gross typo (or a time slip)it is consistent is what the post should say, and will do presently. CB	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 62, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9662557244300842, "perplexity": 306.73178913522713}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-26/segments/1466783394987.40/warc/CC-MAIN-20160624154954-00160-ip-10-164-35-72.ec2.internal.warc.gz"}
http://proceedings.mlr.press/v101/kato19a.html	# Learning Weighted Top-$k$ Support Vector Machine Tsuyoshi Kato, Yoshihiro Hirohashi ; Proceedings of The Eleventh Asian Conference on Machine Learning, PMLR 101:774-789, 2019. #### Abstract Nowadays, the top-$k$ accuracy is a major performance criterion when benchmarking multi-class classifier using datasets with a large number of categories. Top-$k$ multiclass SVM has been designed with the aim to minimize the empirical risk based on the top-$k$ accuracy. There already exist two SDCA-based algorithms to learn the top-$k$ SVM, enjoying several preferable properties for optimization, although both the algorithms suffer from two disadvantages. A weak point is that, since the design of the algorithms are specialized only to the top-$k$ hinge, their applicability to other variants is limited. The other disadvantage is that both the two algorithms cannot attain the optimal solution in most cases due to their theoritical imperfections. In this study, a weighted extension of top-$k$ SVM is considered, and novel learning algorithms based on the Frank-Wolfe algorithm is devised. The new learning algorithms possess all the favorable properties of SDCA as well as the applicability not only to the original top-$k$ SVM but also to the weighted extension. Geometrical convergence is achieved by smoothing the loss functions. Numerical simulations demonstrate that only the proposed Frank-Wolfe algorithms can converge to the optimum, in contrast with the failure of the two existing SDCA-based algorithms. Finally, our analytical results for these two studies are presented to shed light on the meaning of the solutions produced from their algorithms.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8881585001945496, "perplexity": 888.9240273523842}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579251773463.72/warc/CC-MAIN-20200128030221-20200128060221-00173.warc.gz"}
http://math.stackexchange.com/questions/454871/alternating-vector-between-bases-makes-it-still-a-basis	Alternating vector between bases makes it still a basis? Let $V$ be a space over field $F$. $B=\{ v_1, v_2,...,v_n\}$, $C\{ u_1, u_2,...,u_n\}$ are bases of $V$. Show there is $i \in \{1,2,...,n \}$ so that the set $\{ v_2,...,v_n, u_i\}$ is a basis of $V$. I tried proof by contradiction and my intuition tells me that's the correct approach, I just got stuck. Showing it's linear independent is sufficient to prove it's a basis if I'm not mistaken (due to number of vectors). Thanks in advance for any assistance! P.S. I would like if someone had a better idea of how to word my title. - The question as stated is incorrect, since any two bases have the same cardinality. So if $\{v_1, \dotsc, v_n\}$ is a basis, then $\{v_1, \dotsc, v_n, u_i\}$ cannot be a basis, since it has one more element. You may mean $\{v_1, \dotsc, v_{n-1}, u_i\}$ is a basis? – Eric Auld Jul 29 '13 at 14:05 As written, your claim is false: If $\dim_F V=n$, no set of $n+1$ vectors can be a basis. – Andrea Mori Jul 29 '13 at 14:06 Oops, fixed! Thank you. – ohad Jul 29 '13 at 14:07 Don't forget that a set of linearly independent vectors always can be extended to a basis. – Sigur Jul 29 '13 at 14:09 I edited my answer because I accidentally changed the notation in the question. 1.: Every $u_i$ can be written as a linear combination of the $v_j$. Prove that there is an $i$ such that the coefficient $c_1$ of $v_1$ in $$u_i=\sum_{j=1}^n c_j v_j$$ is not zero. If there wasn't one then the $u_i$ would be contained in the span of $v_2,\ldots, v_n$ so they couldn't be a basis. 2.: Show that the vectors $v_1,\ldots,v_{n-1},u_i$ are linearly independent. - As for 1., there must be such non-zero coefficient, otherwise $\{u_1,...,u_n\}$ is linearly dependent because there is $u_i=0$ for some $i$. Is that correct? And now you just say $u_i=c_1v_1+...+c_nv_n$. And so $$d_2v_2+...+d_nv_n+u_i=0 \rightarrow$$ $$d_2v_2+...+d_nv_n+c_1v_1+...+c_nv_n=0 \rightarrow$$ $$c_1v_1+(c_2+d_2)v_2+...+(c_n+d_n)v_n=0$$ And since $\{v_1,..v_n\}$ is linearly independent all the coefficients are zero. So the original expression is linearly independent and therefore a basis. Is this correct? – ohad Jul 29 '13 at 14:19 @ohad: Sorry I made a slight change in notation. I wanted to show that $v_1,\ldots,v_{n-1},u_i$ is linearly independent! – Michalis Jul 29 '13 at 14:23 Michalis isn't it equivalent in method? But is my way of solution generally correct? – ohad Jul 29 '13 at 14:25 @ohad: For the first step your argument is not quite correct. I was going for the following: If $c_n$ is zero for all $u_i$ then all $u_i$ would be in the span of $v_1,\ldots,v_{n-1}$ so they can't form a basis. You started well in the second step except that I wanted you to show that $v_1,\ldots,v_{n-1},u_i$ are linearly independent. Please adjust your solution, if you want I'll explain my concerns with your solution in the chat. – Michalis Jul 29 '13 at 14:25 If $\{v_1,...,v_n\}$ is a basis then $\{v_2,...,v_n\}$ are linearly independent. Now, what does it mean that $\{v_2,...,v_n,u_i\}$ is not a basis? What if this happens for all $i$? We claim that there's a vector $u_i$ such that $$u_i=\sum_{k=1}^n\alpha_k v_k,\quad \alpha_1\neq 0$$ otherewise the family $(v_2,\ldots,v_n)$ span all the vectors $(u_1,\ldots,u_n)$ which's a contradiction. I think that the rest of reasoning is clear.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9360479116439819, "perplexity": 164.6811878148316}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-27/segments/1435375098685.20/warc/CC-MAIN-20150627031818-00017-ip-10-179-60-89.ec2.internal.warc.gz"}
https://www.groundai.com/project/x-shooter-study-of-accretion-in-chamaeleon-i/	# X-Shooter study of accretion in Chamaeleon I1 ###### Key Words.: Stars: pre-main sequence - Stars: variables: T Tauri - Accretion, accretion disks - Protoplanetary disks - open clusters and associations: individual: Chamaeleon I We present the analysis of 34 new VLT/X-Shooter spectra of young stellar objects in the Chamaeleon I star-forming region, together with four more spectra of stars in Taurus and two in Chamaeleon II. The broad wavelength coverage and accurate flux calibration of our spectra allow us to estimate stellar and accretion parameters for our targets by fitting the photospheric and accretion continuum emission from the Balmer continuum down to 700 nm. The dependence of accretion on stellar properties for this sample is consistent with previous results from the literature. The accretion rates for transitional disks are consistent with those of full disks in the same region. The spread of mass accretion rates at any given stellar mass is found to be smaller than in many studies, but is larger than that derived in the Lupus clouds using similar data and techniques. Differences in the stellar mass range and in the environmental conditions between our sample and that of Lupus may account for the discrepancy in scatter between Chamaeleon I and Lupus. Complete samples in Chamaeleon I and Lupus are needed to determine whether the difference in scatter of accretion rates and the lack of evolutionary trends are not influenced by sample selection. ## 1 Introduction During the first Myr of evolution toward the main sequence, the disk around young stellar objects (YSOs) evolves and is dispersed mainly by disk accretion processes and photoevaporation (Alexander et al., 2014). The importance and the properties of disk accretion are constrained by studying objects located in different star-forming regions and at different evolutionary phases. Observations of accretion in YSOs mostly focus on determining the amount of material accreted onto the central star per unit time. This mass accretion rate (, e.g., Hartmann et al., 1998) may be compared to the stellar and disk properties to test the predictions of models of disk evolution. Relations between the accretion luminosity () and stellar luminosity (, e.g., Natta et al., 2006; Clarke & Pringle, 2006; Rigliaco et al., 2011a; Manara et al., 2012) and between and stellar mass (, e.g., Muzerolle et al., 2003; Mohanty et al., 2005; Natta et al., 2006; Herczeg & Hillenbrand, 2008; Antoniucci et al., 2011, 2014; Manara et al., 2012; Alcalá et al., 2014; Ercolano et al., 2014; Frasca et al., 2015) are frequently derived to examine the dependence of accretion on the properties of the central star. Accretion has sometimes been found to correlate with age (e.g., Hartmann et al., 1998; Sicilia-Aguilar et al., 2010; Antoniucci et al., 2014), although ages of individual young stars in a given star-forming region are very uncertain as a result of both observational and model-dependent uncertainties (e.g., Soderblom et al., 2014), including a dependence of age estimate on the spectral type (e.g., Herczeg & Hillenbrand, 2015). Measurements of are obtained spectroscopically using direct or indirect methods. Flux-calibrated spectra that span a wide wavelength range and include the Balmer continuum region yield direct estimates of the while also allowing for the incorporation of veiling into spectral type and extinction measurements. This approach is thought to lead to more reliable estimates for the stellar mass and radius, which are necessary ingredients when estimating and ages (Manara et al. 2013b; Herczeg & Hillenbrand 2014, hereafter HH14). While the entire UV spectrum is obtained only from space telescopes and is then used to derive for relatively small samples of YSOs (e.g., Gullbring et al., 2000; Ingleby et al., 2013), recent instruments mounted on very large telescopes give the community access to large quantity of spectra in the region of the visible UV from Earth, namely from 300 nm to 400 nm, which is sufficient for capturing the Balmer jump. Modeling the excess continuum emission at these wavelengths yields direct measurements of . This method has been used in the past on a substantial number of objects (e.g., Gullbring et al., 1998; Herczeg & Hillenbrand, 2008; Ingleby et al., 2014), but often using non-simultaneous and low-resolution spectra at optical wavelengths to determine the photospheric properties of the target. On the other hand, the indirect method of measuring the luminosity of various emission lines () and converting this into through calibrated - relations (e.g., Muzerolle et al., 1998; Fang et al., 2009) can be efficiently used on spectra covering a relatively narrow wavelength range, such as those offered by multi-object spectrographs. Recent studies have also demonstrated that derived from several emission lines measured simultaneously lead to results compatible, although with a larger scatter, with more direct methods (Rigliaco et al., 2012; Alcalá et al., 2014). Therefore, measuring from the luminosity of emission lines is a method that is being extensively used in the literature (e.g., Mohanty et al., 2005; Natta et al., 2006; Fang et al., 2009; Costigan et al., 2012; Antoniucci et al., 2011, 2014; Biazzo et al., 2014). The second-generation VLT instrument X-Shooter allows us to obtain simultaneous spectra of the whole wavelength range from 300 nm to 2500 nm at medium resolution and to thus derive directly from the UV-excess and stellar properties from the photospheric features of the spectra. Manara et al. (2013b) have developed a new method to model all the components of the observed X-Shooter spectra of YSOs to directly derive simultaneously with the stellar parameters and extinction. First studies of accretion with this instrument focused on individual targets (e.g., Rigliaco et al., 2011b; Stelzer et al., 2013) or on a few very low-mass stars and BDs in -Orionis (Rigliaco et al., 2012). The largest sample studied to date with this instrument is a set of 36 YSOs in the Lupus I and III clouds (Alcalá et al., 2014). This study found a significantly smaller spread of values of at any given with respect to previous works. This result opened new questions in the field, such as whether Lupus is a special region where the initial conditions of star formation are similar for different objects, or if the instrinsic YSO variability in this region is inherently small. At the same time, the narrow distribution derived using X-Shooter spectra started to show the advantage of high-quality and broad-band spectra in obtaining more precise accretion rates. A large sample of 22 transitional disks (Manara et al., 2014) has also been studied with this instrument and the method by Manara et al. (2013b), but larger samples of known full disks in various star-forming regions studied with the same method are needed to determine the general evolution of accretion in YSOs. In this paper we present a study of a large sample of 34 Class II YSOs in the Chamaeleon I star-forming region ( = 160 pc, Luhman, 2008) observed with X-Shooter, and of a few additional targets in Taurus and Chamaeleon II. These results are used to verify whether the small spread of found in Lupus is confirmed in another star-forming region and to extend the comparison between accretion and stellar properties to solar-mass YSOs, which were not targeted by Alcalá et al. (2014). The paper is organized as follows. First, the observations and data reduction procedures are explained in Sect. 2, then the method used to derive stellar and accretion parameters is presented and compared with literature estimates in Sect. 3. The results are then presented in Sect. 4 and are discussed in Sect. 5, including a comparison with other studies with similar methods in other star-forming regions. Finally, Sect. 6 presents the main conclusions of this work. ## 2 Observations and data reduction ### 2.1 Main sample The sample of pre-main sequence (PMS) stars presented in this study was selected from Luhman (2004, 2007), Spezzi et al. (2008), and Alcalá et al. (2008) and has been observed with X-Shooter (Vernet et al., 2011) on the Very Large Telescope (Paranal, Chile) between January 17 and 19, 2010. X-Shooter provides simultaneous medium-resolution ( 5000 - 18000, depending on the spectral arm and the slit width) spectroscopic observations between 300-2500 nm. It consists of three independent cross-dispersed echelle spectrographs, referred to as the UVB, VIS, and NIR arms, with the following spectral coverage: UVB (300-559.5 nm), VIS (559.5-1024 nm), and NIR (1024-2480 nm). The observations were executed by nodding the telescope along a direction perpendicular to the slit. Before the nodding observation, each target was observed in stare mode using a large slit width of 15 - 5″. Large-slit spectra have been used to properly flux calibrate the spectra obtained with narrow slits, which are used for the analysis because they lead to a higher spectral resolution. The observing log is reported in Table LABEL:tab::log and includes the time of observation, the exposure times, the number of nodding positions, and the width of the slits used for the high-resolution observations. The spectra were reduced with the ESO X-Shooter pipeline (Modigliani et al., 2010) version v1.3.2 following a standard reduction scheme: bias subtraction (stare mode only), flat-fielding, wavelength calibration, background removal (stare mode only), and spectrum extraction. The latter is performed manually on the rectified spectrum, and after the extraction of each component of a binary system when both were included in the slit and spatially separated. For the spectra taken in the nodding mode, the background removal is obtained by the difference of spectra pairs. The UVB and VIS arms are corrected for atmospheric extinction using the atmospheric transmission curve of Paranal by Patat et al. (2011). The spectrophotometric standard star GD-71 was observed at the beginning of each night. The spectral response function of the instrument, which gives the conversions from ADU to erg cm s Å, was obtained by dividing the observed spectrum by the tabulated spectrum provided with the X-Shooter pipeline. This has been derived from observations of GD-71 for each night and each arm independently. The response function for the UVB and VIS arms does not show variations during the three nights of observation. Small variations are observed for the NIR arm, and the response function is taken as the average of the three nights. The overall flux calibration accuracy was tested on telluric standard star observations and is stable with a 2% accuracy. Finally, the narrow-slit spectra are scaled upward to match the large-slit spectra. The scaling factor is computed as the ratio of the spectral continuum of the two spectra. More information on the reduction and flux calibration of the spectra obtained in this program will be discussed in Rugel et al., in prep. In total, 40 bona-fide Class II members of the Chamaeleon I star-forming region were observed. Here we discuss the properties of 34 of these objects. The other six objects are transitional disks sources that were analyzed by Manara et al. (2014) with the same method as was implemented in this paper, and they are included in our analysis. The membership of 22 YSOs in the Chamaeleon I sample has been confirmed by recent proper motion studies (Lopez Martí et al., 2013). Following Luhman (2008, and references therein), the adopted distances to objects in the Chamaeleon I region is 160 pc. In addition to these targets, two objects located in the Chamaeleon II cloud (=178 pc, Luhman, 2008), both confirmed members according to Lopez Martí et al. (2013), and four targets located in the Taurus molecular cloud (?)=131-140 pc,][]Herczeg14 were observed, and their accretion properties are reported here. These targets were observed with the same observing strategy as the Chamaeleon I objects and thus are interesting objects with which to compare the accretion properties of YSOs in different star-forming regions. ### 2.2 Additional templates used for the analysis The analysis presented here makes use of several photospheric templates observed with X-Shooter to estimate the stellar parameters of the targets. These templates are non-accreting Class III YSOs collected and characterized mainly by Manara et al. (2013a). This set of templates is almost complete for objects with SpT in the M-subclass, but includes only few objects with SpT in the K-subclass (one K5, two K7 YSOs). Thus, it is useful to include some additional templates here. Three of them have been discussed in Manara et al. (2014) and have SpT G4, G5, and K2. Finally, another target, HBC407, is included here. This object is a Class III YSO (Luhman et al., 2010) with SpT K0 (HH14), which was observed during the program Pr.Id.094.C-0913 (PI Manara). Here we discuss the observations, data reduction, and classification for this target. HBC407 was observed in service mode during the night of December 8, 2014 with the following strategy. First, a complete nodding cycle ABBA was performed using the narrower slits 0.5″-0.4″-0.4″in the three arms, respectively, with an exposure time for each exposure of 430 s in the UVB arm, 375 s in the VIS arm, and three subintegrations of 125s each in the NIR arm. Then, the object was observed immediately after this with the larger slits (5.0″) with an AB nodding cycle with single exposures of 45 s in the UVB and VIS arms and three times 15 s in the NIR arm. The latter was used to obtain a proper flux calibration of the spectra without slit losses. This spectrum was reduced using the X-Shooter pipeline version 2.5.2, and flux calibration of the spectra was made by the pipeline using the standard star FEIGE-110. We then manually rescaled the spectra obtained with the narrow slits to those obtained with the large slits and found that they agree very well with the available photometry. Telluric correction was carried out using the telluric standard Hip036392 and the procedure described by Alcalá et al. (2014). This object has been classified as a K0 YSO with extinction =0.8 mag by HH14. Using the spectral indices discussed in that work, in particular using the R5150 index, we obtain the same SpT for this target. We then compared the observed spectrum with a BT-Settl AGSS2009 synthetic spectrum (Allard et al., 2011) at =5100 K and log=-4.0 and found a best agreement by applying a reddening correction of =0.8 mag using the extinction law by Cardelli et al. (1989) and =3.1. This is consistent with a SpT between G8 and K1 according to different SpT- relations (HH14 and Kenyon & Hartmann (1995), respectively). Here a SpT K0 is assumed for this target. Finally, we calculated the luminosity of the target from the observed spectrum and the synthetic spectrum following the same procedure as Manara et al. (2013a), and, assuming =140 pc (HH14), we derive log(/) = 0.45 for this target, the same value as HH14. However, the position of this target is below the main sequence. This could suggest that it is not a member of the Taurus region. Nevertheless, this uncertainty does not influence the estimates of stellar parameters for other targets based on this template because the information on the distance is taken into account in the procedure. The spectrum of this target shows no signs of accretion in the typical lines tracing accretion in YSOs (e.g., H), but a prominent LiI 670.8 nm absorption line, and narrow chromospheric emission lines as a reversal in the core of the photospheric absorption lines in the Ca IR triplet. ## 3 Analysis method Here we briefly explain our method to self-consistently derive stellar and accretion parameters from the X-Shooter spectra, and we report our results. All the objects discussed here have also been previously characterized by other authors (e.g., Luhman, 2007, see Table 3) using red optical and infrared spectra. The comparison with these results is also presented in this section. ### 3.1 Determining the stellar and accretion parameters To determine the stellar and accretion parameters, we used the method described in Manara et al. (2013b), which determines the SpT, , , and by finding the best fit among a grid of models. This grid includes different photospheric templates (see Sect. 2.2), several slab models to reproduce the excess continuum emission due to accretion, and extinction parametrized with the reddening law by Cardelli et al. (1989) with =3.1 and values of from 0 mag to 10 mag with steps of 0.1 mag. The slab model has been used by Valenti et al. (1993) and Herczeg & Hillenbrand (2008), for example, and it is not a physical model of the accretion region, unlike the shock models by Calvet & Gullbring (1998), but provides an accurate bolometric correction to determine the total excess flux that is due to accretion. The best fit is found by comparing the flux of the observed spectrum and that of each model in various continuum regions and in different features of the spectra, and by minimizing a distribution to determine the best match. The points included in the fitting procedure range from 330 nm to 715 nm. A proper fit of the Balmer continuum is needed to determine and the veiling of the spectra. The Balmer continuum is well determined even with spectra with very low signal-to-noise ratio (S/N) on the continuum at 330 nm. The spectra of most of the targets discussed here have enough signal to perform the analysis. Only spectra with no signal in the Balmer continuum regions, such as those of Cha H 9 and Cha H 1 in this work, are not suitable for the method described here. More discussion of these and other individual targets is available in Appendix B. The stellar parameters (, ) are derived from the properties of the best-fit template and from the slab model parameters as described by Manara et al. (2013b), and are reported in Table 1. The best fit of the UV excess for all the targets are shown in Fig. LABEL:fig:best_fits-LABEL:fig::best_fits4. We have compared our direct measurements of from the excess Balmer continuum emission to alternative estimates from line luminosities in the same spectrum. The agreement is good when using the - relations by Alcalá et al. (2014), with only small differences for the objects with SpT K2. For further analyses of the emission lines present in the spectra, we refer to Fedele et al. (in prep.). The HR diagram (HRD) of the Chamaeleon I sample is shown in Fig. 1 overplotted on the isochrones from the evolutionary models by Baraffe et al. (1998). This evolutionary model was chosen for consistency with previous analyses of X-Shooter spectra, so that the comparison is less model dependent. Most of the objects are located on this diagram around the 3 Myr isochrone, with a few of objects located closer to 10 Myr, and only three objects are located slightly below the 30 Myr isochrone, but still above the 30 Myr isochrone from other models (e.g., Siess et al., 2000). The stellar mass () was obtained by comparing the position in the HRD to the Baraffe et al. (1998) evolutionary tracks. Finally, the stellar radius () was derived from and , and from the usual relation =1.25/(G) (e.g., Gullbring et al., 1998). The distribution of the derived parameters SpT, , and for the Chamaeleon I targets is shown in Fig. 2. As expected, most object have low 3 mag. The Chamaeleon I sample discussed here, together with the six transitional disks in Chamaeleon I discussed by Manara et al. (2014), includes more than 40% of the known Class II and transitional disks YSOs in the region, with higher completeness levels at SpT earlier than M3, corresponding to 0.5 , than at later SpT. This sample is thus large enough to discuss general properties of YSOs with 0.5 in the Chamaeleon I region alone. ### 3.2 Non-accreting targets in Chamaeleon I All targets analyzed here have excess dust emission in Spitzer photometry and spectroscopy (e.g., Luhman et al., 2008b; Manoj et al., 2011), indicating the presence of a disk in the system. Some of these, however, show no evidence of ongoing accretion from the lack of measurable excess in the Balmer continuum. In this section we discuss the properties of another proxy of accretion present in our spectra, the H line profile. The equivalent width (EW) and width at 10% of the peak (W) of the H line can be used to distinguish objects with ongoing accretion, although with discrepant results with respect to the Balmer continuum excess in some cases. According to White & Basri (2003), an object is defined as an accretor when W270 km/s, and EW is higher than a threshold that depends on the SpT of the target. According to the White & Basri (2003) criterion, seven targets in the sample are non-accreting: T4, ISO-ChaI 52, T33A, T45a, T54-A, Hn17, and Hn18. The fitting of the observed spectrum was also performed for these non-accreting YSOs. In most cases, the accretion excess in the Balmer continuum with respect to the photosphere is very small, within the noise of the spectra, or absolutely negligible, and this is compatible with the non-accreting status of these targets as suggested by the H line profile. The value of derived in these cases is to be considered as an upper limit on the accretion rate. However, it seems that the excess in the Balmer continuum is non-negligible for ISO-ChaI 52, in particular, although we measure W190 km/s and EW = -10.6 Å. This object has been reported to accrete at a higher rate by Antoniucci et al. (2011), but their result is based on the flux of only few near-infrared emission lines, and they did not self-consistently derive the stellar parameters of the targets. It is also possible that the object is now accreting at a lower rate than in the past. For another non-accreting target, T4, Frasca et al. (2015) reported a higher value of W390 km/s than was measured from our spectra, W230 km/s. This could be a variable object with a negligible accretion rate at the time of our observations. ### 3.3 Targets in Chamaeleon II and Taurus While the vast majority of the targets discussed in this work are located in the Chamaeleon I region, four additional targets are located in Taurus and two are located in Chamaeleon II. These were analyzed with the same method as described in Sect. 3.1, and their properties are reported in Table 2. All these targets have a detectable accretion excess in the Balmer continuum and are also accretors according to the EW and W of the H line criteria. Two objects, namely FN-Tau and Sz50, have W smaller than 270 km/s, but their EWs are larger than the threshold for their SpT. Only Hn24 presents a negligible excess in the Balmer continuum. This object has a very wide W of more than 300 km/s. We report the accretion rate detected for this target, but the small excess and the small EW of the H line indicate that it might be non-accreting. ### 3.4 Comparison with other methods The stellar parameters (SpT, , ) for all the targets discussed here have previously been derived by several studies (e.g., Luhman, 2007; Spezzi et al., 2008; Daemgen et al., 2013; Frasca et al., 2015, cf. Table 3-4). Accretion rate estimates are only available for some targets (e.g., Antoniucci et al., 2011, 2014; Frasca et al., 2015). Here we compare these results with our estimates. Differences between our results and those in the literature are typically less than a spectral subclass in SpT and mag in , both within the uncertainties. The largest differences in SpT occur for K-type stars, probably because our moderate-resolution blue spectra include better temperature diagnostics (e.g., Covey et al., 2007) than the low-resolution red spectra used in most previous studies of the region. The scarcity of templates in this work in the early-K spectral range may also contribute to this difference. Differences in estimates may be related to different methods, as Luhman (2007) used infra-red photometry and a different to derive extinction, while here the whole optical X-Shooter spectrum was used and is derived together with SpT and . The results derived with the method used here were also checked against other methods to derive SpT and from the observed spectra. In particular, the SpT determined here are also compared with those obtained using the spectral indices by ?][]Herczeg14, by Riddick et al. (2007), and the TiO index by Jeffries et al. (2007). In general, the agreement is very good with only a few exceptions in cases where different indices would lead to different results. Similarly, derived here agrees well, with differences in all the cases but one smaller than a factor 1.6, with those obtained using the bolometric correction by HH14 after removing veiling. Finally, Fig. 3 shows the comparison between the values of for Chamaeleon I targets derived here and those from the POISSON survey (Antoniucci et al., 2011, 2014) and the Gaia-ESO survey (GES, Frasca et al., 2015). The POISSON survey used infrared spectra and the luminosity of the Br and Pa emission lines to derive , and obtained assuming the stellar parameters for the targets from Luhman (2007). These values of are different from those derived here in most cases. The reason for the difference is the use of a single indirect accretion indicator, and also the relation between the luminosity of the Br line and they used, which differs from the more recent one by Alcalá et al. (2014). The estimates of from the GES were all derived using the EW and 10% width of the H emission line. Although our accretion rates generally agree with those from GES, some large discrepancies for individual objects are present. The H line, and especially its 10% width, is known not to be the best tracer of an accretion rate (e.g., Costigan et al., 2012), although it is widely used because it is the brightest emission line in YSOs. This comparison presented here confirms that single values of from this indicator alone should be considered with caution. ## 4 Results The dependence of the accretion rates on the stellar parameters is presented here. The relation between and is first discussed for the Chamaelon I sample and in comparison with other large X-Shooter sample of YSOs, then the - relation is presented. The results presented here are to be considered as only representative for the range of that is covered with high completeness by this survey, namely at 0.5. ### 4.1 Accretion luminosity and stellar luminosity dependence The vs relation for the Chamaeleon I sample is shown in Fig. 4. The targets mostly occupy the region between = and =0.01, although not uniformly at all stellar luminosity. Indeed, objects with 0.1 have / ratios ranging from 0.01 up to 1, or even more in two cases. These two strongly accreting targets both have . In contrast, objects with low stellar luminosity (0.1) have 0.1, and the objects with lowest luminosity have even 0.01. However, the sample discussed here comprises few objects in this range. Non-accreting objects, as expected, are located in the lower part of the plot, and they always have 0.01. The positions of the targets on the - plane was fitted with the statistics package ASURV (Feigelson & Nelson, 1985) run in the IRAF3 environment, considering the value of for the non-accreting objects as an upper limit. With the whole sample, we then find a best-fit relation of logLacc=(1.58±0.24)⋅logL⋆−(1.16±0.21) (1) with a standard deviation around the best fit of 0.89. If only accreting objects are analyzed, the best-fit line is logLacc=(1.72±0.18)⋅logL⋆−(0.75±0.16) (2) with a standard deviation of 0.62. The latter relation is shown in Fig. 4. In Fig. 5 the values of vs for the Chamaeleon I targets are compared with those in Taurus and Chamaeleon II analyzed here, with the samples of Class II YSOs in the Lupus clouds analyzed by Alcalá et al. (2014), and with transitional disks in various regions, including Chamaeleon I, by Manara et al. (2014). These samples of X-Shooter spectra comprise the largest set of flux-calibrated broadband spectra of young stars used to simultaneously measure accretion, photospheric properties, and extinction. The distribution of stellar parameters for the objects in this sample is different, and comparisons can only be performed by extrapolating the results to other stellar parameters, for instance, ranges. The Taurus targets analyzed here follow the - distribution of the Lupus targets, while the Chamaeleon II targets are located in the lower part of the distribution. With respect to the best fit of the sample of Alcalá et al. (2014) (shown in the figure as a black dotted line, from Natta et al. 2014), the Chamaeleon I targets follow a steeper slope of vs . However, the two values are compatible. Indeed, the slope of the best fit for this relation using the Chamaeleon I data alone and excluding non-accreting objects is 1.70.2, while the slope derived from the Lupus sample is 1.50.2. In the Chamaeleon I sample there are more strongly accreting YSOs with than in the Lupus sample. This difference in the fraction of strongly accreting YSOs in the Chamaeleon I sample compared to that of Alcalá et al. (2014) may be the result of different selection criteria (see discussion in Sect. 5.2). In any case, the sample analyzed here confirms that the relation between and has a slope significantly steeper than 1 and shows that there are several very strongly accreting YSOs in Chamaeleon I. In contrast, no strongly accreting YSOs are present in the Lupus sample analyzed by Alcalá et al. (2014). The distribution of the accreting transitional disks located in the Chamaeleon I region shown in Fig. 5 also seems to follow the best fit of the - relation of the Class II YSOs in this region fairly well, with the only exception of CHXR22E, which is a non-accreting transitional disks that has been discussed by Manara et al. (2014). At the same time, the transitional disks from other regions are also mostly located in the same part of the - plane as the majority of Class II YSOs, mostly following the locus of Lupus targets. ### 4.2 Mass accretion rate and stellar mass dependence The dependence of on in the Chamaeleon I sample is shown in Fig. 6. All the non-accreting objects are located in the lower part of the distribution. Only ISO-ChaI 237 has a similar as non-accreting objects, and it could also be borderline accretor. A tight correlation between and is present, which is a well-known result in the literature. The best fit for this relation including non-accreting objects, again considered as upper limit on , is log˙Macc=(1.70±0.44)⋅logM⋆−(8.65±0.22), (3) with a standard deviation of 1.04. When excluding the non-accreting YSOs, the best fit is log˙Macc=(1.94±0.34)⋅logM⋆−(8.21±0.17), (4) with a smaller standard deviation of 0.75. However, several targets are outliers in this plot. Excluding Cha-H1, which is the only brown dwarf in the sample, and ISO-ChaI 237, whose position in the plot is in the same region as the non-accreting YSOs, the following best fit of the - relation for the accretors in Chamaeleon I is obtained: log˙Macc=(1.83±0.35)⋅logM⋆−(8.13±0.16) (5) with a smaller standard deviation around the best fit of 0.67. Clearly, these two outliers influence the results substantially. This best fit and its standard deviation are shown in Fig. 6. Figure 7 shows the - relation for the same samples as in Fig. 5, that is, the Chamaeleon I targets and those in Taurus and Chamaeleon II analyzed here, the Lupus sample by Alcalá et al. (2014), and the transitional disks by Manara et al. (2014). The few measurements in Taurus and Chamaeleon II are consistent with the relations found in Chamaeleon I and Lupus. The slope of this relation is very similar between the Chamaeleon I sample (1.830.35), when excluding the outliers and non-accreting YSOs, and the Lupus sample (1.810.20). The most relevant difference is then the larger spread around the best fit that is found in the Chamaeleon I sample (0.67 dex) with respect to the spread of the Lupus targets (only 0.4 dex). In particular, there are some objects with higher than that of objects with the same in Lupus at 0.2, but also several objects with lower than the Lupus sample at 1 . It is also clear from Fig. 7 that there are several accreting Class II YSOs in Chamaeleon I with similar to those of transitional disks, in particular similar to the transitional disks in Chamaeleon I, but also to those in other regions. ## 5 Discussion ### 5.1 Accretion rates in full and transitional disks The results presented in Sect. 4 show that and scale with the stellar properties, respectively and , with a high degree of correlation. In particular, the samples presented here and those from the literature include objects at different evolutionary stages of the disks, mainly Class II YSOs, that is, objects surrounded by a full disk, and transitional disks. Although the comparison sample of transitional disks by Manara et al. (2014) included here is small and biased toward known accretors, the comparison of in transitional disks and full disks presented here shows no substantial differences. The values of are similar for full and transitional disks even at . According to Najita et al. (2015), this is a result of the selected comparison sample. Indeed, they suggested that age differences between the targets might result in smaller differences in , as is expected to decrease with age because of the viscous evolution of the disks (e.g., Hartmann et al., 1998). However, Fig. 7 shows that transitional disks located in Chamaeleon I have similar to full disks in the same star-forming region. A difference between the full- and transitional-disk Chamaeleon I samples is that there are no transitional disks that are as strongly accreting as the strongest accretor from the sample of full disks. These results confirm that the sample of accreting transitional disks by Manara et al. (2014) has similar accretion properties as full disks. However, no transitional disks in this sample have been identified as strong accretors. A possible difference between our analysis and the analysis by Najita et al. (2015) may be the prevalence of strong accretors in their sample. Again, this result is based on a small and biased sample of transitional disks and might not be valid for the whole class of objects. ### 5.2 Empirical differences between Chamaeleon I and Lupus The results shown in Sects. 4.1 and 4.2 suggest that there are two main differences in the accretion rate properties of Chamaeleon I and Lupus accreting YSOs: the Chamaeleon I sample comprises more strongly accreting YSOs with , and the spread of values of at any given in the Chamaeleon I sample is larger than the one found by Alcalá et al. (2014) for objects located in the Lupus clouds and is more compatible with the 0.8 dex found for Taurus members by Herczeg & Hillenbrand (2008). In any case, this spread of is still much smaller than was found in previous studies (e.g., Muzerolle et al., 2003; Mohanty et al., 2005; Natta et al., 2006; Manara et al., 2012; Ercolano et al., 2014). Before discussing the possible reasons for the difference in the accretion properties of objects in the two largest X-Shooter samples, it is important to remark that the Chamaeleon I and the Lupus X-Shooter samples both include only 40-50% of the targets surrounded by an optically thick disk (i.e., Class II or transitional disks) in each of these regions. Therefore, selection effects are present in both samples, and some differences might be ascribed to this difference in the intrinsic stellar properties of the targets. In particular, the distribution of in the two samples is different: more than 70% of the Lupus objects have 0.5 , while 60% of the Chamaeleon I targets have 0.5 . It is possible that the - relation is tighter at lower , as recent results in low-mass stars and brown dwarfs (BDs) in -Ophiuchus suggest (Manara et al., 2015), and this would explain why there is a smaller spread of in the Lupus sample. This means that it might be more plausible to have YSOs accreting with at higher masses, which would explain why more strongly accreting YSOs are present in the Chamaeleon I sample. Both hypotheses can be tested only with more complete samples in various star-forming regions. The selection criteria of the two samples might likewise bias the results. The Lupus targets were selected mostly to be low-mass stars with low or negligible extinction, while the Chamaeleon I targets sampled mostly objects with higher masses and bright, thus it might have been easier to include strongly accreting targets in the latter. In the following, we discuss some possible reasons for this difference in the context of accretion and disk evolution theories. ### 5.3 Constraints on accretion and disk evolution theories The slope of the - relation is a proxy for determining the main driver of the dispersal of protoplanetary disks (Clarke & Pringle, 2006). The facts that the slope is significantly higher than 1 and that there is no evidence of an upper locus of objects with slope unity in the - plane disagrees with the expectations formulated by Hartmann et al. (2006). However, this locus of strongly accreting objects might not be observed in the samples analyzed here because the samples are incomplete. In the context of disk dispersal by photoevaporation, the slopes derived for both the Chamaeleon I and Lupus samples are more compatible with X-ray photoevaporation models (Ercolano et al., 2014) than EUV photoevaporation models (Clarke & Pringle, 2006). If the larger spread of values in Chamaeleon I with respect to Lupus is driven by the different range of the two samples, this spread is indeed larger for higher mass stars than for BDs; this could be explained with the models by Alexander & Armitage (2006). They explain the - relation as a consequence of initial conditions at formation of the disk. To reproduce the observed slope2, they suggest that , thus that higher mass stars evolve faster. However, observing this effect in a single star-forming region implies the presence of significant age spreads within that region and that the individual stellar ages may be reliably measured. However, ages of individual young stars are unreliable at present (Soderblom et al., 2014). In the context of viscous evolution theory (e.g., Hartmann et al., 1998), a difference in age could explain the differences in the number of strong accretors and in the spread of values between Chamaeleon I and Lupus. The difference in age between the Lupus and Chamaeleon I regions is probably smaller than 1-2 Myr, if there is any difference at all (e.g., Luhman et al., 2008b; Alcalá et al., 2014). Viscous evolution models (e.g., Hartmann et al., 1998) predict a typical decrease of by 0.3 dex between 2 and 3 Myr for a typical low-mass YSO. This value is higher than the differences in the intercepts of the - relations in the two samples and might explain some observed differences. However, the uncertainty on the age estimates and the possibility of non-coevality of the targets in each region lead us to consider this effect as a minor, if not negligible, explanation of the observed differences. As variability is often referred to as a possible explanation for the observed spread of values, it is worth asking if it can play a role in explaining the differences between Lupus and Chamaeleon I discussed here. Costigan et al. (2012) have shown that the typical variability of in Chamaeleon I targets is generally 0.4 dex. Independent studies in other star-forming regions also confirm this finding (e.g., Venuti et al., 2014). Therefore, this effect can be relevant to explain some, even most, of the observed spread of values. If the main factor that explains the observed spread is variability, then this would suggest that the typical rate of accretion variability is lower for objects located in the Lupus cloud than for those in the Chamaeleon I region. We also consider whether the observed differences are due to the different environment in which the targets discussed here are located. Padoan et al. (2005) suggested that the observed are a consequence of Bondy-Hoyle accretion of material surrounding the YSO onto the central object. Such a description would imply a strong dependence of the values of on the amount of pristine material in the region. However, the mean value of is very similar in the Chamaeleon I and Lupus regions, the largest difference is the scatter of the values. Therefore, this hypothesis could be tested by checking the amount of gas present in the immediate vicinity of individual YSOs and not at the level of the whole cloud. Dullemond et al. (2006) also suggested that the spread of values in the - relation is related to different initial conditions at the formation of disks, in particular different initial core rotation rates. If this is the case, then it might be possible that there has been a larger spread of core rotation rates in Chamaeleon I than in Lupus, or, on the other hand, that higher mass stars have a larger spread of this value at formation. Finally, it is also possible that the YSO population of Lupus is different from the populations in other regions. For example, Galli et al. (2015) have found that the lifetimes of disks may be shorter in this region than the lifetimes for objects located in the Taurus star-forming region. Although this result relies on individual age estimates, it suggests that objects located in different environments might follow different evolutionary paths. ## 6 Conclusions We have presented the analysis of 34 X-Shooter spectra of YSOs in the Chamaeleon I star-forming region, together with four targets in Taurus and two in Chamaeleon II. We have derived their stellar and accretion parameters self-consistently from the same simultaneous spectra ranging from the UV excess, a direct accretion tracer, to the region at 700 nm, which is full of molecular features that are proxies of SpT of the targets. We distinguished between clearly accreting YSOs and those that do not fulfill the criteria to be considered accretors. The dependence of accretion on the stellar parameters was tested with the logarithmic - and the - relations, which both show a very good correlation when considering only the accreting targets. The slopes of these relations are 1.720.18 and 1.830.35, respectively. These values are compatible with those found in other star-forming regions, such as Lupus. More evolved objects, that is, transitional disks, also located in the Chamaeleon I region follow the locus of full disks in these two relations very nicely as well. The largest discrepancy between our results and previous results with similar methods in Lupus is the larger spread of the - relation around the best fit, and, in particular, the larger amount of strongly accreting YSOs we found. These differences might suggest that there are intrinsic differences in the evolution of protoplanetary disks in these two regions, but they might also be ascribed to the fact that these two samples cover only 50% of the accreting YSOs in these regions and span a different range of stellar masses. The Chamaeleon I sample analyzed here is mostly composed of YSO with masses closer to solar mass, while the Lupus sample comprises many lower mass stars. We might thus see a different behavior in a different stellar mass range. Our results suggest that the accretion properties of accreting YSOs located in the Chamaeleon I and in the Lupus regions might be different. This difference will be tested when complete X-Shooter samples in both star-forming regions, and possibly in more regions, will be available. At the same time, we await the results from current ALMA surveys of Chamaeleon I and Lupus that will better determine the disk properties of these targets, as this could help us in understanding the differences between the measured accretion rates for objects with similar masses. Finally, with the data release of Gaia we will have the possibility to refine the precision of our measurements with the newly determined distances to individual objects in these nearby star-forming regions. ###### Acknowledgements. DF acknowledges support from the Italian Ministry of Science and Education (MIUR), project SIR (RBSI14ZRHR) and from the ESTEC Faculty Visiting Scientist Programme. We thank the referee for the valuable comments that helped to improve the quality of this paper. We acknowledge various discussions with Michael Rugel, Timo Prusti, Juan Alcalá. We thank Kevin Covey and Adam Kraus for help in preparing the proposal. This research made use of Astropy, a community-developed core Python package for Astronomy at http://www.astropy.org, of APLpy, an open-source plotting package for Python hosted at http://aplpy.github.com, and of the SIMBAD database, operated at CDS, Strasbourg, France Additional information from the literature on the targets discussed here is reported in Tables 3-4. ## Appendix B Comments on individual targets As discussed in Sect. 3.1, stellar and accretion properties of the targets discussed here were determined following the procedure described by Manara et al. (2013b). In a few cases, however, the quality of the spectra was not good enough for this procedure to work, or the properties of the target did not lead to satisfyingly good fits. In these cases slightly different procedures were adopted. Additional points were included in the fit of some targets, in particular at 400 nm and between 500 nm and 600 nm. In two cases, namely when fitting Cha H1 and Cha H9, the procedure was totally different. These cases are described in the following. ### b.1 Chamaeleon I targets T3: this target has a best fit with templates with SpT K7. However, the derived is low and results in a position on the HRD slightly below the 30 Myr isochrone according to the Baraffe et al. (1998) models. Even including additional points in the fit did not change the results. However, the fit is good, the SpT is compatible with that from the literature (M0) and that from the TiO index by Jeffries et al. (2007) (K6) and is slightly later than the result with the indices by HH14 (K4.2). Similarly, the estimated here agrees well with previous values in the literature. When using the bolometric correction by HH14 to determine , this is only 1.2 times higher than the one derived from the best-fit template. This would make the object only slightly younger, thus still older than 20 Myr. CR Cha: This object was best fit with a K0 template. Previous works reported it as an object with SpT K2, but the fit with a K2 template is not good. The value of SpT derived using the indices by HH14 is K1, but no templates with this SpT are available. Given that the position of this target on the HRD is outside the range of the evolutionary models by Baraffe et al. (1998), the stellar parameters were derived using evolutionary models by Siess et al. (2000). CT Cha A: the best fit of this target was obtained by including additional points at 400 nm and between 500 nm and 600 nm. If these additional points were not included, the best fit would have been with a K2 template, but it would have been worse and leading to a significantly older age. However, the other stellar and accretion parameters would be compatible within the uncertainties. When comparing determined from the best fit with that obtained using the bolometric correction by HH14, the latter would lead to a value higher by a factor 1.8. The reason for this discrepancy is possibly the high veiling in this object due to very intense accretion. Cha H1: None of the photospheric templates available in this study has the same SpT as this target. Indeed, literature estimates and spectral indices lead to an M7-M8 SpT. This object is also known to have negligible extinction (Luhman, 2007), and it seems not to accrete at a high rate from the emission lines present in the spectrum. Therefore, it is possible to adopt the SpT from the spectral indices for this target, assume =0 mag, determine using the bolometric correction by HH14, and from the luminosity of the emission lines present in the spectrum and the relations by Alcalá et al. (2014) to convert these luminosity in . This was the procedure adopted here to derive the stellar parameters reported in Table 1 and used in this work. Cha H9: The spectrum of this object has almost no flux in the whole UVB arm, thus the procedure adopted in this work for the other targets was not applicable. This target is probably accreting at a very low rate according to its H line. Indeed, the EW is 20 Å, while the 10% width of the line is 200 km/s. The stellar parameters were thus derived in the following way. First, the SpT was determined using several spectral indices, which led to an estimate of SpT M5.5. Then, the VIS part of the spectrum was compared to a photospheric template with SpT M5.5 that was reddened with increasing values of in steps of 0.1 mag until a best match was found. The best agreement between the two spectra is with =4.8 mag, which is higher than that reported in the literature. Finally, was determined using the bolometric correction by HH14 and from the only emission line detected with high S/N in the spectrum, which is the H line. As the width of this line is quite small, it should be noted that the derived here might be an upper limit of the real , as this object might be non-accreting. Sz22: This object is a very complex multiple system. Only the close binary at the center (Schmidt et al., 2013) is included in the slit. This object is a confirmed member of Chamaeleon I by Lopez Martí et al. (2013) from proper motion studies. The best fit was obtained by including all the additional points listed above and with a value of much lower than the one reported in the literature (Luhman, 2007; Schmidt et al., 2013). No signs of the companion were found in the overall shape of the spectrum. The derived age is older than 10 Myr, but still compatible with younger ages according to other evolutionary models. ESO H 562: The spectrum of this target has a low S/N (1) in the UVB arm until 400 nm. It is reported to be a close binary (0.28″ separation) of two objects with the same SpT (M0.5, Daemgen et al. 2013). Therefore, both objects have probably been included in the slit and are not resolved. The best fit was obtained with SpT M1 and a = 0.11 . The same is obtained when using the bolometric correction by HH14. This places the object low on the HRD, at an age of 24 Myr. The =3.4 mag derived here is smaller than the 4 mag reported by Daemgen et al. (2013), while the determined here is the same as that reported by Daemgen et al. (2013) for component B. It is thus possible that in the VIS and UVB arm the spectrum of component B dominates, as Daemgen et al. (2013) reported that component A has =10.5 mag and it is as bright as component B in the near-infrared. The latter has only =4.3 mag, according to Daemgen et al. (2013), thus at optical wavelengths component B is significantly brighter than component A. T33-B: The best fit was made with a template with SpT K0, but this led to a low implying an age40 Myr according to the evolutionary models by Baraffe et al. (1998). A similar was determined when using the bolometric correction by HH14. It should be noted that this component of the system T33 is composed of two objects separated by 0.11″ (Daemgen et al., 2013), thus not resolved by our observations. From near-infrared spectra, Daemgen et al. (2013) reported a SpT M0.5 for these targets. However, both the fit and the spectral indices lead to a much earlier SpT and to an similar to that of component A. Cha H6: the best fit for this target was with the photospheric template with SpT M6.5, but this was obtained from a spectrum with a low S/N in the UVB. However, the spectral indices also confirm that the SpT of this target is around M6. The value of determined from the fit places the object immediately on the 1 Myr isochrone of the evolutionary models by Baraffe et al. (1998), while the from the bolometric correction by HH14 leads to a smaller . As reported earlier on, this difference might be ascribed to the SpT-Teff relation used by HH14, which differs from that by Luhman et al. (2003), which was used here, at this late stellar type. T44: This object is a very strong accretor almost totally veiled, and it is known to launch a jet (Robberto et al., 2012). The results of the fit are not perfect, for example, the slope of the Balmer continuum is not well reproduced. However, even including additional points in the fit the accretion parameters change within the uncertainties, while the stellar parameters, in particular the SpT, might change. T45a: This is a non-accreting target, but the profile of the H line is unusual for a non-accreting object. It seems rather narrow (150-170 km/s) and weak, but has a strong self-absorption at almost zero velocity. The excess in the UV is almost zero, and the very low, consistent with being non-accreting. ISO-ChaI 237: The spectrum of this target has a low S/N in the UVB arm, thus the best fit is slightly uncertain, but leads to a SpT compatible with the values from spectral indices and from the literature. This best fit leads to a low in the - plane, in the region where non-accreting targets are located. It is difficult to determine if this object is non-accreting because its spectrum presents a broad H line but with strong self-absorption and, thus, a small EW. T49: This object is one with stronger accretion than most in the sample analyzed here. The spectrum is highly veiled, but a best fit was found and confirmed, also including additional points. T54-A: This is a non-accreting target, composed of two components separated by 0.24″, thus not resolved by our observations (Daemgen et al., 2013). Component A should, however, dominate the emission in this wavelength range. The SpT derived here agrees with that from the literature and from spectral indices by HH14. ## Appendix C Best fit Available on online version. ## Appendix D Observation log ### Footnotes 1. thanks: This work is based on observations made with ESO Telescopes at the Paranal Observatory under programme ID 084.C-1095 and 094.C-0913. 2. Non-accreting objects according to the H line width at 10% of the peak and H line equivalent width. All stellar parameters have been derived using the Baraffe et al. (1998) evolutionary models apart from objects with a symbol, for which the Siess et al. (2000) models were used. 3. IRAF is distributed by National Optical Astronomy Observatories, which is operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation. 4. Spectral types, extinction, disk classification, accretion indication, and binarity are adopted from the following studies: 1. Luhman (2007); 2. Luhman et al. (2008b); 3. Luhman (2004); 4. Costigan et al. (2012); 5. Daemgen et al. (2013); 6. Manoj et al. (2011); 7. Kraus & Hillenbrand (2007); 8. Ghez et al. (1997); 9. Nguyen et al. (2012); 10. ?; 11. Schmidt et al. (2013) . Complete name on SIMBAD for T# is Ass Cha T 2# 5. Spectral types, extinction, disk classification, accretion indication, and binarity are from the following studies: 1. Herczeg & Hillenbrand (2014); 2. Furlan et al. (2011); 3. Spezzi et al. (2008); 4. Alcalá et al. (2008) ### References 1. Alcalá, J. M., Spezzi, L., Chapman, N., et al. 2008, ApJ, 676, 427 2. Alcalá, J. M., Natta, A., Manara, C. F., et al. 2014, A&A, 561, A2 3. Alexander, R. D., & Armitage, P. J. 2006, ApJ, 639, L83 4. Alexander, R., Pascucci, I., Andrews, S., Armitage, P., & Cieza, L. 2014, Protostars and Planets VI, 475 5. Allard, F., Homeier, D., & Freytag, B. 2011, Astronomical Society of the Pacific Conference Series, 448, 91 6. Antoniucci, S., García López, R., Nisini, B., et al. 2011, A&A, 534, A32 7. Antoniucci, S., García López, R., Nisini, B., et al. 2014, A&A, 572, A62 8. Baraffe, I., Chabrier, G., Allard, F., & Hauschildt, P. H. 1998, A&A, 337, 403 9. Biazzo, K., Alcalá, J. M., Frasca, A., et al. 2014, A&A, 572, A84 10. Calvet, N., & Gullbring, E. 1998, ApJ, 509, 802 11. Cardelli, J. A., Clayton, G. C., & Mathis, J. S. 1989, ApJ, 345, 245 12. Clarke, C. J., & Pringle, J. E. 2006, MNRAS, 370, L10 13. Costigan, G., Scholz, A., Stelzer, B., et al. 2012, MNRAS, 427, 1344 14. Covey, K. R., Ivezić, Ž., Schlegel, D., et al. 2007, AJ, 134, 2398 15. Daemgen, S., Petr-Gotzens, M. G., Correia, S., et al. 2013, A&A, 554, AA43 16. Dullemond, C. P., Natta, A., & Testi, L. 2006, ApJ, 645, L69 17. Ercolano, B., Mayr, D., Owen, J. E., Rosotti, G., & Manara, C. F. 2014, MNRAS, 178 18. Fang, M., van Boekel, R., Wang, W., et al. 2009, A&A, 504, 461 19. Feigelson, E. D., & Nelson, P. I. 1985, ApJ, 293, 192 20. Frasca, A., Biazzo, K., Lanzafame, A. C., et al. 2015, A&A, 575, A4 21. Furlan, E., Luhman, K. L., Espaillat, C., et al. 2011, ApJS, 195, 3 22. Galli, P. A. B., Bertout, C., Teixeira, R., & Ducourant, C. 2015, A&A, 580, A26 23. Ghez, A. M., McCarthy, D. W., Patience, J. L., & Beck, T. L. 1997, ApJ, 481, 378 24. Gullbring, E., Hartmann, L., Briceño, C., & Calvet, N. 1998, ApJ, 492, 323 25. Gullbring, E., Calvet, N., Muzerolle, J., & Hartmann, L. 2000, ApJ, 544, 927 26. Ingleby, L., Calvet, N., Herczeg, G., et al. 2013, ApJ, 767, 112 27. Ingleby, L., Calvet, N., Hernández, J., et al. 2014, ApJ, 790, 47 28. Kenyon, S. J., & Hartmann, L. 1995, ApJS, 101, 117 29. Kraus, A. L., & Hillenbrand, L. A. 2007, ApJ, 662, 413 30. Hartmann, L., Calvet, N., Gullbring, E., & D’Alessio, P. 1998, ApJ, 495, 385 31. Hartmann, L., D’Alessio, P., Calvet, N., & Muzerolle, J. 2006, ApJ, 648, 484 32. Herczeg, G. J., & Hillenbrand, L. A. 2008, ApJ, 681, 594 33. Herczeg, G. J., Cruz, K. L., & Hillenbrand, L. A. 2009, ApJ, 696, 1589 34. Herczeg, G. J., & Hillenbrand, L. A. 2014, ApJ, 786, 97 35. Herczeg, G. J., & Hillenbrand, L. A. 2015, ApJ, 808, 23 36. Jeffries, R. D., Oliveira, J. M., Naylor, T., Mayne, N. J., & Littlefair, S. P. 2007, MNRAS, 376, 580 37. Lopez Martí, B., Jimenez Esteban, F., Bayo, A., et al. 2013, A&A, 551, A46 38. Luhman, K. L., Stauffer, J. R., Muench, A. A., et al. 2003, ApJ, 593, 1093 39. Luhman, K. L. 2004, ApJ, 602, 816 40. Luhman, K. L. 2007, ApJS, 173, 104 41. Luhman, K. L. 2008, Handbook of Star Forming Regions, Volume II, 169 42. Luhman, K. L., Allen, L. E., Allen, P. R., et al. 2008b, ApJ, 675, 1375 43. Luhman, K. L., Allen, P. R., Espaillat, C., Hartmann, L., & Calvet, N. 2010, ApJS, 186, 111 44. Manara, C. F., Robberto, M., Da Rio, N., et al. 2012, ApJ, 755, 154 45. Manara, C. F., Testi, L., Rigliaco, E., et al. 2013a, A&A, 551, A107 46. Manara, C. F., Beccari, G., Da Rio, N., et al. 2013b, A&A, 558, A114 47. Manara, C. F., Testi, L., Natta, A., et al. 2014, A&A, 568, AA18 48. Manara, C. F., Testi, L., Natta, A., & Alcalá, J. M. 2015, A&A, 579, A66 49. Manoj, P., Kim, K. H., Furlan, E., et al. 2011, ApJS, 193, 11 50. Modigliani, A., Goldoni, P., Royer, F., et al. 2010, Proc. SPIE, 7737 51. Mohanty, S., Jayawardhana, R., & Basri, G. 2005, ApJ, 626, 498 52. Muzerolle, J., Hartmann, L., & Calvet, N. 1998, AJ, 116, 2965 53. Muzerolle, J., Hillenbrand, L., Calvet, N., Briceño, C., & Hartmann, L. 2003, ApJ, 592, 266 54. Najita, J. R., Andrews, S. M., & Muzerolle, J. 2015, MNRAS, 450, 3559 55. Natta, A., Testi, L., & Randich, S. 2006, A&A, 452, 245 56. Natta, A., Testi, L., Alcalá, J. M., et al. 2014, A&A, 569, AA5 57. Nguyen, D. C., Brandeker, A., van Kerkwijk, M. H., & Jayawardhana, R. 2012, ApJ, 745, 119 58. Padoan, P., Kritsuk, A., Norman, M. L., & Nordlund, Å. 2005, ApJ, 622, L61 59. Patat, F., Moehler, S., O’Brien, K., et al. 2011, A&A, 527, A91 60. Riddick, F. C., Roche, P. F., & Lucas, P. W. 2007, MNRAS, 381, 1067 61. Rigliaco, E., Natta, A., Randich, S., Testi, L., & Biazzo, K. 2011a, A&A, 525, A47 62. Rigliaco, E., Natta, A., Randich, S., et al. 2011b, A&A, 526, L6 63. Rigliaco, E., Natta, A., Testi, L., et al. 2012, A&A, 548, A56 64. Robberto, M., Spina, L., Da Rio, N., et al. 2012, AJ, 144, 83 65. Schmidt, T. O. B., Vogt, N., Neuhäuser, R., Bedalov, A., & Roell, T. 2013, A&A, 557, AA80 66. Sicilia-Aguilar, A., Henning, T., & Hartmann, L. W. 2010, ApJ, 710, 597 67. Siess, L., Dufour, E., & Forestini, M. 2000, A&A, 358, 593 68. Soderblom, D. R., Hillenbrand, L. A., Jeffries, R. D., Mamajek, E. E., & Naylor, T. 2014, Protostars and Planets VI, 219 69. Spezzi, L., Alcalá, J. M., Covino, E., et al. 2008, ApJ, 680, 1295 70. Stelzer, B., Alcalá, J. M., Scholz, A., et al. 2013, A&A, 551, A106 71. Valenti, J. A., Basri, G., & Johns, C. M. 1993, AJ, 106, 2024 72. Venuti, L., Bouvier, J., Flaccomio, E., et al. 2014, A&A, 570, A82 73. Vernet, J., Dekker, H., D’Odorico, S., et al. 2011, A&A, 536, A105 74. White, R. J., & Basri, G. 2003, ApJ, 582, 1109 113549	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8903520703315735, "perplexity": 2257.8145994086844}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-47/segments/1542039743282.69/warc/CC-MAIN-20181117040838-20181117062838-00451.warc.gz"}
http://mathhelpforum.com/advanced-algebra/203638-need-quick-help-proof-about-linear-transformations.html	# Math Help - Need [Quick] Help for Proof About Linear Transformations 1. ## Need [Quick] Help for Proof About Linear Transformations Hi, I am TA'ing a graduate course on linear algebra this semester, and we have come up to a theorem that I cannot prove myself without resorting to Schur's Lemma and going into issues of density of invertible matrices, which is too advanced at this point in the class. The problem is as follows: Let $T:V \rightarrow V$ be a linear operator on the finite-dimensional vector space $V$ over $F$; let $dim(V)=n$. Then, if for all bases of $V$, the matrix representation of $T$ is the same, then $T = \lambda I$, for some $\lambda \in F$. Of course, I should have asked sooner, but the class meets in 12 hours from now =P So if anyone can wheel off an elementary proof pretty quickly, it would be awesome. 2. ## Re: Need [Quick] Help for Proof About Linear Transformations I think that if you just permute the basis elements you'll actually have to permute both the rows and columns. I'm not entirely sure about this though. Next, if the above is true, you can cyclically shift the basis elements by one position, obtaining that all the diagonal entries are the same. Then certain off-diagonal sections are individually equal to some constants. The goal would be to show these constants are all the same. I can imagine that using different permutations you can see that all the off-diagonal entries are the same. This is not much more so far, but it might be a good start. 3. ## Re: Need [Quick] Help for Proof About Linear Transformations Quick observations and a proof except for the sole case of $\mathbb{F} = \mathbb{Z}/2\mathbb{Z}$. 1) $\exists \lambda \ni T_{ii} = \lambda \forall i$ That's because, given any basis, you can simply permute its order in how you define T in that basis. 2) TS = ST for all S that's an F-linear isomorphism of V. ( for all matrices S in GL(V, F), if you first nail down an initial prefered basis) That's because that's how you change bases. 3) This problem seems like it might have a nice proof by induction on the dimension of V. Finding an n-1 subspace W such that T(W) contained in W seemed to be the key, and is doable somehow, but I just found this more direct proof, and so stopped searching for such an elegant proof. 4) A proof for $\mathbb{F} \ne \mathbb{Z}/2\mathbb{Z}$: Fix an initial basis. Choose any n elements $b_1, b_2, ... b_n$ in $\mathbb{F}-\{0\}$. Let $S = diag(b_1, b_2, ... b_n)$. Then obviously $S \in GL(V, \mathbb{F})$. Have $(TS)_{ij} = \Sigma_{k=1}^n T_{ik}S_{kj} = \Sigma_{k=1}^n T_{ik}(\delta_k^jS_{jj}) = T_{ij}S_{jj} = b_jT_{ij}$. Similarly $(ST)_{ij} = \Sigma_{k=1}^n S_{ik}T_{kj} = \Sigma_{k=1}^n (\delta_k^iS_{ii})T_{kj} = S_{ii}T_{ij} = b_iT_{ij}$. Since $TS = ST$, have $(TS)_{ij} = (ST)_{ij} \ \forall i, j \in \{1, 2, ..., n\}$. Thus $b_jT_{ij} = b_iT_{ij} \ \forall i, j \in \{1, 2, ..., n\}$. But then $(b_j - b_i)T_{ij} = 0 \ \forall i, j \in \{1, 2, ..., n\}$, so whenever $b_j \ne b_i$, have $T_{ij} = 0$. ----- Note that provided $\mathbb{F} \ne \mathbb{Z}/2\mathbb{Z}$ (as I'll assume from now on), $\mathbb{F}$ has at least 2 non-zero elements. That's required for what follows. Now, choose any pair $k, l \in \{1, 2, ..., n\}$ such that $k \ne l$. Will show $T_{kl} = 0$ by choosing an appropriate diangonal matrix S. First choose $b_k \ne b_l$, both in $\mathbb{F}-\{0\}$ (we know that's possible), and then fill out the other $b_i$'s as needed (all 1's works) to make that diagonal matrix $S$. Using such an $S$, it follows by the argument above that $T_{kl} = 0$. Thus $T_{ij} = 0 \ \forall i \ne j, i, j \in \{1, 2, ..., n\}$. By permuting any basis for $V$, it's clear that the diagonal entries of $T$ must all agree. Thus, for $\mathbb{F} \ne \mathbb{Z}/2\mathbb{Z}$, such a $T$ must be of the form $T = \lambda I$ for some $\lambda \in \mathbb{F}$. ----- 5) For $\mathbb{F} = \mathbb{Z}/2\mathbb{Z}$, and $n = \dim V = 2$, there are only 4 vectors, and by simple enumeration of possibilities for T and bases you can show that T must be the identity. It's a lot to write in LaTex, so I won't bother. Surely you could do this using elementary matricies, and probably in that generality avoid my $\mathbb{F} \ne \mathbb{Z}/2\mathbb{Z}$ constraint. Since the result popped out so easily for diagonal matricies, I didn't bother. 4. ## Re: Need [Quick] Help for Proof About Linear Transformations just a sketch off the top of my head: call the matrix representation of T, A. then PA = AP, for ANY invertible matrix P (which we can regard as a "change of basis" matrix). if det(A) ≠ 0, then A lies in the center of GL(n,F), which consists of matrices of the form λI (this is not hard to prove), for λ ≠ 0 in F. so the only thing left to prove is that if det(A) = 0, A = 0. note that if there is some u with Au ≠ 0, and and some v ≠ 0 with Av = 0, extending {v} to a basis of V and extending {u+v} to a basis of V should yield different matrix representations of A. since det(A) = 0, there is such a v, hence there can be no u, and that should do it. 5. ## Re: Need [Quick] Help for Proof About Linear Transformations Originally Posted by Deveno just a sketch off the top of my head: Thanks; I did find a proof (actually two) last night pretty much immediately after I posted this, and the one that I presented as a solution was pretty much along the same lines as this. From the PA = AP you can get around the issue of getting stuck in the special case of $GL_n(F)$ by proceeding from $[T]_{B_x, B_x} = [T]_{B_y, B_y}$ for all x, y: $\sum_{i, j} \lambda_{i,j} P_{B_i, B_j}[T]_{B_x, B_x} = \sum_{i, j} [T]_{B_x, B_x}(\lambda_{i,j} P_{B_i, B_j}) \Rightarrow$ $[\sum_{i, j} \lambda_{i,j} P_{B_i, B_j}] \cdot [T]_{B_x, B_x} = [T]_{B_x, B_x} \cdot [\sum_{i, j} (\lambda_{i,j} P_{B_i, B_j})]$ Where the lambdas are arbitrary scalars, and just note that of course these P's span $F^{n \times n}$ given that they contain the permutation matrices and proceed in the obvious way. I also found an induction on cofactor expansions, but I think that is more abstract, so I went with this more computational proof.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 49, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9320701956748962, "perplexity": 377.7056513153371}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-32/segments/1438042987034.19/warc/CC-MAIN-20150728002307-00094-ip-10-236-191-2.ec2.internal.warc.gz"}
https://www.physicsforums.com/threads/find-y-f-dx-0-0-given-f-x-y.545634/	Find ∂/∂y ∂F/dx (0,0) given F(x,y) • Start date • #1 807 8 Attached. Homework Equations Just taking derivatives. The Attempt at a Solution Basically, at first I was thinking I can differentiate because the original function is continuous at (0,0) so I do and get: http://www.wolframalpha.com/input/?i=d/dy+(d/dx+(xy(8x^2+%2B+4y^2)/(x^2%2By^2))). Correct me if I am wrong but I was thinking I had to differentiate this twice partially as the Wolfram Alpha link shows and then plug in (0,0) to the answer that Wolfram Alpha gave but that is also not defined. I tried taking the limit with y = 0 and x = 0 respectively and found that the limit as (x,y) => (0,0) of the differentiated part does not exist. I then thought that wait, having F(0,0) = 0 be defined does not mean that it is continuous so I shouldn't be able to differentiate in the first place. I am now completely confused as to what I need to do. Attachments • Q5.jpg 19.8 KB · Views: 386 • #2 179 0 Try d/dx (xy(8x^2+4y^2)/(x^2+y^2)) for x=0 and you get 4y Next try d/dy (xy(8x^2+4y^2)/(x^2+y^2)) for y=0 and you get 8x So the second derivatives will give you different results. • Last Post Replies 1 Views 2K • Last Post Replies 11 Views 11K • Last Post Replies 11 Views 808 • Last Post Replies 9 Views 943 • Last Post Replies 7 Views 5K • Last Post Replies 4 Views 1K • Last Post Replies 3 Views 2K • Last Post Replies 14 Views 5K • Last Post Replies 3 Views 806 • Last Post Replies 2 Views 7K	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9450625777244568, "perplexity": 1344.7503347044812}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038077818.23/warc/CC-MAIN-20210414125133-20210414155133-00601.warc.gz"}
http://mathhelpforum.com/advanced-algebra/179558-basis-perpendicular-vector.html	# Thread: Basis for a perpendicular vector 1. ## Basis for a perpendicular vector Problem: Find a basis for $W^{\perp}$, where $W = span\left \{ \begin{bmatrix}1\\ 2\\ 3\\4\end{bmatrix},\begin{bmatrix}5\\ 6\\ 7\\8\end{bmatrix} \right \}$ Attempt: First I tried taking an arbitrary 4x1 matrix, and dotting it with a vector from the span of W. Then I realized that it must be orthogonal to all of the vectors in the span of W, so I trashed that idea. I know that $W^{\perp}$ is the kernel of the orthogonal projection onto W. So therefore, $W^{\perp} = proj_{W}(\vec{x}) = (\vec{u}_{1} \cdot \vec{x})\vec{u}_{m}+...+(\vec{u}_{m} \cdot \vec{x})\vec{u}_{m}$ This is where I get lost, and I'm not sure how to continue or even if I'm on the right track. Any help is greatly appreciated. Here is my work: 2. suppose v = (v1,v2,v3,v4) in R^4 is perpendicular to (1,2,3,4) and (5,6,7,8). then v.(a(1,2,3,4) + b(5,6,7,8)) = a(v.(1,2,3,4)) + b(v.(5,6,7,8)) = a0 + b0 = 0 + 0 = 0. so it suffices to pick v perpendicular to each element of the basis {(1,2,3,4), (5,6,7,8)} of W. you should be able to create a system of 2 linear equations in 4 variables, which will reduce to a system in two parameters. chose those two parameters such that you get 2 linearly independent vectors in R^4, and you're done. 3. Sorry, I'm having a little trouble understanding the formatting. Is v a column vector with values v1->v4? And where are 'a' and 'b' coming from in the second line? 4. yes, v is a column vector with coordinates v1,v2,v3,v4 in the standard basis. a and b are just arbitrary real numbers. any element of W, which is spanned by those two vectors, is a linear combination of the two, some scalar multiple of the first, plus some scalar multiple of the second. the point is, if you get a vector perpendicular to both, it will be perpendicular to any linear combination of the two. so v.(1,2,3,4) = 0 is the linear equation: v1 + 2v2 + 3v3 + 4v4 = 0 and v.(5,6,7,8) = 0 is the linear equation: 5v1 + 6v2 + 7v3 + 8v4 = 0. you need to solve these 2 equations simultaneously. well, as you might expect, the dimension of W┴, is going to be 2, so you should wind up with a solution set that looks like: (v1(s,t), v2(s,t), v3(s,t), v4(s,t)), where each vj(s,t) is some linear expression in s and t. or, to put it another way, you can eliminate one variable, and express a second variable in terms of the remaining 2. back-substitution will then give the "eliminated" variable in terms of those same two "undetermined" variables. converting to unit vectors isn't necessary, you're looking for orthognal vectors, not orthonormal ones.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 4, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8504117727279663, "perplexity": 392.27327800701977}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084890514.66/warc/CC-MAIN-20180121100252-20180121120252-00761.warc.gz"}
http://mathhelpforum.com/advanced-algebra/183401-homomorphism-2-a.html	1. ## Homomorphism #2 Is this correct anyone; C is the group of all complex numbers under addition O : C ----> C z ----> z + iz For all z1,z2 E C O (z1+z2) = z1 + z2 + iz1 + iz2 = z1+iz1 + z2+iz2 = O (z1) + O (z2) Hence O satisfies the homomorphism property and so is a homomorphism. 2. ## Re: Homomorphism #2 Originally Posted by Arron Is this correct anyone; C is the group of all complex numbers under addition O : C ----> C z ----> z + iz For all z1,z2 E C O (z1+z2) = z1 + z2 + iz1 + iz2 = z1+iz1 + z2+iz2 = O (z1) + O (z2) Hence O satisfies the homomorphism property and so is a homomorphism. It's pretty easy to show that multiplication by any complex number is a group homomorphism of $\mathbb{C}$ , and your function is only multiplication by $1+i$... Tonio 3. ## Re: Homomorphism #2 explicitly, let $w \in \mathbb{C}$ be given, and define: $h_w:\mathbb{C} \rightarrow \mathbb{C}$ by $h_w(z) = zw$ then $h_w(z_1+z_2) = (z_1+z_2)w = z_1w+z_2w = h_w(z_1) + h_w(z_2)$ this is what distributivity tells us in a commutative ring R: that multiplication by a fixed element r is an additive endomorphism. a slightly more archaic way of saying this, is that multiplication is compatible with addition. 4. ## Re: Homomorphism #2 I am trying to find the image and kernal of this homomorphism, am I correct in saying The identity element in (C, +) is 0, so Ker O = {z E C : O (z) =0} = (z E C : z+iz = 0} = {0} Because z + iz = z for each z E C, the function O is onto and Im (O) = C. 5. ## Re: Homomorphism #2 saying what you wish to be true, is not what constitutes a "proof" or demonstration. in particular, if you claim that ker(O) = {0}, you have to give a reason for it. one possible justification is that O(z) = z+iz = z(1+i). so if O(z) = z(1+i) = 0, then z(1+i)((1-i)/2)) = 0((1-i)/2) --> z = 0. this shows that ker(O) is a subset of {0}, and obviously we have 0 is an element of ker(O), so ker(O) = {0}. now, your argument that O is onto is just not one at all. in the first place, z+ iz DOES NOT EQUAL z (unless z = 0). to prove O is onto, given any w in C, you have to FIND some z in C with O(z) = w. to do this, you first set w = z + zi = z(1+i), and try to solve for z. then z = w/(1+i). does this work? well: O(w/(1+i)) = (w/(1+i)) + (w/(1+i))i = (w/(1+i))(1+i) = w((1+i)/(1+i)) = w(1) = w. THAT shows O is onto, so that Im(O) = C. 6. ## Re: Homomorphism #2 Hi, Could you not have divided each side by (1 + i), instead of multiplying by (the conjugate/2)? Thank you, GaryOg. 7. ## Re: Homomorphism #2 one could do many things. for example, one could use the fact that since C is a field, it is a fortiori, an integral domain, so: z(1+i) = 0, implies one of z or 1+i is 0. since 1+i is not 0, z must be. dividing by 1+i is the same as multiplying by 1/(1+i). since (1+i)((1-i)/2)) = (1/2)(1+i)(1-i) = (1/2)(12 - i2) = (1/2)(1 - (-1)) = (1/2)(1 + 1) = (1/2)(2) = 1, we see that 1/(1+i) IS (1-i)/2. in other words, "dividing by 1+i" is exactly what i did.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 6, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9271878600120544, "perplexity": 2863.607602928363}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948609934.85/warc/CC-MAIN-20171218063927-20171218085927-00021.warc.gz"}
http://mathhelpforum.com/calculus/199126-gradient-vectors.html	1. ## Gradient Vectors I apologize upfront if this question is too vague but it's only because my understanding of it is too vague as well. Anyway, here goes… I'm currently studying partial derivatives and gradient vectors but am having difficulty visualizing the orientation of the gradient vector relative to its originating function. I've assumed that all vectors originate at (0,0,0) and extend outward into any direction, but if at any given point along an (x,y,z) function there exists a vector indicating the maximum slope and direction at that point, how does that vector originate at (0,0,0) for all possible slopes and directions? I'm imagining tiny, little arrows pointing here and there scattered throughout the air in a room, all indicating a change in temperature, but, of course, they cannot all be originating from (0,0,0). How can this be? If anyone can help, thank you very much. 2. ## Re: Gradient Vectors Even if the vectors don't originate at the origin you can express them as if they do. For the sake of my diagram, let go back to two dimensions for a bit: Consider this situation, we have a vector AB and we want to represent it in terms of vectors from the origin, the following property of vectors can be used: $\tiny \dpi{300} \fn_cm \underset{AB}{\rightarrow} = \mathbf{b}- \mathbf{a}$ $\tiny \dpi{300} \fn_cm \underset{AB}{\rightarrow} = \underset{OB}{\rightarrow}- \underset{OA}{\rightarrow}$ So back to three dimensions: $\tiny \dpi{300} \fn_cm \underset{AB}{\rightarrow} = \begin{pmatrix}x_{b} \\ y_{b} \\ z_{b} \end{pmatrix} - \begin{pmatrix}x_{a} \\ y_{a} \\ z_{a} \end{pmatrix}$ Where AB is a direction vector from the point A to the point B and x(a), y(a), z(a) etc. are values for the direction vector from the origin to the points A and B ( therefore also the position vectors of A and B ). I hope this is what you were asking. =) 3. ## Re: Gradient Vectors thanks! I guess I misunderstood a vector as something that always originates at the origin. now the concept makes perfect sense.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 3, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.870246946811676, "perplexity": 385.86897932171576}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-47/segments/1510934806388.64/warc/CC-MAIN-20171121132158-20171121152158-00749.warc.gz"}
http://nowmag.org/23-section-modulus-i-beam-pictures/	# 23+ Section Modulus I Beam Pictures 23+ Section Modulus I Beam Pictures. You can specify the beam's density, young's modulus, and poisson's ratio or shear modulus in the stiffness and inertia section of the block dialog box. To define section modulus, it may be defined as the ratio of total moment resisted by the section to the stress in the extreme fibre which is equal to. Other geometric properties used in design include area for tension and shear, radius of gyration for compression, and moment of inertia and polar moment of inertia for stiffness. Or is there loss involved when doubling a beam? The section modulus is a geometric indicator of the efficiency of the beam system design. ### Or is there loss involved when doubling a beam? The elastic section modulus is used to calculate the elastic design resistance for bending or to calculate the stress at the extreme fibre of the section due to a moment. Elastic section modulus is defined as z = i / y, where i is the second moment of area (or izz moment of inertia) and y is the distance from the neutral axis to any given fibre. The section of the beam consists of material 1 with elastic modulus e1 and material 2 with elastic modulus e2. The maximum bending stress may then be written as.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9982465505599976, "perplexity": 879.5947636980931}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178358033.38/warc/CC-MAIN-20210226234926-20210227024926-00575.warc.gz"}
https://ham.stackexchange.com/questions/953/what-is-the-point-of-a-swr-power-meter-where-the-needles-cross-over-each-other	# What is the point of a SWR/power meter where the needles cross over each other? There are SWR/power meters with two separate gauges, and others that have them mounted so the gauge needles cross over each other. Aside from looking cool, does the cross-over point of the two meters provide specific or useful information? A cross-meter is capable of showing you three measurements simultaneously: • Output Power • Reflected Power • SWR From this image by Axel Schwenke on Wikipedia, you can see that the needle on the left indicates forward power, and the needle on the right indicates reflected power. The observed intersection of the two needles can be used to indicate the SWR of the system. The lines in the middle of the meter indicate what the SWR is for a given forward and reflected power reading. • Exactly. It prevents having to flip a switch between FWD/REV in order to read the SWR. And as an added bonus indicates power. Upvote. – SDsolar Apr 29 '17 at 21:28 A properly calibrated cross-needle power meter such as e.g. the MFJ-842 actually tells you something more than just the forward and reflected power, which as you point out can just as easily be indicated by two separate instruments. The intersection of the needles gives you a pretty good indication of the actual standing wave ratio or SWR because the SWR is directly related to the relationship between the forward and reflected power. So, there's more utility to this than simply "saving space" or "looking cool". Of course, you could have a third meter showing the resultant SWR based on forward and reflected power indicated on two separate instruments, but why bother when you can put all the information into one and at the same time reduce the risk of measurement inconsistencies?	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.867299497127533, "perplexity": 1081.6820705649661}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487612537.23/warc/CC-MAIN-20210614135913-20210614165913-00352.warc.gz"}
https://math.stackexchange.com/questions/3776724/accumulation-values-of-a-sequence	# Accumulation Values of a sequence $$a_n=(1+ \frac{1}{n})^ {-4+n}$$ How do you find all the accumulation points of a sequence? I normally firstly look of the first members of the sequence and try to find a subsequences. Then calculate their limit and then I have the accumulation points. I know that the sequence is increasing and that the limit of $$a_n= e$$. But I could not get further. Could you give me any tips for the problem and in general, how you proceed? And maybe is a stupid question, but if you have the limit of sequence is it so that the subsequences have the same limit as the limit of the sequence? As your example,the only accumulation value is $$e$$	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 3, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9282657504081726, "perplexity": 128.39633628810714}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439738819.78/warc/CC-MAIN-20200811180239-20200811210239-00462.warc.gz"}
http://mathhelpforum.com/calculus/170345-analysis-crossbow-action.html	# Thread: Analysis of crossbow action 1. ## Analysis of crossbow action This is a home-engineering problem, but there's not much mechanics involved. It's exclusively trig/calculus. I haven't done any of that since I left school more than 40 years ago, so I'm brushing away the cobwebs. I've spent two days trying to puzzle this out, but I just can't crack it. It's to do with the kind of Roman crossbow called an "inswinger". I'll point it to shoot skywards, so we can talk about "vertical" and "horizontal", to make things easier. You have two axles 2 feet apart. The bow arms attached to the axles are rigid levers driven by torsion springs. The arms are 1 foot in length, and the bowstring (2 ft long) is strung between the tips. In the rest position, with the bowstring taut, the arms point directly upwards at the 12 o'clock position. To cock the bow you pull the string down, forcing the levers inwards and downwards until they point directly at each other - a 90 deg turn. The tips can be regarded as touching at that point. And obviously then the angles subtended by the two halves of the string is zero. On release, the arms fly upwards and outwards, pulling the bowstring taut while propelling the missile up the arrow shute. Supposing a constant velocity from the arm tips (to simplify things), I need to calculate the missile velocity at any point during the release. And that is a function of the changing geometrical relationships in the configuration. I can see that the velocity ratio missile/armtip is 1:1 at the start, and something indeterminately large at the finish (infinite in theory). But finding a formula to fix the missile's position at any point is defeating me. I can simplify a little by ignoring everything to the right of the arrow chute, because it's just the mirror image of the left side. If I draw a line between the axles and call that the "baseline", I can be sure about just one position, apart from the starting and finishing positions. It's where the arm tip is at 60 deg to the baseline. Since the left half of the baseline is the same length as the arm, and the left half of the string is the same length, you have an equilateral triangle. So at that point the missile is positioned exactly on the baseline. But after that, I can't use the baseline as a reference, because the line between the missile and the axle is no longer horizontal, and no longer the same length. The only line that isn't moving is the chute, so it might be better to deal with a right-angle triangle involving that. The arm-tip is at A, missile at B, and the horizontal from A crosses the chute at C. The angle between the arm and the baseline is $\theta$. The angle of the string to the chute is $\phi$. The missile is getting its velocity ( $\text{v}_m$) from two sources. Call the tip's linear velocity $\text{v}_t$. The vertical component of that, $\text{v}_v$, does not cause the shape of the triangle ABC to change. It just raises it. So there is a 1:1 relationship. One inch of upward movement of the tip causes one inch of movement of the missile. So I've got the beginnings of an equation. $\text{v}_m = \text{v}_v+$...something. ...where $\text{v}_v$ must be $\text{v}_t \text{ cos }\theta$, and "something" is the movement of the missile caused by the horizontal movement of the tip ( $\text{v}_h$), which changes the angle of the string, thus forcing the missile up the chute. And $\text{v}_h=\text{v}_t\text{ sin } \theta$. But I can't just plug it into the equation, the way I can with $\text{v}_v$. There is no 1:1 relationship. It depends on $\phi$. But how do you calculate that? What is the relationship between $\phi$ and $\theta$? That's where I'm stuck. 2. Oh my... getting a diagram oh what you are trying to describe would be very helpful. To be honest, you lost me at "The bow arms attached to the axles are rigid levers driven by torsion springs".. 3. I would include a diagram if I knew how to do it. I had enough trouble with the LATEX codes! But the torsion springs are twisted bundles of fibres wrapped around the axle. Does that clarify it? 4. Okay, I did a google search and the first image that I think is in your problem is that: Is that close? Except I'm not sure which part is which without any label... I could add some labels to clear it up, but without your help here, I don't think that it'll be possible, 5. Yes, that's it, basically. But the arms in that diagram start below what I described as the "baseline" (the line drawn between the two axles). My model starts with the arm-tips exactly on the baseline, and they finish at the 12 o'clock position, unlike the one in the diagram, which stops 10 degrees or so before that. Also the arms on mine each occupy half the width between the axles, with their tips just touching in the cocked position, so the angle between the two halves of the string would be 0 at the starting point. 6. Lol, seems I'm more practising my image editing skills Okay, is that better? (and my labels good?) Assume the vertical arm is at 12 o'clock (it's 4 degrees too much to the left though) EDIT: Oh, I don't know whether there are 2 strings or not... and from your description, it appears to have only 1...? 7. I think you might have misinterpreted the first diagram. No, there aren't two strings - or four arms. That diagram just shows the arms and string in both the starting and finishing positions. There are just two arms and one string. Each arm travels through 94 degrees on that model. On mine it travels through 90 deg. Your amended diagram is correct, except that you have written "baseline" in the wrong position. The "baseline" is a line drawn directly between the two axles. And the arm is also 1 ft long, so the tips are touching at the start position, and the missile would start 1ft below the baseline, since the string is 2ft long and is folded double at that point. 8. Ok, I think that I understand now, and I see how it is. (they could've put arrows in the first diagram, I really thought there were 4 arms!) So, if I understand well, the relationship is 1:1 from the 'set' position until the missile reaches the baseline (ie, exactly between the axles), in terms of position. And the first part is trying to find the position from the 'set' position up to the baseline? EDIT: Going to bed now. A little past midnight. 9. Well, no. The 1:1 velocity ratio between the missile and the arm tips only applies at the start position, when they're moving in the same direction. By the time the missile has reached the baseline, the relationship has changed. Looking only at the left half of the diagram, when the missile is directly between the axles, the arm is at 60 deg. And at that point the arm, half-string and half-baseline make the three equal sides (1 ft each) of an equilateral triangle. That's the only position I can be sure of in terms of a geometric relationship. What happens after that is something I need a formula for. 10. Okay, if I understood well now: At the start, the velocity $v_m$ of the missile is equal to the velocity of the arm, $v_a$ So, at time t = 0 and for very small values of t, $v_m \approx v_a$ Assuming constant acceleration for the missile and constant velocity for the arm tip and considering, the arm tip moves $\dfrac{\pi}{3}\ ft$ while the missile moves 1 ft and we have: $\dfrac{\pi}{3} = v_a t$ $1 = \dfrac12 at^2$ $1 = \dfrac12 a\left(\dfrac{\pi}{3v_a}\right)^2$ $a = \dfrac{18v_a\ ^2}{\pi^2}$ So, the function modelling the velocity at any particular time between the point where the missile is at rest and the point where it is between the axle is: $v_m = \dfrac{18v_a\ ^2t}{\pi^2}$ EDIT: If I consider it all, from the start to the finish, the arm does a quarter of a turn (and 1 ft vertically) while the missile does 2 ft, seems very much like the position of the missile follows: $s_m = 2 \sin(\theta)$ for $0 \leq \theta \leq \dfrac{\pi}{2})$ And to get the velocity at any point, we take the derivative: $v_m = 2 \cos(\theta)$ Hm... not convincing. I'll try get where you were stuck, ie at phi and theta, notice that: $\sin\phi + \cos\theta = 1$ 11. Originally Posted by Unknown008 At the start, the velocity $v_m$ of the missile is equal to the velocity of the arm, $v_a$ So, at time t = 0 and for very small values of t, $v_m \approx v_a$ Assuming constant acceleration for the missile and constant velocity for the arm tip and considering, the arm tip moves $\dfrac{\pi}{3}\ ft$ while the missile moves 1 ft and we have: $\dfrac{\pi}{3} = v_a t$ $1 = \dfrac12 at^2$ $1 = \dfrac12 a\left(\dfrac{\pi}{3v_a}\right)^2$ $a = \dfrac{18v_a\ ^2}{\pi^2}$ So, the function modelling the velocity at any particular time between the point where the missile is at rest and the point where it is between the axle is: $v_m = \dfrac{18v_a\ ^2t}{\pi^2}$ You can assume constant velocity of the arm-tip, because I stated that as a given. But it isn't, in fact, true. The arm starts from rest, so it obviously does accelerate. But at the moment all I'm interested in is the relative velocity of the missile to the arm, so we can work with a constant arm velocity without invalidating the logic. However, you cannot assume constant acceleration of the missile. That definitely isn't true. Most of its acceleration will occur in the last few degrees of movement of the arm. You could assume it over a very short distance, but then your $\dfrac{\pi}{3}$ formula is not true over a short distance. EDIT: If I consider it all, from the start to the finish, the arm does a quarter of a turn (and 1 ft vertically) while the missile does 2 ft, seems very much like the position of the missile follows: $s_m = 2 \sin(\theta)$ for $0 \leq \theta \leq \dfrac{\pi}{2})$ And to get the velocity at any point, we take the derivative: $v_m = 2 \cos(\theta)$ Hm... not convincing. I'll try get where you were stuck, ie at phi and theta, notice that: $\sin\phi + \cos\theta = 1$ Bingo! Well spotted! I was looking for a relationship between $\phi$ and $\theta$, and I believe this is it. I have to go out now, but I'll back back online later. In the meantime I'll be thinking about the implications.... 12. I've had to spend some time getting re-aquainted with trigonometry. After all these years most of it has fallen through a hole in my memory. Now that I have a connection between $\theta$ and $\phi$ - i.e, $\cos \theta+\sin \phi=1$ (thanks for that!), I can proceed with the velocity analysis. $V_a$ = velocity of arm tip. $V_v$ = vertical component of $V_a$. $V_h$ = horizontal component of $V_a$. $V_m$=velocity of missile $V_m_1$=velocity of missile due to $V_v$ $V_m_2$=velocity of missile due to $V_h$ $V_m=V_m_1+V_m_2$ - or, to put it in words, the velocity of the missile due to the vertical movement of the arm must be added to the velocity of the missile due to the horizontal movement of the arm. The vertical part is easily solved. Since the vertical movement doesn't change the angle of the string, the relationship is 1:1. $V_m_1=V_v$ So, $V_m=V_v+V_m_2$ And, $V_v=V_a \cos \theta$, giving, $V_m=V_a \cos \theta + V_m_2$ $V_m_2$, the vertical movement of the missile produced by the horizontal movement of the arm, $V_h$, must be governed by the ratio of the vertical and horizontal sides of the ABC triangle. And since it increases as the angle increases, it must be horizontal/vertical. That resolves as, $V_m_2 = V_h \tan \phi$ So our main equation becomes, $V_m=V_a \cos \theta + V_h \tan \phi$ And $V_h=V_a \sin \theta$ giving $V_m=V_a \cos \theta + V_a \sin \theta .\tan \phi$ I have to get rid of $\phi$ and have everything expressed in terms of $\theta$, using $\cos \theta + \sin \phi=1$. That will involve some conversions. I've had to look up trigonometric identities on the internet, because I've forgotten them all. This one looks promising: $\tan \phi = \frac{\sin \phi}{\sqrt{1-\sin^2 \phi}}$ It will allow me to convert my $\tan \phi$ to an expression involving $\cos \theta$ $\sin \phi = 1-\cos \theta$ Giving, $\tan \phi = \frac{1-\cos \theta}{\sqrt {1-(1-\cos \theta)^2}}$ = $\frac{1-\cos \theta}{\sqrt{1-(1-2\cos \theta+\cos^2 \theta)}}$ = $\frac{1-\cos \theta}{\sqrt {2\cos \theta-\cos^2 \theta}}$ So our main equation for missile velocity against arm velocity is, $V_m=V_a \cos \theta + V_a \sin \theta .\frac{1-\cos \theta}{\sqrt {2\cos \theta-\cos^2 \theta}}$ That's where I'll stop for the moment, because the next step is calculus. I have a book on calculus, which has my name on the flyleaf, and a Malawi address where I haven't lived since 1965. I don't believe I've opened it with a purpose since the late '60s. So it will take me some time to go through it and learn all over again how to differentiate an expression like the one I have to deal with. It'll probably take me a few days, or even a week. But I don't want to embark on it with the wrong equation. I'd appreciate a check on what I've done so far, because even my basic algebra is pretty rusty, and I could have made mistakes. 13. Okay, this is become complicated for me... Calculus? If it's about the derivative or integration, wolfram does it for you http://www.wolframalpha.com/input/?i=\frac{1-cos\theta}{\sqrt{2cos\theta+-+cos^2\theta}} 14. Okay, thanks unknown008, I'll take it from here.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 78, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8776703476905823, "perplexity": 515.5943028832754}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-17/segments/1492917119838.12/warc/CC-MAIN-20170423031159-00261-ip-10-145-167-34.ec2.internal.warc.gz"}
http://beltoforion.de/article.php?a=tides_explained&hl=en&p=orbit&s=idPageTop	# Tides explained On the origin of a global ocean phenomenon ## Perigean Spring Tides In this section we will briefly investigate another special type of tide called the perigean spring tide. Sometimes they are also referred to as king tide. These are naturally occuring tides that are higher than usual. To understand their origin we first have to look at the orbit of the Moon. ### A closer look at the orbit of the Moon The tidal models discussed so far were based on simplified circular orbits. Although this has been a justifiable simplification it is a tad too simple to explain the finer details of tides. Now lets have a closer look at the realities of the Moon and its orbit and how this is affecting the tides. We already learned about Spring tides and Neap tides. Now lets look into a special type of spring tide called the perigean Spring Tide. To understand perigean spring tides lets first look at the orbit of the Moon. Moon orbits the Earth in prograde direction. Prograde direction means this is the same direction that earth is spinning. It completes one revolution with respect to the stars in 27.32 days. This is also called a siderial period. Both Earth and Moon are orbiting a common center of mass (the barycenter). The Orbit of the Moon is distinctly elliptical. Its average eccentricity is 0.0549 which brings the ellipse pretty close to a cirlce. The greatest distance between Moon and Earth is 405400 km (Apogee) the shortest distance is 362600 km (Perigee). The following image shows a schematic of the Moons orbit. Perigean Spring Tides occur when Sun, Earth and Moon are arranged in a line and the Moon is it its Perigee at the same time. In other words: The Moon is either full or new and it is closest to Earth [6]. Such an arrangement happens three to four times per year. The difference in water level between ordinary Spring Tides and Perigean Spring Tides is usually a couple of centimeters with a maximum difference of 15 cm being on record for parts of the Alaska coast. When seen with respect to the tidal range of 9 m in the same area the effect is rather small (<2 %). ## Tilt of the Moon Orbit Another property having an influence on the height of local tides is the tilt of the axis of the earth and the tilt of the Moon orbit against the ecliptic. For those who have not heard the term before: That is the plane in which the earth orbits the sun. The Earth axis is tilted 23.44° against the ecliptic. The Moons orbit is tilted 5.14° against the ecliptic. In this chapter we will focus on the effect caused by the tilt of the Moons orbit. It is rather trivial to imagine, that spring tides are a bit stronger if the Moon happens to be located at a point in its orbit where it intersects with the ecliptic. Only then Sun, Moon and Earth are located exactly at a straight line. This is also the only arrangement under which lunar or solar eclipses will occur. From that we can draw the conclusion that at the times of lunar or solar eclipse tides are a bit stronger stronger than usual. You might also like:	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.847998321056366, "perplexity": 573.9960357449645}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676593223.90/warc/CC-MAIN-20180722120017-20180722140017-00597.warc.gz"}
http://math.stackexchange.com/questions/138916/questions-on-limits-of-the-sequence-a-n-frac-1nn1n	# Questions on limits of the sequence $a_{n}=\frac{(-1)^{n}n+1}{n}$ Given $a_{n}=\frac{(-1)^{n}n+1}{n}$, compute $$\lim\limits{\inf(a_{n})}$$ $$\lim\limits{\sup(a_{n})}$$ $$\inf\{a_{n}\}$$ $${\sup(a_{n})}$$ My attempt: I tried taking different values for the sequence and reached the following results: $$\lim\limits{\inf(a_{n})}=1$$ $$\lim\limits{\sup(a_{n})}=-1$$ $$\inf\{a_{n}\}=3/2$$ $${\sup(a_{n})}=-1$$ The teacher told me to do it more formally, anyone can help me please? - Remember that $\liminf a_n \leq \limsup a_n$. – Siminore Apr 30 '12 at 15:21 You have accidentally swapped $\inf$ and $\sup$ in all of the above, but otherwise these would be the correct answers. You just need to prove it, that is provide a rigorous justification. – Eric Naslund Apr 30 '12 at 15:24 It's all correct, just remember that sup is the large one and inf the small. – Ekuurh Apr 30 '12 at 15:24 Note that you've interchanged '(lim)inf' and '(lim)sup' throughout your question. $$a_n=\begin{cases} 1+\frac1n,&\text{if }n\text{ is even}\\\\ -1+\frac1n,&\text{if }n\text{ is odd}\;. \end{cases}$$ Let's look first at $\sup_n a_n$. When $n$ is odd, $a_n\le 0$, and when $n$ is even, $a_n>0$, so in order to find $\sup_n a_n$ we need only look at the even-numbered terms: all of them are larger than any of the odd-numbered terms. But the subsequence $\langle a_{2n}:n\in\Bbb Z^+\rangle$ is clearly decreasing, so for every $n\in\Bbb Z^+$ we have $a_n\le a_2=\frac32$. Thus, $\sup_n a_n=\max_n a_n=a_2=\frac32$. You can make a similar argument for $\inf_n a_n$. Now let's look at $\limsup_n a_n$. By definition $$\limsup_n a_n=\lim_{n\to\infty}\sup_{k\ge n}a_k\;,\tag{1}$$ so for each $n\in\Bbb Z^+$ we need to see what $\sup\{a_k:k\ge n\}$ is. For this we can reason just as I did above to find that $$\sup_{k\ge n}a_k=\begin{cases} a_n,&\text{if }n\text{ is even}\\ a_{n+1},&\text{if }n\text{ is odd}\;,\tag{2} \end{cases}$$ but I'll leave the details to you. Once you've shown $(2)$, you just have to evaluate $(1)$, and you should find that it's simply $$\lim_{n\to\infty}a_{2n}=\lim_{n\to\infty}\left(1+\frac1n\right)=1\;.$$ (I doubt that you'll be asked for a formal proof that this limit really is $1$.) The arguments that you need for $\liminf_n a_n$ are very similar. - To make it more formal just use the definitions that you have for lim inf, lim sup, inf, and sup. Show that your answers meet those definitions. -	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.943026065826416, "perplexity": 209.22214130117453}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1409535921869.7/warc/CC-MAIN-20140901014521-00014-ip-10-180-136-8.ec2.internal.warc.gz"}
https://avenuewebmedia.com/ppw0vlhg/measurement-definition-in-physics-962447	Glossary. Osmium and iridium are the densest known elements at standard conditions for temperature and pressure, but certain chemical compounds may be denser. A physical quantity (like length) has to be measured with respect to some fixed quantity. Any quantity that can be measured is called a physical quantity. For example, when you measure how far and how fast a hockey puck slid, you need to measure both the distance in centimeters and the time in seconds. 1 In science, a measurement is a collection of quantitative or numerical data that describes a property of an object or event. The symbol most often used for density is ρ (the lower case Greek letter rho). The study of matter, forces, and energy through applied mathematics. A significant figure is a digit within a number that is expected to be accurate. The density of a material varies with temperature and pressure. Albert Einstein (1879–1955) put forward the theory of relativity, which gives the same result as Newton's laws of motion at slow speeds, but is more accurate as speeds increase towards that of light. The matter has … While it was previously defined as a specific volume of water (i.e. Measurement error is the amount of inaccuracy. For example, we use metres to measure the length of a piece of cloth, kilometres to measure the distance from one place to another, millimetres to measure the thickness of the hair, and so on. m Here we will measure and compare the same kind of physical quantities with the help of numbers and units. Units and Dimensions System of Units. ∗ \| Definition, Type, Formulas, Units – Motion in a Straight Line. We are giving a detailed and clear sheet on all Physics Notes that are very useful to understand the Basic Physics Concepts. Originally defined as the amount of time the earth needs to make 1/86,400 of a rotation, it is now defined as 9,192,631,770 oscillations of a cesium-133 atom. mass Get Physics and Measurement, Physics Chapter Notes, Questions & Answers, Video Lessons, Practice Test and more for CBSE Class 10 at TopperLearning. V The formula for density is Density Formula, Density is the amount of mass per volume, or the quantity of mass per unit volume of a substance. A second is a unit of time. You may also see conversion factors for weight and mass (e.g. Define measurement. We all know that a car is much heavier than a bicycle but exactly how many times? Spell. These numbers and units together form a physical quantity. ü Independent of time 2. Increasing the temperature of a substance (with a few exceptions) decreases its density by increasing its volume. Definition: "MEASUREMENT" is the determination of the size or magnitude of something. {\displaystyle {\text{unbalanced force}}={\frac {\text{length}}{\text{time}}}{\text{mass}}}. Flashcards. Physics and Measurements Revision notes by Vedantu. Here is an example. This system is called the International System of Units or the SI units. It is certainly different from accuracy. Physics in raw terms is the study of everything around us. An ampere (A) is a measure for electric current. The current system of units has three standard units: the meter, kilogram, and second. Terms in this set (29) Definition of physics. Definition of measurement. For example, Isaac Newton (1642–1727) found the laws behind the motion of bodies, which we now use to design rockets that travel to the moon and other planets. The same principle applies; the numbers are removed, and the units are verified to be equal on both sides of the equation. In general, an operation performed on two numbers will result in a new number. The funda- Measurement is an integral part of Physics like any other scientific subject. Types of Speed in Physics. Click here to download the pdf version of "MEASUREMENT 1 - Form 1 Physics Notes", and read the full contents of this page OR . Measurements in Physics. When you are presented with an equation, dimensional analysis is performed by stripping the numerical components and leaving only the unit types (length, mass, or time). (1)- the unit of measurement. This can be easily answered if we have chosen a standard mass and call it unit mass. To simplify comparisons of density across different systems of units, it is sometimes replaced by the dimensionless quantity "relative density" or "specific gravity", i.e. Measurements in Physics.nb A physical quantity (like length) has to be measured with respect to some fixed quantity. When doing calculations, you should only keep at most 1 doubtful digit; while it is acceptable to keep them when using a handheld calculator or computer, the final answer should be adjusted to reflect the correct number of significant digits. The density (or more precisely, the volumetric mass density) of a substance is its mass per unit volume. Uniform Speed Definition: While that cannot apply in all situations, these factors may be used in some limited scopes. {\displaystyle {\text{weight force (weight)}}=9.8m/s^{2}{\text{mass}}}, unbalanced force Density is an intensive property in that increasing the amount of a substance does not increase its density; rather it increases its mass. CLICK HERE to get "MEASUREMENT 1 - Form 1 Physics Notes" on Whatsapp. The small reading reduces the error of the calculation. Learn physics terms and definitions units measurement with free interactive flashcards. Measurement is one of the fundamental processes of science science [Lat. Gravity. Measurement is the process of finding the length, size, or quantity of a substance. Increasing the pressure on an object decreases the volume of the object and thus increases its density. A meter is a unit of length, currently defined as the distance that light travels in 1/299,792,458 of a second. Solution: We know that 1000 m = 1 km Therefore, 2000 m = 2 km Therefore, the distance between the two houses is 2 km. Thus a relative density less than one means that the substance floats in water. We are giving a detailed and clear sheet on all Physics Notes that are very useful to understand the Basic Physics Concepts. The act of measuring or the process of being measured. M.K.S.A = Meter-Kilogram-Second-Ampere Unit. In early times, people used different body parts like hand span, cubit, and fathom to measure length. Thus a slug in equal to 14.5 kilograms, or 32.174 pounds. where ρ is the density, m is the mass, and V is the volume. The SI base units of length, mass, and time are the meter (m), kilogram (kg), and second (s), respectively. Since this comparison cannot be perfect, measurements inherently include error, which is how much a measured value deviates from the true value. Centimetre (cm) and millimetre (mm) are used to measure shorter distances while kilometre (km) is used to measure longer distances. The unit standard of time (and space) should be the same in 1050 as it is in 1012. This is the most affordable option. Express the distance between their houses in kilometres (km). The reciprocal of the density of a substance is occasionally called its specific volume, a term sometimes used in thermodynamics. You also use the standard English system of inches and feet and so on — that’s the FPI (foot-pound-inch) system. C.G.S = Cent-Gram-Second System. 2. Kids try to compare their height, size of candy, size of dolls and amount of toys they have. For example: Take a book and use ruler (scale) to find its length. Different materials usually have different densities, and density may be relative to buoyancy, purity and packaging. The following definitions are given in the ISO Guide to the Expression of Uncertainty in Measurement. All these happen even before they know math. Example 1: Raju and his friend Akhil live 2000 m from each other. Kids try to compare their height, size of candy, size of dolls and amount of toys they have. Zeros at the end of the number but before the decimal point are not included as significant figures (although exceptions may occur.). in physics. 2 1. 20 It is an intriguing fact that some physical quantities are more fundamental than others and that the most fundamental physical quantities can be defined only in terms of the procedure used to measure them. OR . Mathematically, density is defined as mass divided by volume:[1], ρ Each division is called millimetre. Standard Units of Measurements Units that have a fixed quantity and do not vary from person to person and place to place are called standard units. Uniform Speed Definition: The SI unit of length is metre. Measurement is an integral part of human race, without it there will be no trade, no statistics. This variation is typically small for solids and liquids but much greater for gases. = In some cases (for instance, in the United States oil and gas industry), density is loosely defined as its weight per unit volume,[2] although this is scientifically inaccurate – this quantity is more properly called specific weight. Note: A slug is also an animal similar to a snail. The comparison of an unknown size to an accepted value or standard. Isaac Newton found the laws of motion in the 17th century. In the FPS system, one pound is the equivalent of 4.45 Newtons. If an exact number is used, it should have the same number of digits as the estimated number. Physics - Measurement Units - The following table illustrates the major measuring units in physics − These three units form the mks-system or the metric system. The units in which they are measured are thus called fundamental units.In this textbook, the fundamental physical quantities are taken to be length, mass, time, and electric … Scales are used to measure. What is your definition for an object? A measurement is made by comparing a quantity with a standard unit. Measurement is a process of detecting an unknown physical quantity by using standard quantity. What is Mass Measured in Physics? Learn. It may also be used to determine the type of unit used for an unknown variable. Creative Commons Attribution-ShareAlike License. length For that reason, physics uses a number of measurement systems, such as the CGS (centimeter-gram-second) system and the MKS (meter-kilogram-second) system. In other words, the closeness of the measured value to a standard or true value. 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Measurement can also be defined as "Comparison of an unknown quantity with some known quantity of the same kind". km Measurement is the numerical quantitation of the attributes of an object or event, which can be used to compare with other objects or events. C.G.S = Cent-Gram-Second System. Measurement is the process of finding the length, size, or quantity of a substance. Test. The word Physics originates from the Greek word Physis, which means nature. measurement synonyms, measurement pronunciation, measurement translation, English dictionary definition of measurement. ∗ mass A kelvin (K) is a measure for temperature. Precision is a measure of how well a result can be determined (without reference to a theoretical or true value). m A conversion factor is a ratio between two compatible units. For example, the foot (which is literally the length measurement of an average human foot) is around 25 – 30 cm. The following definitions are given in the ISO Guide to the Expression of Uncertainty in Measurement. In contrast, a doubtful figure is a digit that might not be correct. Types of Speed in Physics. Created by. You get all topics/full set papers at a lesser price than if you bought per paper/ per topic. Every time you cook from a recipe or stand on a scale, you're taking a measurement. Physicists also revise the laws of physics from time to time according to experimental results. What is Speed in Physics? In physics, most measurements have units, such as meters or seconds. One would know a simple ruler or tape could be used to measure small distances, your height, and possibly much more. We also use units like inches and yards which are still in use but they are not the … A kilogram is a unit of mass. ( For a Student and Employee), Thank You Letter for Job Interview, Friend, Boss, Support \| Appreciation and Format of Thank You Letter, How To Write a Cover Letter \| Format, Sample and Important Guidelines of Cover letter, How to Address a Letter \| Format and Sample of Addressing a Letter, Essay Topics for High School Students \| Topics and Ideas of Essay for High School Students. = 1 kilogram is "close enough" to 2.2 pounds) based on external factors. Many additional terms relevant to the field of measurement are given in a companion publication to the ISO Guide, entitled the International Vocabulary of Basic and General Terms in Metrology, or VIM.Both the ISO Guide and VIM may be readily purchased. Another Simple Measurement - Another way a measuring instrument influences a measurement. Dimensional analysis is used to determine if an equation is dimensionally correct. Precision is the amount of information whose conveyance takes place by a number in terms of its digits. Based on either limitations of the measuring instruments or from statistical fluctuations in the quantity being measured [Baird, 2]. 3. These laws work at normal speeds, but when an object's speed approaches the speed of light, these laws fails. This causes it to rise relative to more dense unheated material. The lux (lx) is a measure unit for luminous flux per unit area. Some common standard units of length are inch, millimetre, centimetre, and kilometre. Indicates the precision of a measurement [Bevington, 2]. F.P.S = Foot-Pound-Second System. However, these units are not reliable as the length of body parts varies from person to person. If the object or quantity to be measured is not accessible for direct comparison, it is converted or “transduced” into an analogous measurement signal. A system of measuring: measurement in miles. Measuring Common Objects - An introduction to measurement in physics. As a general rule, any non-zero digit shown is a significant figure. 2 1. 1 pound (lb) is equal to 456 grams. A fixed quantity with respect to which a physical quantity is measured is called a unit. Philosophy of physics - Philosophy of physics - The measurement problem: The field of quantum mechanics has proved extraordinarily successful at predicting all of the observed behaviours of electrons under the experimental circumstances just described. A mole (mol) is the amount of substance (based on number of atoms rather than mass.). HelpYouBetter » Physics » Units and Measurements » Measurement in Physics & SI units of Measurement. The accuracy of the system is classified into three types as follows: 1. ∗ These formulas are then used by other physicists and engineers to predict results of their experiments. The scope and application of measurement are dependent on the context and discipline. In most materials, heating the bottom of a fluid results in convection of the heat from the bottom to the top, due to the decrease in the density of the heated fluid. In physics and measurement, we will get to know about physical quantities and their different types. Measurement. are examined. and as such, the unit of force involves multiplying length and mass, and dividing by the square of the time. A unit is used as a standard of measurement. Point Accuracy The accuracy of the instrument only at a particular poin… The general logico-algebraic approach to quantum/classical physics is justified as a special case of measurement theory. / SI units are a metric system of units, meaning values can be calculated by factors of 10. Math provides a great way to study about anything, that's why we see computers involved in almost anything because they are good at math. 9.8 Measurement of an object consists of: It is the degree of consistency and agreement among independent measurements of the same quantity; also the reliability or reproducibility of the result. Depending on the type of physical quantity, intensity can … What is unit in physics? s santejm. Therefore, people realized the need for. All these happen even before they know math. Zeros that appear after the decimal point and are at the end of the number are also significant. = The most common measurement system in introductory physics is the MKS system. One metre is divided into 100 equal divisions, each called centimetre which is again divided into equal divisions. (All but this last definition suggest that the uncertainty includes an estimate of the precision and accuracy of the measured value.) Your measurements are the sizes of various parts of your body, especially your chest, waist, and hips, that you refer to when you want to buy clothes. The main objective of physics is to ﬁnd the limited num-ber of fundamental laws that govern natural phenomena and to use them to develop theories that can predict the results of future experiments. Discusses measurement, qualitative versus quantitative measurements, the SI units for measurement, prefixes in the SI system, and unit conversion. Many additional terms relevant to the field of measurement are given in a companion publication to the ISO Guide, entitled the International Vocabulary of Basic and General Terms in Metrology, or VIM.Both the ISO Guide and VIM may be readily purchased. Commonly used units of length: 10 millimetres = 1 centimetre (cm) 100 centimetres = 1 metre (m) 1000 metres = 1 kilometre (km) A unit can be converted to another. Accuracy refers to the closeness of the measurements related to a specific value. Here is a brief of the topics covered under the topic of Physics and Measurement shared by our faculties. Depending on the size of the object, we need to measure, we have to choose an appropriate unit. To find out, you have to convert the miles into kilometers. If both numbers are exact, the new number should be calculated fully (within reason). The Lumen (lm) is a measure unit for total amount light visible for the human eye emitted by a source. Measurement begins with a definition of the quantity that is to be measured, and it always involves a comparison with some known quantity of the same kind. scientia=knowledge]. Math is built into our brains even before we start to learn it. This new number should have the same number of significant digits as the least accurate number. Suppose the length was 20 cm. What Is Measurement In Physics. The predictions that quantum physics makes are in general probabilistic.The mathematical tools for making predictions about what measurement outcomes may occur were developed during the 20th century and make use of linear algebra and functional analysis. Significant figures are relevant in measured numbers, general estimates or rounded numbers. Here is a brief of the topics covered under the topic of Physics and Measurement shared by our faculties. It exists independently and is autonomous of every other boundary, such as the temperature, pressure, and the area of the object in space. PLAY. M.K.S = Meter-Kilogram-Second System. The adoption of SI units in 1960 made it easier for scientists of different countries to communicate their results to one another. For example, the force of gravity may appear as the following: weight force (weight) Units and Dimensions You can see the philosophy of measurement in the little kids who don't even know what math is. {\displaystyle \rho ={\frac {m}{V}},}. = Match. This unit of measurement is still in use nowadays. Math is built into our brains even before we start to learn it. time miles 0.621 The ability to take accurate measurements is an important skill we all use in our lives. If the object or quantity to be measured is not accessible for direct comparison, it is converted or “transduced” into an analogous measurement signal. Mass is a fundamental characteristic property of matter. Indeed, it has proved extraordinarily successful at predicting all of the observed behaviours of all physical systems … The measurement means the action of measuring something or measurement is defined as the process of determining the value of an unknown quantity by comparing it with some pre-defined standard. miles Yet Another Simple Measurement - Try some more-sophisticated analysis as a class, and explore random and systematic uncertainties. You can see the philosophy of measurement in the little kids who don't even know what math is. For the sake of uniformity, scientists all over the world have adopted a common set of units. ü Independent of time 2. 1 liter or a 10 cm³ cube), its current definition is based on a prototype platinum-iridium cylinder. STUDY. \| Definition, Type, Formulas, Units – Motion in a Straight Line. Write. This results in order of magnitude calculation. The units in which they are measured are thus called fundamental units.In this textbook, the fundamental physical quantities are taken to be length, mass, time, and electric current. M.K.S.A = Meter-Kilogram-Second-Ampere Unit. Possibly the oldest discipline in Physics could be astronomy. 32.2 In quantum physics, a measurement is the testing or manipulation of a physical system in order to yield a numerical result. Precision shows the closeness of two or more measurements that they have to each other. n. 1. Math p… M.K.S = Meter-Kilogram-Second System. = Measurement is an integral part of human race, without it there will be no trade, no statistics. The most common measurement system you see in introductory physics is the meter-kilogram-second (MKS) system, referred to as SI (short for Système International d’Unités, the International System of Units), but you may also come across the foot-pound-second (FPS) system. In Physics, we do have certain scales for certain quantities which we will see very shortly. This page was last edited on 17 January 2021, at 01:28. km And measurement would be the assignment of a number to a physical quantity of a physical system. Measurement and Measurement Units in Physics. Physics. The order of magnitude gives the approximate idea of the powers of 10 .Any number in the form a10b [ here a multiplied by 10.. And 10raised to the power b]if a >or = (10)^0.5 the a become 1 and b is not changed but when a>(10)^0.5 then a is taken as 10 so power of b increases by 1. pounds and kilograms). In a pure substance, the density has the same numerical value as its mass concentration. Click Create Assignment to assign this modality to your LMS. Choose from 500 different sets of physics terms and definitions units measurement flashcards on Quizlet. Since ancient times, people have used several ways to measure length. The ability of the instrument to measure the accurate value is known as accuracy. ike all other sciences, physics is based on experimental observations and quan-titative measurements. People in different countries may be using a different set of standard measurement units. Multiplication by conversion factors allows for quantities to change units. Accuracy is obtained by taking small readings. Since ancient times, people have used several ways to measure length. 2 The most common measurement system in introductory physics is the MKS system. By comparing that unknown quantity with some standard quantity of equal nature, known as measurement unit. Physics - Measurement Units - The following table illustrates the major measuring units in physics − Physics and Measurements Revision notes by Vedantu. Measurement begins with a definition of the quantity that is to be measured, and it always involves a comparison with some known quantity of the same kind. (2)- the number of units the object measures. Unit analysis is similar to dimensional analysis, except that it uses units instead of the basic dimensions. The current metric system also includes the following units: A slug is a unit of mass in the FPS (foot-pound-second) system. It is defined as the force required to accelerate 1 pound of mass to 1 ft/s2, where These factors rely on equivalence (e.g. System of Units. Measurement is an integral part of Physics like any other scientific subject. Adopting standard units of measurement does not solve the problem. Physicists try to express everyday happenings in concise mathematical formulas. {\displaystyle 20{\text{ miles}}=20{\text{ miles}}{\frac {1{\text{ km}}}{0.621{\text{ miles}}}}=32.2{\text{ km}}}. Measurements in Physics.nb Wikipedia has related information at Physics, From Wikibooks, open books for an open world, Estimates and Order-of-Magnitude calculation, https://en.wikibooks.org/w/index.php?title=Fundamentals_of_Physics/Physics_and_Measurement&oldid=3799878. For example, the metric system, created by the French in 1790, is a standard set of units. What is Speed in Physics? miles The predictions that quantum physics makes are in general probabilistic. Problems such as the definition of systems, the significance of observations, numerical scales and observables, etc. If the car is 300 times heavier than the u… the ratio of the density of the material to that of a standard material, usually water. The unit standard of time (and space) should be the same in 1050 as it is in 1012. In quantum physics, a measurement is the testing or manipulation of a physical system in order to yield a numerical result. F.P.S = Foot-Pound-Second System. Glossary. 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Quantity by using standard quantity words, the density, m is the determination of the measured value.....	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8960320949554443, "perplexity": 815.5378971668491}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038077336.28/warc/CC-MAIN-20210414064832-20210414094832-00634.warc.gz"}
https://en.wikipedia.org/wiki/Stone%E2%80%93Weierstrass_theorem	# Stone–Weierstrass theorem In mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on a closed interval [a, b] can be uniformly approximated as closely as desired by a polynomial function. Because polynomials are among the simplest functions, and because computers can directly evaluate polynomials, this theorem has both practical and theoretical relevance, especially in polynomial interpolation. The original version of this result was established by Karl Weierstrass in 1885 using the Weierstrass transform. Marshall H. Stone considerably generalized the theorem (Stone 1937) and simplified the proof (Stone 1948). His result is known as the Stone–Weierstrass theorem. The Stone–Weierstrass theorem generalizes the Weierstrass approximation theorem in two directions: instead of the real interval [a, b], an arbitrary compact Hausdorff space X is considered, and instead of the algebra of polynomial functions, approximation with elements from more general subalgebras of C(X) is investigated. The Stone–Weierstrass theorem is a vital result in the study of the algebra of continuous functions on a compact Hausdorff space. Further, there is a generalization of the Stone–Weierstrass theorem to noncompact Tychonoff spaces, namely, any continuous function on a Tychonoff space is approximated uniformly on compact sets by algebras of the type appearing in the Stone–Weierstrass theorem and described below. A different generalization of Weierstrass' original theorem is Mergelyan's theorem, which generalizes it to functions defined on certain subsets of the complex plane. ## Weierstrass approximation theorem The statement of the approximation theorem as originally discovered by Weierstrass is as follows: Weierstrass Approximation Theorem. Suppose f is a continuous real-valued function defined on the real interval [a, b]. For every ε > 0, there exists a polynomial p(x) such that for all x in [a, b], we have \| f (x) − p(x)\| < ε, or equivalently, the supremum norm \|\| f − p\|\| < ε. A constructive proof of this theorem using Bernstein polynomials is outlined on that page. ### Applications As a consequence of the Weierstrass approximation theorem, one can show that the space C[a, b] is separable: the polynomial functions are dense, and each polynomial function can be uniformly approximated by one with rational coefficients; there are only countably many polynomials with rational coefficients. Since C[a, b] is Hausdorff and separable it follows that C[a, b] has cardinality equal to 20 — the same cardinality as the cardinality of the reals. (Remark: This cardinality result also follows from the fact that a continuous function on the reals is uniquely determined by its restriction to the rationals.) ## Stone–Weierstrass theorem, real version The set C[a, b] of continuous real-valued functions on [a, b], together with the supremum norm \|\| f \|\| = supaxb \| f (x)\|, is a Banach algebra, (i.e. an associative algebra and a Banach space such that \|\| fg\|\| ≤ \|\| f \|\|·\|\|g\|\| for all f, g). The set of all polynomial functions forms a subalgebra of C[a, b] (i.e. a vector subspace of C[a, b] that is closed under multiplication of functions), and the content of the Weierstrass approximation theorem is that this subalgebra is dense in C[a, b]. Stone starts with an arbitrary compact Hausdorff space X and considers the algebra C(X, R) of real-valued continuous functions on X, with the topology of uniform convergence. He wants to find subalgebras of C(X, R) which are dense. It turns out that the crucial property that a subalgebra must satisfy is that it separates points: a set A of functions defined on X is said to separate points if, for every two different points x and y in X there exists a function p in A with p(x) ≠ p(y). Now we may state: Stone–Weierstrass Theorem (real numbers). Suppose X is a compact Hausdorff space and A is a subalgebra of C(X, R) which contains a non-zero constant function. Then A is dense in C(X, R) if and only if it separates points. This implies Weierstrass’ original statement since the polynomials on [a, b] form a subalgebra of C[a, b] which contains the constants and separates points. ### Locally compact version A version of the Stone–Weierstrass theorem is also true when X is only locally compact. Let C0(X, R) be the space of real-valued continuous functions on X which vanish at infinity; that is, a continuous function f is in C0(X, R) if, for every ε > 0, there exists a compact set KX such that \| f \| < ε on X \ K. Again, C0(X, R) is a Banach algebra with the supremum norm. A subalgebra A of C0(X, R) is said to vanish nowhere if not all of the elements of A simultaneously vanish at a point; that is, for every x in X, there is some f in A such that f (x) ≠ 0. The theorem generalizes as follows: Stone–Weierstrass Theorem (locally compact spaces). Suppose X is a locally compact Hausdorff space and A is a subalgebra of C0(X, R). Then A is dense in C0(X, R) (given the topology of uniform convergence) if and only if it separates points and vanishes nowhere. This version clearly implies the previous version in the case when X is compact, since in that case C0(X, R) = C(X, R). There are also more general versions of the Stone–Weierstrass that weaken the assumption of local compactness.[1] ### Applications The Stone–Weierstrass theorem can be used to prove the following two statements which go beyond Weierstrass's result. • If f is a continuous real-valued function defined on the set [a, b] × [c, d] and ε > 0, then there exists a polynomial function p in two variables such that \| f (x, y) − p(x, y) \| < ε for all x in [a, b] and y in [c, d]. • If X and Y are two compact Hausdorff spaces and f : X × YR is a continuous function, then for every ε > 0 there exist n > 0 and continuous functions f1, ..., fn on X and continuous functions g1, ..., gn on Y such that \|\| f − ∑ fi gi \|\| < ε. The theorem has many other applications to analysis, including: ## Stone–Weierstrass theorem, complex version Slightly more general is the following theorem, where we consider the algebra C(X, C) of complex-valued continuous functions on the compact space X, again with the topology of uniform convergence. This is a C-algebra with the -operation given by pointwise complex conjugation. Stone–Weierstrass Theorem (complex numbers). Let X be a compact Hausdorff space and let S be a subset of C(X, C) which separates points. Then the complex unital -algebra generated by S is dense in C(X, C). The complex unital -algebra generated by S consists of all those functions that can be obtained from the elements of S by throwing in the constant function 1 and adding them, multiplying them, conjugating them, or multiplying them with complex scalars, and repeating finitely many times. This theorem implies the real version, because if a sequence of complex-valued functions uniformly approximate a given function f, then the real parts of those functions uniformly approximate the real part of f. As in the real case, an analog of this theorem is true for locally compact Hausdorff spaces. ## Stone–Weierstrass theorem, quaternion version Following John C.Holladay (1957) : consider the algebra C(X, H) of quaternion-valued continuous functions on the compact space X, again with the topology of uniform convergence. If a quaternion q is written in the form q=a+ib+jc+kd then the scalar part a is the real number (q-iqi-jqj-kqk)/4. Likewise being the scalar part of -qi,-qj and -qk : b,c and d are respectively the real numbers (-qi-iq+jqk-kqj)/4, (-qj-iqk-jq+kqi)/4 and (-qk+iqj-jqk-kq)/4. Then we may state : Stone–Weierstrass Theorem (quaternion numbers). Suppose X is a compact Hausdorff space and A is a subalgebra of C(X, H) which contains a non-zero constant function. Then A is dense in C(X, H) if and only if it separates points. ## Stone-Weierstrass Theorem, C-algebra version The space of complex-valued continuous functions on a compact Hausdorff space X i.e. C(X, C) is the canonical example of a unital commutative C-algebra ${\displaystyle {\mathfrak {A}}}$. The space X may be viewed as the space of pure states on ${\displaystyle {\mathfrak {A}}}$, with the weak-* topology. Following the above cue, a non-commutative extension of the Stone–Weierstrass theorem, which has remain unsolved, is as follows: Conjecture. If a unital C-algebra ${\displaystyle {\mathfrak {A}}}$ has a C-subalgebra ${\displaystyle {\mathfrak {B}}}$ which separates the pure states of ${\displaystyle {\mathfrak {A}}}$, then ${\displaystyle {\mathfrak {A}}={\mathfrak {B}}}$. In 1960, Jim Glimm proved a weaker version of the above conjecture. Stone-Weierstrass theorem (C-algebras).[2] If a unital C-algebra ${\displaystyle {\mathfrak {A}}}$ has a C-subalgebra ${\displaystyle {\mathfrak {B}}}$ which separates the pure state space (i.e. the weak- closure of the pure states) of ${\displaystyle {\mathfrak {A}}}$, then ${\displaystyle {\mathfrak {A}}={\mathfrak {B}}}$. ## Lattice versions Let X be a compact Hausdorff space. Stone's original proof of the theorem used the idea of lattices in C(X, R). A subset L of C(X, R) is called a lattice if for any two elements f, gL, the functions max{ f, g}, min{ f, g} also belong to L. The lattice version of the Stone–Weierstrass theorem states: Stone–Weierstrass Theorem (lattices). Suppose X is a compact Hausdorff space with at least two points and L is a lattice in C(X, R) with the property that for any two distinct elements x and y of X and any two real numbers a and b there exists an element f ∈ L with f (x) = a and f (y) = b. Then L is dense in C(X, R). The above versions of Stone–Weierstrass can be proven from this version once one realizes that the lattice property can also be formulated using the absolute value \| f \| which in turn can be approximated by polynomials in f. A variant of the theorem applies to linear subspaces of C(X, R) closed under max (Hewitt & Stromberg 1965, Theorem 7.29): Stone–Weierstrass Theorem. Suppose X is a compact Hausdorff space and B is a family of functions in C(X, R) such that 1. B separates points. 2. B contains the constant function 1. 3. If f ∈ B then af ∈ B for all constants aR. 4. If f, gB, then f + g, max{ f, g} ∈ B. Then B is dense in C(X, R). More precise information is available: Suppose X is a compact Hausdorff space with at least two points and L is a lattice in C(X, R). The function φ ∈ C(X, R) belongs to the closure of L if and only if for each pair of distinct points x and y in X and for each ε > 0 there exists some f ∈ L for which \| f (x) − φ(x)\| < ε and \| f (y) − φ(y)\| < ε. ## Bishop's theorem Another generalization of the Stone–Weierstrass theorem is due to Errett Bishop. Bishop's theorem is as follows (Bishop 1961): Let A be a closed subalgebra of the Banach space C(X, C) of continuous complex-valued functions on a compact Hausdorff space X. Suppose that f ∈ C(X, C) has the following property: f \|SAS for every maximal set SX such that all real functions of AS are constant. Then f ∈ A. Glicksberg (1962) gives a short proof of Bishop's theorem using the Krein–Milman theorem in an essential way, as well as the Hahn–Banach theorem : the process of Louis de Branges (1959). See also Rudin (1973, §5.7). ## Nachbin's theorem Nachbin's theorem gives an analog for Stone–Weierstrass theorem for algebras of complex valued smooth functions on a smooth manifold (Nachbin 1949). Nachbin's theorem is as follows (Llavona 1986): Let A be a subalgebra of the algebra C(M) of smooth functions on a finite dimensional smooth manifold M. Suppose that A separates the points of M and also separates the tangent vectors of M: for each point mM and tangent vector v at the tangent space at m, there is a fA such that df(x)(v) ≠ 0. Then A is dense in C(M). • Müntz–Szász theorem. • Bernstein polynomial. • Runge's phenomenon shows that finding a polynomial P such that f (x) = P(x) for some finely spaced x = xn is a bad way to attempt to find a polynomial approximating f uniformly. However, as is shown in Walter Rudin's Principles of Mathematical Analysis, one can easily find a polynomial P uniformly approximating f by convolving f with a polynomial kernel. • Mergelyan's theorem, concerning polynomial approximations of complex functions. ## Notes 1. ^ Willard, Stephen (1970). General Topology. Addison-Wesley. p. 293. ISBN 0-486-43479-6. 2. ^ Glimm, James (1960). "A Stone-Weierstrass Theorem for C*-algebras". Annals of Mathematics. Second Series. 72 (2): 216–244 [Theorem 1]. doi:10.2307/1970133. JSTOR 1970133. ## References ### Historical works The historical publication of Weierstrass (in German language) is freely available from the digital online archive of the Berlin Brandenburgische Akademie der Wissenschaften: • K. Weierstrass (1885). Über die analytische Darstellbarkeit sogenannter willkürlicher Functionen einer reellen Veränderlichen. Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin, 1885 (II). Erste Mitteilung (part 1) pp. 633–639, Zweite Mitteilung (part 2) pp. 789–805. Important historical works of Stone include: • Stone, M. H. (1937), "Applications of the Theory of Boolean Rings to General Topology", Transactions of the American Mathematical Society, Transactions of the American Mathematical Society, Vol. 41, No. 3, 41 (3): 375–481, doi:10.2307/1989788, JSTOR 1989788. • Stone, M. H. (1948), "The Generalized Weierstrass Approximation Theorem", Mathematics Magazine, 21 (4): 167–184, doi:10.2307/3029750, JSTOR 3029750; 21 (5), 237–254.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 10, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9735526442527771, "perplexity": 428.04213039316835}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988720475.79/warc/CC-MAIN-20161020183840-00226-ip-10-171-6-4.ec2.internal.warc.gz"}
https://brilliant.org/discussions/thread/help-me-with-this-4/?sort=new	# Help me with this! The semicircle above has a radius of 7. Find X. Note by Interliser 727 2 years ago This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science. When posting on Brilliant: • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused . • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone. • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge. MarkdownAppears as italics or _italics_ italics bold or __bold__ bold - bulleted- list • bulleted • list 1. numbered2. list 1. numbered 2. list Note: you must add a full line of space before and after lists for them to show up correctly paragraph 1paragraph 2 paragraph 1 paragraph 2 [example link](https://brilliant.org)example link > This is a quote This is a quote # I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world" # I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world" MathAppears as Remember to wrap math in $$ ... $$ or $ ... $ to ensure proper formatting. 2 \times 3 $2 \times 3$ 2^{34} $2^{34}$ a_{i-1} $a_{i-1}$ \frac{2}{3} $\frac{2}{3}$ \sqrt{2} $\sqrt{2}$ \sum_{i=1}^3 $\sum_{i=1}^3$ \sin \theta $\sin \theta$ \boxed{123} $\boxed{123}$ Sort by: Not enough info. I would expect to be given the length of the other horizontal. Then it would be easy. - 2 years ago Yes, it not possible to get the value of X. link - 2 years ago BTW, is it possible? - 2 years ago Yes it is possible to calculate but check whether all the information is required or not . I think some information is missing . - 2 years ago I don't really know, but it only gave me the radius.I'll check again later - 2 years ago If we know the length of the chord we can easily find the answer. - 2 years ago 2nd cord? - 2 years ago Yes the chord above the diameter - 2 years ago It says the second cord is the bottom of an imaginary square and the surface area of two connected in form of up and down cube created with the imaginary square is 50. - 2 years ago	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 8, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9696703553199768, "perplexity": 3145.1182523954553}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439738609.73/warc/CC-MAIN-20200810042140-20200810072140-00237.warc.gz"}
https://www.physicsforums.com/threads/helium-baloon-question.185140/	# Homework Help: Helium Baloon question 1. Sep 17, 2007 ### Kleptoma 1. The problem statement, all variables and given/known data A 1.10 g balloon is filled with helium gas until it becomes a 20.0 cm-diameter sphere. What maximum mass can be tied to the balloon (with a massless string) without the balloon sinking to the floor? Density of Air: 1.28 kg/m^3 Density of Helium gas : 0.18 kg/m^3 Volume of a sphere: V = 4/3pir^3 g = 9.80 m/s^2 2. Relevant equations Buoyant Force (Fb) = p (density) * V * g Density = mass/volume Fb = Wo = p (density of object) * V (Volume of object) * g 3. The attempt at a solution Fb = Weight of the object + Mg (M is the mass to be solved) (1.23)(4/3pi(0.1m)^3)(9.8) = (0.18)(4/3pi(0.1m)^3)(9.8) + M(9.8) M = 4.40 * 10^-3kg I am uncertain of whether the buoyant force is the force of air pushing upwards on the balloon, or whether it is the helium gas in the balloon. Nonetheless, I used air as the density to solve for the buoyant force. Another possibility would be that: T = Fb - mg T = (1.23)(4/3pi(0.1m)^3)(9.8) - (0.0011kg)(9.8) T = 0.0397 N Therefore mass of the object = 0.0397N / 9.8 = 4.05 * 10^-3 I am doubtful of which solution is correct and if either solution is actually correct. Last edited: Sep 17, 2007 2. Sep 17, 2007 ### Kurdt Staff Emeritus The weight that the balloon can hold will be the buoyant force, minus the weight of the balloon. The buoyant force is equal to the weight of the displaced air. 3. Sep 17, 2007 ### Kleptoma Thanks Kurdt. I know that the buoyant force is equal to the weight of the displaced air, which is represented by (1.23)(4/3pi(0.1m)^3)(9.8). But I'm not sure how to calculate the weight of the balloon. Do I use the density of the helium gas and multiply it by Volume and gravity (Weight = DVg) or simply use Weight = mg 4. Sep 17, 2007 ### Kurdt Staff Emeritus You'll need to use the density multiplied by the volume, and of course add it to the weight of the balloon.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9336279034614563, "perplexity": 1329.8532253471235}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267867493.99/warc/CC-MAIN-20180625053151-20180625073151-00521.warc.gz"}
https://answers.yahoo.com/question/index?qid=20200119160117AA96qXb	# the voltage? i dont get why the 6 ohm is (+) though it is written that vo=-6i Relevance • 4 weeks ago i is defined as clockwise. That means it enters the 6Ω on the right side. The convention I was taught is that positive current enters the + side of a resistance. In this problem the equations are written using the opposite convention for KVL Thus the left side of the 6Ω is the negative side of voltage = 6i. But that is the opposite of the defined Vo. Thus Vo = -6i. They did this to trip you up obviously. They could have defined Vo as being + on the right side to match the assumed current so that Vo = 6i but they didn't to complicate the problem.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9762142300605774, "perplexity": 596.0983330202312}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875142603.80/warc/CC-MAIN-20200217145609-20200217175609-00123.warc.gz"}
http://math.stackexchange.com/questions/253546/expected-value-linearity	# Expected value linearity From the book A First Course in Probability 8th ed by Ross, expected value linearity is defined as $E[aX + b] = aE[X] + b$ where a and b are both constants and X is a random variable. However a random process defined $X(t) = Kcos(wt)$ where K is a random variable has an expected value of $E[X(t)] = E[K]cos(wt)$. My question is how is this true when cosine is a function of t and not a constant. Thanks. - @AndréNicolas. Even if the distribution of $K$ involved $t$ as a parameter, you would still be able to get $\cos(w t)$ outside of the expectation. – Learner Dec 8 '12 at 4:17 Yes, thank you, it was very imprecisely phrased. – André Nicolas Dec 8 '12 at 4:38 At this stage of your study, heuristically, you could consider constants everything that is not random. In the question, $X(t)=K \cos(w t)$ is a random function of $t$ only because $K$ is random. When you take the expectation, you could ignore everything that is not random. Were either $w$ or $t$ or both random, you would not be able to get $\cos( w t)$ outside of the expectation. I am assuming this is the same case for variance? $Var(X(t)) = cos(wt)^2Var(K)$ ? – wi1 Dec 8 '12 at 6:14 Yes. Remember that $Var(X)=E[X^2]-E[X]^2$. So the rules that apply to the expectation follow for the variance from that formula. – Learner Dec 8 '12 at 6:22	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9564734697341919, "perplexity": 229.74603956587694}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-41/segments/1412037663460.43/warc/CC-MAIN-20140930004103-00410-ip-10-234-18-248.ec2.internal.warc.gz"}
https://lttt.vanabel.cn/category/mathnotes	## Poincare Conjecture and Elliptization Conjecture 1. Poincare Conjecture Theorem 1 (Poincare Conjecture). If a compact three-dimensional manifold $M^3$ has the property that every simple closed curve within the manifold can be deformed continuously to a point, does it follow that $M^3$ is homeomorphic to the sphere $S^3$? 2. Thurston Elliptization Conjecture Theorem 2 (Thurston Elliptization Conjecture). Every closed 3-manifold with finite fundamental group has a metric of constant positive curvature and hence is homeomorphic to a quotient $S^3/\Gamma$, where $\mathrm{\Gamma} \subset \mathrm{SO(4)}$ is a finite group of rotations that acts freely on $S^3$. The Poincare Conjecture corresponds to the special case where the group $\Gamma \cong \pi_1(M^3)$ is trivial. ## Principle Bundle, Associated bundle, Gauge Group and Connections Let $M$ be compact riemannian manifold without boundary, and $G$ be a compact Lie group. A principle $G$-bundle over $M$, denoted as $P(M,G)$ is a manifold $P$ with a free right action $P\times G\ni(p,g)\mapsto pg\in P$ of $G$ such that $M=P/G$, and $P$ is locally trivial, i.e., for every point $x\in M$, there is a neighbourhood $U$ such that the primage $\pi^{-1}(U)$ of the canonical projection is isomorphic to $U\times G$ in the sense that it preserver the fiber and $G$-equivariant, more precisely, there is a diffeomorphism $\Phi:\pi^{-1}(U)\to U\times G$ such that $\Phi(p)=(\pi(p),\phi(p))$ and $\phi:\pi^{-1}(U)\to G$ satisfying $\phi(pg)=(\phi(p))g$ for all $p\in\pi^{-1}(U)$ and $g\in G$. Now for any other manifold $F$ on which $G$ acts on the left $G\times F\ni(g,f)\mapsto gf\in F$, the associated fiber bundle $P\times_\rho F$ is the quotient space $P\times F/\sim$, where $[p,f]\sim[pg,g^{-1}f]$, for all $g\in G$. With the projection $\tilde\pi:[p,g]\mapsto\pi(p)$ this is a principle bundle over $M$ with typical fiber $F$. The local trivialization is induced by the one $\Phi:\pi^{-1}(U)\to U\times G$ as Continue Reading ## [转载]What’s a Gauge? From: Terence Tao’s blog: What’s a Gauge. Gauge theory” is a term which has connotations of being a fearsomely complicated part of mathematics – for instance, playing an important role in quantum field theory, general relativity, geometric PDE, and so forth. But the underlying concept is really quite simple: a gauge is nothing more than a “coordinate system” that varies depending on one’s “location” with respect to some “base space” or “parameter space”, a gauge transform is a change of coordinates applied to each such location, and a gauge theory is a model for some physical or mathematical system to which gauge transforms can be applied (and is typically gauge invariant, in that all physically meaningful quantities are left unchanged (or transform naturally) under gauge transformations). By fixing a gauge (thus breaking or spending the gauge symmetry), the model becomes something easier to analyse mathematically, such as a system of partial differential equations (in classical gauge theories) or a perturbative quantum field theory (in quantum gauge theories), though the tractability of the resulting problem can be heavily dependent on the choice of gauge that one fixed. Deciding exactly how to fix a gauge (or whether one should spend the gauge symmetry at all) is a key question in the analysis of gauge theories, and one that often requires the input of geometric ideas and intuition into that analysis. Continue Reading ## 标准的椭圆理论:一个能量不等式 Proposition 1. 假设$u$是方程 $$0=\Delta u-\frac{1}{2}x\cdot \nabla u.$$ $$\int_{\|x\|< r}e^{-\frac{\|x\|^2}{4}}\|\nabla u\|^2\rd x\leq\frac{c}{r^2}\int_{r< \|x\|< 2r}e^{-\frac{\|x\|^2}{4}}u^2\rd x,\quad\forall r >0.$$	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9827127456665039, "perplexity": 204.3103435270857}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560627999291.1/warc/CC-MAIN-20190620230326-20190621012326-00163.warc.gz"}
http://physics.stackexchange.com/questions/33881/applying-uncertainty-principle-and-the-difference-in-delta-x	# Applying uncertainty principle and the difference in $\Delta x$ These two questions seem to be very similar, but the textbook uses a bit different methods for calculating $\Delta x$ of uncertainty principle. Question A) Suppose that there is a room with the same length of its side (x-axis, y-axis, z-axis). A ball is 100g. In this case, the solution says that I should put $\Delta x$ as 15m. Question B) Find the kinetic energy of an electron bound by nucleus using uncertainty principle. Assume that the nucleus has the side of $1.0 \times 10^{-14} m$. In this case, however, the solution says that I should put $\Delta x$ as the half of $1.0 \times 10^{-14}m$. What is the difference between these two cases? (I know that the second case is only approximation.) - For a particle in a cubic box the solution is fairly simple because the wavefunction separates into functions of $x$, $y$ and $z$, and you can calculate $\Delta x$ exactly. The Wikipedia article on the particle in a box gives: $$\Delta x^2 = \frac{L^2}{12} \left( 1 - \frac{6}{n^2\pi^2} \right)$$ though I get $\Delta x \approx 18m$ not $15m$. The calculation for a spherical box is a lot harder, and in any case a spherical box is only an approximation to reality. This is presumably why you've been told to just assume $\Delta x$ is half the size of the nucleus rather than trying to calculate it.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9648720622062683, "perplexity": 145.83008076642813}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-22/segments/1432207929956.54/warc/CC-MAIN-20150521113209-00334-ip-10-180-206-219.ec2.internal.warc.gz"}
https://www.ideals.illinois.edu/handle/2142/69516	## Files in this item FilesDescriptionFormat application/pdf 8324594.pdf (3MB) (no description provided)PDF ## Description Title: Ss/tdma Time Slot Assignment With Generalized Switching Modes Author(s): Lewandowski, James Louis Department / Program: Computer Science Discipline: Computer Science Degree Granting Institution: University of Illinois at Urbana-Champaign Degree: Ph.D. Genre: Dissertation Subject(s): Computer Science Abstract: A number of problems which arise from the time slot assignment problem for satellite communications using the satellite-switched time division multiple access (SS/TDMA) method are studied. Given an m x n real matrix T, and a set of (0, 1)-matrices {Z(,1), Z(,2), (.)(.)(.) , Z(,(nu))}, the optimization problem we wish to solve is to find non-negative real constants (gamma)(,1), (gamma)(,2), ... , (gamma)(,(nu)) so that(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI)is minimized, and where the inequality is implied for each corresponding entry. Although this problem can be posed as a linear program, more efficient algorithms are presented here for several special cases. We first consider the case where each Z(,i) has a specified number of ones in each row and column. For two vectors (rho) = {(rho)(,1), (rho)(,2), (.)(.)(.) , (rho)(,m)} and (lamda) = {(lamda)(,1), (lamda)(,2), (.)(.)(.) , (lamda)(,n)} of positive integers where(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI)we define the set U((rho), (lamda)) to be the set of all (0, 1)-matrices which have exactly (rho)(,i) ones in the i('th) row for all i, and exactly (lamda)(,j) ones in the j('th) column for all j. A complete characterization of the class of real matrices which can be written as a positive sum of matrices in U((rho), (lamda)) is given. This result provides a generalization of the Birkhoff-Von Neumann Theorem. Also for a given T, an algorithm is presented which will solve, if possible, the above optimization problem for this set of (0, 1)-matrices. The optimization problem does not always have a feasible solution, and a complete characterization of the conditions needed for the existence of one is given. In the above expression, using these (0, 1)-matrices, (nu) can be as large as O(n('2)). In SS/TDMA applications, it is desirable to have a small value for (nu). As a second problem we consider a fixed set Z = {Z(,1), Z(,2), (.)(.)(.) , Z(,l)} for (0, 1)-matrices, where l is proportional to O(n). Given such a set of matrices, which satisfy certain constraints, we have an efficient algorithm which solves the above optimization problem for these matrices. Issue Date: 1983 Type: Text Description: 120 p.Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1983. URI: http://hdl.handle.net/2142/69516 Other Identifier(s): (UMI)AAI8324594 Date Available in IDEALS: 2014-12-15 Date Deposited: 1983	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9013755321502686, "perplexity": 1916.574058667042}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547583705091.62/warc/CC-MAIN-20190120082608-20190120104608-00185.warc.gz"}
https://figshare.com/articles/Systematic_Structural_Coordination_Chemistry_of_i_p_i_-_i_tert_i_-Butyltetrathiacalix_4_arene_1_Group_1_Elements_and_Congeners/3601098	## Systematic Structural Coordination Chemistry of p-tert-Butyltetrathiacalix[4]arene: 1. Group 1 Elements and Congeners 2001-01-13T00:00:00Z (GMT) by Determinations of the crystal structures of complexes of the alkali metal ions with, in the case of Li, the dianion and, in the cases Na−Cs, the monoanion of p-tert-butyltetrathiacalix[4]arene have shown that both the sulfur atoms which form part of the macrocyclic ring, as well as the pendent phenolic/phenoxide oxygen donor atoms, are involved in coordination to these metals. Although the Li and Na complex structures are similar to those of the corresponding complexes of p-tert-butylcalix[4]arene, there is no similarity in the structures of the Cs complexes, with the present structure showing no evidence of polyhapto Cs+−π interactions. Instead, the complex crystallizes as a ligand-bridged (S-, O-donor) aggregate of three Cs ions, solvent molecules, and four calixarenes, somewhat like the Rb complex, though here four Rb ions are present, and higher in aggregation than the K+ complex, where two K+ ions are sandwiched between two calixarene moieties. The triethylammonium complex of the thiacalixarene monoanion, though formally analogous in that it involves a monocation, has a simpler structure than any of the alkali metal derivatives, based formally on proton coordination (H-bonding). However, interestingly, it can be isolated in both solvated (dmf, dmso) and unsolvated forms, as indeed can the “free”, p-tert-butyltetrathiacalix[4]arene ligand itself.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9191970229148865, "perplexity": 4499.4861663398615}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579251802249.87/warc/CC-MAIN-20200129194333-20200129223333-00034.warc.gz"}
https://planetmath.org/transcendentalroottheorem	# transcendental root theorem Suppose a constant $x$ is transcendental over some field $F$. Then $\sqrt[n]{x}$ is also transcendental over $F$ for any $n\geq 1$. ###### Proof. Let $\overline{F}$ denote an algebraic closure of $F$. Assume for the sake of contradiction that $\sqrt[n]{x}\in\overline{F}$. Then since algebraic numbers are closed under multiplication (and thus exponentiation by positive integers), we have $(\sqrt[n]{x})^{n}=x\in\overline{F}$, so that $x$ is algebraic over $F$, creating a contradiction. ∎ Title transcendental root theorem TranscendentalRootTheorem 2013-03-22 14:04:23 2013-03-22 14:04:23 mathcam (2727) mathcam (2727) 8 mathcam (2727) Theorem msc 11R04	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 11, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9982970356941223, "perplexity": 590.4761445845774}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178361723.15/warc/CC-MAIN-20210228175250-20210228205250-00219.warc.gz"}
https://dev.thep.lu.se/yat/changeset/1153/trunk/doc	# Changeset 1153 for trunk/doc Ignore: Timestamp: Feb 26, 2008, 3:01:02 AM (13 years ago) Message: Removing text not being true anymore File: 1 edited ### Legend: Unmodified r1125 and calculating the weighted median of the distances. \section Distance \section Kernel \subsection polynomial_kernel Polynomial Kernel The polynomial kernel of degree \f$N\f$ is defined as \f$(1+)^N\f$, where \f$\f$ is the linear kernel (usual scalar product). For the weighted case we define the linear kernel to be \f$=\sum {w_xw_yxy}\f$ and the case we define the linear kernel to be \f$=\frac{\sum {w_xw_yxy}}{\sum{w_xw_y}\f$ and the polynomial kernel can be calculated as before \f$(1+)^N\f$. Is this kernel a proper kernel (always being semi positive definite). Yes, because \f$\f$ is obviously a proper kernel as it is a scalar product. Adding a positive constant to a kernel yields another kernel so \f$1+\f$ is still a proper kernel. Then also \f$(1+)^N\f$ is a proper kernel because taking a proper kernel to the \f$Nth\f$ power yields a new proper kernel (see any good book on SVM). \f$(1+)^N\f$. \subsection gaussian_kernel Gaussian Kernel We define the weighted Gaussian kernel as \f$\exp\left(-\frac{\sum w_xw_y(x-y)^2}{\sum w_xw_y}\right)\f$, which fulfills the conditions listed in the introduction. Is this kernel a proper kernel? Yes, following the proof of the non-weighted kernel we see that \f$K=\exp\left(-\frac{\sum w_xw_yx^2}{\sum w_xw_y}\right)\exp\left(-\frac{\sum w_xw_yy^2}{\sum w_xw_y}\right)\exp\left(\frac{\sum w_xw_yxy}{\sum w_xw_y}\right)\f$, which is a product of two proper kernels. \f$\exp\left(-\frac{\sum w_xw_yx^2}{\sum w_xw_y}\right)\exp\left(-\frac{\sum w_xw_yy^2}{\sum w_xw_y}\right)\f$ is a proper kernel, because it is a scalar product and \f$\exp\left(\frac{\sum w_xw_yxy}{\sum w_xw_y}\right)\f$ is a proper kernel, because it a polynomial of the linear kernel with positive coefficients. As product of two kernel also is a kernel, the Gaussian kernel is a proper kernel. \section Distance We define the weighted Gaussian kernel as \f$\exp\left(-N\frac{\sum w_xw_y(x-y)^2}{\sum w_xw_y}\right)\f$. \section Regression	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9832596182823181, "perplexity": 1627.9631450611585}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046151531.67/warc/CC-MAIN-20210724223025-20210725013025-00570.warc.gz"}
https://astarmathsandphysics.com/university-maths-notes/matrices-and-linear-algebra/3899-rank-and-nullity-of-a-matrix.html?tmpl=component&print=1&page=	## Rank and Nullity of a Matrix The Rank of a matrix is the number of non zero rows in the row reduced form of the matrx and the nullity is the number of zero rows in the row reduced form Example: $\mathbf{M} = \left( \begin{array}{ccc} 1 & 2 & 1 \\ 1 & -1 & 0 \\ -1 & -2 & -1 \\ 2 & 1 & 1 \end{array} \right)$ $\left( \begin{array}{ccc} 1 & 2 & 1 \\ 0 & -3 & -1 \\ 0 & 0 & 0 \\ 0 & 3 & -1 \end{array} \right) \begin{array}{c} \\ R2-R1 \\ R3+R1 \\R4-2*R1 \end{array}$ $\left( \begin{array}{ccc} 1 & 2 & 1 \\ 0 & -3 & -1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array} \right) \begin{array}{c} \\ \\ \\R4-R2 \end{array}$ There are two non zero rows so the rank of the matrix is 2 and there are two zero rows so the nullity is 2.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8786711692810059, "perplexity": 67.18095746342695}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187823153.58/warc/CC-MAIN-20171018214541-20171018234541-00283.warc.gz"}
http://link.springer.com/article/10.1007%2Fs00382-013-2009-0	Date: 28 Dec 2013 # Impact of bio-physical feedbacks on the tropical climate in coupled and uncoupled GCMs Rent the article at a discount Rent now * Final gross prices may vary according to local VAT. ## Abstract The bio-physical feedback process between the marine ecosystem and the tropical climate system is investigated using both an ocean circulation model and a fully-coupled ocean–atmosphere circulation model, which interact with a biogeochemical model. We found that the presence of chlorophyll can have significant impact on the characteristics of the El Niño-Southern Oscillation (ENSO), including its amplitude and asymmetry, as well as on the mean state. That is, chlorophyll generally increases mean sea surface temperature (SST) due to the direct biological heating. However, SST in the eastern equatorial Pacific decreases due to the stronger indirect dynamical response to the biological effects outweighing the direct thermal response. It is demonstrated that this biologically-induced SST cooling is intensified and conveyed to other tropical-ocean basins when atmosphere–ocean coupling is taken into account. It is also found that the presence of chlorophyll affects the magnitude of ENSO by two different mechanisms; one is an amplifying effect by the mean chlorophyll, which is associated with shoaling of the mean thermocline depth, and the other is a damping effect derived from the interactively-varying chlorophyll coupled with the physical model. The atmosphere–ocean coupling reduces the biologically-induced ENSO amplifying effect through the weakening of atmospheric feedback. Lastly, there is also a biological impact on ENSO which enhances the positive skewness. This skewness change is presumably caused by the phase dependency of thermocline feedback which affects the ENSO magnitude.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8892713189125061, "perplexity": 2632.8383017556716}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-41/segments/1410657124356.76/warc/CC-MAIN-20140914011204-00082-ip-10-196-40-205.us-west-1.compute.internal.warc.gz"}
http://math.stackexchange.com/questions/174628/every-rational-function-which-is-holomorphic-on-riemann-sphere-mathbbc-inf	# Every rational function which is holomorphic on Riemann Sphere($\mathbb{C}_{\infty}$) could any one give me a hint how to show Every rational function which is holomorphic on every point of Riemann Sphere( $\mathbb{C}_{\infty}$) must be constant?(with out applying Maximum Modulas Theorem). Thank you. - Does using the fact that the Riemann sphere is compact count as using the maximum modulus principle? – Zhen Lin Jul 24 '12 at 11:55 ## 1 Answer If a function is analytic on the sphere at $\infty$, it is bounded in an neighborhood of $\infty$. Consequently, it is bounded globally, since the complement of a neighbhorhood at $\infty$ is compact. Now invoke Liouville's theorem; the functon must be constant. - why it is bounded in an neighborhood of $\infty$? – La Belle Noiseuse Jul 24 '12 at 13:42 If it is analytic at $\infty$, it is continuous there and it therefore bounded there. Note that a polynomial has a pole at $\infty$ on the Riemann sphere. – ncmathsadist Jul 24 '12 at 14:38	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9958040714263916, "perplexity": 278.21514020402765}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-32/segments/1438042986615.83/warc/CC-MAIN-20150728002306-00276-ip-10-236-191-2.ec2.internal.warc.gz"}
http://mathhelpforum.com/calculus/166363-example-absolutely-continuity.html	# Math Help - example of absolutely continuity 1. ## example of absolutely continuity I'm studying Real analysis, It's known Lipschitz continuity implies absolutely continuity. and Cantor function is an example for uniformly continuous function but not a absolutely continuous function, i wonder if in between case, if f(x) is a holder continuous which is uniformly continuous but not lipschistz continuous, what's the result? In any reference, the example of absolutely continuous function was Lipshitz but is it true for holder continuous function? 2. Something that is holder but not lipschistz: f(x) = sqrt(x) around x = 0	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9200887680053711, "perplexity": 1472.4387730754752}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1408500823169.67/warc/CC-MAIN-20140820021343-00361-ip-10-180-136-8.ec2.internal.warc.gz"}
https://www.physicsforums.com/threads/box-sliding-up-a-ramp.155134/	# Box Sliding Up a Ramp 1. Feb 7, 2007 ### robbondo 1. The problem statement, all variables and given/known data You are working for a shipping company. Your job is to stand at the bottom of a 8.0-m-long ramp that is inclined at 37 degrees above the horizontal. You grab packages off a conveyor belt and propel them up the ramp. The coefficient of kinetic friction between the packages and the ramp is .30 What speed do you need to give a package at the bottom of the ramp so that it has zero speed at the top of the ramp? 2. Relevant equations $$x = x_0 + v_0 t + (1/2) a t^2$$ $$v = v_0 + a t$$ $$v^2 = v_0^2 + 2 a \Delta x$$ $$\vec{F}_{net} = \Sigma \vec{F} = m \vec{a}$$ 3. The attempt at a solution Well first I drew a digram which gave me a component of $$nsin37$$ as the force acting down the ramp and then of course the nu force of friction. SO, then I wrote out that as $$\vec{F}_{net} = \Sigma \vec{F} = m \vec{a} = -mgsin37 - .30mg + F_t$$ With $$F_t$$ being the force of the throw Then I know that since I'm solving for the initial velocity that should be related to the acceleration and the force through $$v = v_0 + a t$$ but I'm not sure how. I haven't been given the value for the mass, so I'm not sure if I can even solve this problem only using the given variables. Last edited: Feb 7, 2007 2. Feb 7, 2007 Ok, and where's the problem in the text? 3. Feb 7, 2007 ### robbondo oops... fixed it. 4. Feb 7, 2007 ### denverdoc Am I missing something, just below where it is lopped off in the quote. was there an edit? Rob, anyway its a bit like the last one, the masses all divide out.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9267215728759766, "perplexity": 463.6548683587836}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-09/segments/1487501171781.5/warc/CC-MAIN-20170219104611-00199-ip-10-171-10-108.ec2.internal.warc.gz"}
https://math.stackexchange.com/tags/partial-derivative/info	Tag Info For questions regarding partial derivatives. The partial derivative of a function of several variables is the derivative of the function with respect to one of those variables, with all others held constant. The partial derivative of a function of several variables is the derivative of the function with respect to one of the variables, with the others held constant. The partial derivative, like the ordinary derivative, describes the rate of change of a function in a particular direction. If $f = f(a_1, a_2, \dots, a_n)$ is a function of $n$ variables, then the partial derivative of $f$ with respect to the variable $a_i$ can be written as a limit: $$\frac{\partial f}{\partial a_i} = \lim_{h \to 0} \frac{f(a_1, \dots, a_i + h, \dots, a_n) - f(a_1, \dots, a_i, \dots, a_n)}{h}.$$ Alternatively, this quantity can be denoted as $f_{a_i}$. If the function has continuous partial second derivatives, then: $$\frac{\partial}{\partial x_i}\left(\frac{\partial}{\partial x_j}\right)=\frac{\partial}{\partial x_j}\left(\frac{\partial}{\partial x_i}\right)$$ a result known as Schwarz's Theorem.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.924949586391449, "perplexity": 67.00002857007412}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560627998724.57/warc/CC-MAIN-20190618123355-20190618145355-00555.warc.gz"}
http://economicsworlds.blogspot.com/2010/02/concept-of-production-possibility-curve.html	## Tuesday, February 9, 2010 ### Concept of production possibility curve Human wants unlimited,where the means to satisfy those wants are limited.This creates the fundamental economics problem of choosing and allocating scarce resources among competing uses.The production possibility curve of frontier is a tool used to illustrate and explain the problem of scarcity and choice.This problem of choice was illuminated by Paul.A. Samuelson for the first time in terms of production possibility curve (PPC). In the words of David Begg and other -"The production possibility curve (PPC) shows the maximum combination of output that the economy can produce using all available resources. Similarly,according to Richard G.Lipsey and Colin Harbury,"The production possibility curve is a graphic expression of all the combination of goods and service that ca be produced when all resources are fully and efficiently employed. Assumption The production possibility curve is based on following assumption : 1. Two goods :Only two goods (food and shoe) are produced in the economy. 2. Full employment :There is full employment of resources. 3. Supply of factor :The supply of factor is fixed,but can be reallocated in the production of two goods within limit. 4. Production technique :The production technique is given and constant in the low of diminishing return operates. 5. Time period :The time period is short.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9210664629936218, "perplexity": 4253.198981633281}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1398223203841.5/warc/CC-MAIN-20140423032003-00570-ip-10-147-4-33.ec2.internal.warc.gz"}
https://www.physicsforums.com/threads/acceleration-vectors-quick-question.110527/	# Acceleration vectors (quick question) 1. Feb 13, 2006 ### nick727kcin this is the last thing that i dont really get. did i draw these acceleration vectors correctly? thanks 2. Feb 13, 2006 ### Staff: Mentor It depends on the details of the motion, but it looks like you've got the idea. If the car moves at a constant speed around the curve, the only component of acceleration will be perpendicular to the motion (also called "radial" or "centripetal"). But if, as you are showing, the car's speed is increasing, then there will be a parallel (or tangential) component of acceleration as well. 3. Feb 13, 2006 ### Staff: Mentor I almost forgot: One thing that's not accurate in your diagram is the length of the total acceleration vector. The total acceleration is the vector sum of the two components, so it's got to be much longer than what your diagram shows. (Looks to me like the total vector is smaller than the perpendicular component. ) 4. Feb 13, 2006 ### marlon YES. You need to remember this : Velocity is a vector. One can change a vector in TWO ways : 1) one can change the magnitude of the vector 2) one can change the direction of the vector The tangential component does (1) (ie the velocity gets bigger or smaller) The centripetal component does (2) (ie the actual circular orbit) marlon Similar Discussions: Acceleration vectors (quick question)	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9560081362724304, "perplexity": 862.8486902873963}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-13/segments/1490218189589.37/warc/CC-MAIN-20170322212949-00603-ip-10-233-31-227.ec2.internal.warc.gz"}
https://worldwidescience.org/topicpages/n/non-newtonian+fluid+layers.html	#### Sample records for non-newtonian fluid layers 1. Laminar boundary-layer flow of non-Newtonian fluid Science.gov (United States) Lin, F. N.; Chern, S. Y. 1979-01-01 A solution for the two-dimensional and axisymmetric laminar boundary-layer momentum equation of power-law non-Newtonian fluid is presented. The analysis makes use of the Merk-Chao series solution method originally devised for the flow of Newtonian fluid. The universal functions for the leading term in the series are tabulated for n from 0.2 to 2. Equations governing the universal functions associated with the second and the third terms are provided. The solution together with either Lighthill's formula or Chao's formula constitutes a simple yet general procedure for the calculation of wall shear and surface heat transfer rate. The theory was applied to flows over a circular cylinder and a sphere and the results compared with published data. 2. Flow non-normality-induced transient growth in superposed Newtonian and non-Newtonian fluid layers OpenAIRE Ridolfi, Luca; Camporeale, Carlo Vincenzo 2009-01-01 In recent years non-normality and transient growths have attracted much interest in fluid mechanics. Here, we investigate these topics with reference to the problem of interfacial instability in superposed Newtonian and non-Newtonian fluid layers. Under the hypothesis of the lubrication theory, we demonstrate the existence of significant transient growths in the parameter space region where the dynamical system is asymptotically stable, and show how they depend on the main physical parameters... 3. Flow non-normality-induced transient growth in superposed Newtonian and non-Newtonian fluid layers. Science.gov (United States) Camporeale, C; Gatti, F; Ridolfi, L 2009-09-01 In recent years non-normality and transient growths have attracted much interest in fluid mechanics. Here, we investigate these topics with reference to the problem of interfacial instability in superposed Newtonian and non-Newtonian fluid layers. Under the hypothesis of the lubrication theory, we demonstrate the existence of significant transient growths in the parameter space region where the dynamical system is asymptotically stable, and show how they depend on the main physical parameters. In particular, the key role of the density ratio is highlighted. 4. Boundary layer flow on a moving surface in otherwise quiescent pseudo-plastic non-Newtonian fluids Institute of Scientific and Technical Information of China (English) Liancun Zheng; Liu Ting; Xinxin Zhang 2008-01-01 A theoretical analysis for the boundary layer flow over a continuous moving surface in an otherwise quiescent pseudo-plastic non-Newtonian fluid medium was presented. The types of potential flows necessary for similar solutions to the boundary layer equations were determined and the solutions were numerically presented for different values of power law exponent. 5. Bacterial gliding fluid dynamics on a layer of non-Newtonian slime: Perturbation and numerical study. Science.gov (United States) Ali, N; Asghar, Z; Anwar Bég, O; Sajid, M 2016-05-21 Gliding bacteria are an assorted group of rod-shaped prokaryotes that adhere to and glide on certain layers of ooze slime attached to a substratum. Due to the absence of organelles of motility, such as flagella, the gliding motion is caused by the waves moving down the outer surface of these rod-shaped cells. In the present study we employ an undulating surface model to investigate the motility of bacteria on a layer of non-Newtonian slime. The rheological behavior of the slime is characterized by an appropriate constitutive equation, namely the Carreau model. Employing the balances of mass and momentum conservation, the hydrodynamic undulating surface model is transformed into a fourth-order nonlinear differential equation in terms of a stream function under the long wavelength assumption. A perturbation approach is adopted to obtain closed form expressions for stream function, pressure rise per wavelength, forces generated by the organism and power required for propulsion. A numerical technique based on an implicit finite difference scheme is also employed to investigate various features of the model for large values of the rheological parameters of the slime. Verification of the numerical solutions is achieved with a variational finite element method (FEM). The computations demonstrate that the speed of the glider decreases as the rheology of the slime changes from shear-thinning (pseudo-plastic) to shear-thickening (dilatant). Moreover, the viscoelastic nature of the slime tends to increase the swimming speed for the shear-thinning case. The fluid flow in the pumping (generated where the organism is not free to move but instead generates a net fluid flow beneath it) is also investigated in detail. The study is relevant to marine anti-bacterial fouling and medical hygiene biophysics. 6. Rheology and non-Newtonian fluids CERN Document Server Irgens, Fridtjov 2014-01-01 This book gives a brief but thorough introduction to the fascinating subject of non-Newtonian fluids, their behavior and mechanical properties. After a brief introduction of what characterizes non-Newtonian fluids in Chapter 1 some phenomena characteristic of non-Newtonian fluids are presented in Chapter 2. The basic equations in fluid mechanics are discussed in Chapter 3. Deformation kinematics, the kinematics of shear flows, viscometric flows, and extensional flows are the topics in Chapter 4. Material functions characterizing the behavior of fluids in special flows are defined in Chapter 5. Generalized Newtonian fluids are the most common types of non-Newtonian fluids and are the subject in Chapter 6. Some linearly viscoelastic fluid models are presented in Chapter 7. In Chapter 8 the concept of tensors is utilized and advanced fluid models are introduced. The book is concluded with a variety of 26 problems. Solutions to the problems are ready for instructors. 7. Coupling electrokinetics and rheology: Electrophoresis in non-Newtonian fluids. Science.gov (United States) Khair, Aditya S; Posluszny, Denise E; Walker, Lynn M 2012-01-01 We present a theoretical scheme to calculate the electrophoretic motion of charged colloidal particles immersed in complex (non-Newtonian) fluids possessing shear-rate-dependent viscosities. We demonstrate that this non-Newtonian rheology leads to an explicit shape and size dependence of the electrophoretic velocity of a uniformly charged particle in the thin-Debye-layer regime, in contrast to electrophoresis in Newtonian fluids. This dependence is caused by non-Newtonian stresses in the bulk (electroneutral) fluid outside the Debye layer, whose magnitude is naturally characterized in an electrophoretic Deborah number. 8. PKN problem for non-Newtonian fluid OpenAIRE 2012-01-01 The paper presents analytical solution for hydraulic fracture driven by a non-Newtonian fluid and propagating under plane strain conditions in cross sections parallel to the fracture front. Conclusions are drawn on the influence of the fluid properties on the fracture propagation. 9. Electroosmotic mobilities of non-Newtonian fluids CERN Document Server Zhao, Cunlu 2010-01-01 Owing to frequent processing of biofluids in Lab-on-a-chip microfluidic devices, electroosmotic mobilities of non-Newtonian fluids are investigated numerically. The general Cauchy momentum equation governing the electroosmotic velocity is simplified by incorporation of the Gouy-Chapman solution of the Poisson-Boltzmann equation and the Carreau fluid constitutive model. Then the finite element method for solving the simplified version of Cauchy momentum equation is validated through comparisons with two exact solutions, i.e., Newtonian fluids and power-law fluids. Analyses shows that different from Newtonian fluids with a constant dimensionless electroosmotic mobility of unit one, dimensionless electroosmotic mobilities for non-Newtonian Carreau fluids are dependent on four dimensionless groups, such as dimensionless surface zeta potential , Weissenberg number Wi, fluid power-law exponent n and transitional parameter {\\beta}. It is found out that with increasing and decreasing of n and {\\beta}, electroosmotic ... 10. On energy boundary layer equations in power law non-Newtonian fluids Institute of Scientific and Technical Information of China (English) 郑连存; 张欣欣 2008-01-01 The hear transfer mechanism and the constitutive models for energy boundary layer in power law fluids were investigated.Two energy transfer constitutive equations models were proposed based on the assumption of similarity of velocity field momentum diffusion and temperature field heat transfer.The governing systems of partial different equations were transformed into ordinary differential equations respectively by using the similarity transformation group.One model was assumed that Prandtl number is a constant,and the other model was assumed that viscosity diffusion is analogous to thermal diffusion.The solutions were presented analytically and numerically by using the Runge-Kutta formulas and shooting technique and the associated transfer characteristics were discussed. 11. Electrokinetics of non-Newtonian fluids: a review. Science.gov (United States) Zhao, Cunlu; Yang, Chun 2013-12-01 This work presents a comprehensive review of electrokinetics pertaining to non-Newtonian fluids. The topic covers a broad range of non-Newtonian effects in electrokinetics, including electroosmosis of non-Newtonian fluids, electrophoresis of particles in non-Newtonian fluids, streaming potential effect of non-Newtonian fluids and other related non-Newtonian effects in electrokinetics. Generally, the coupling between non-Newtonian hydrodynamics and electrostatics not only complicates the electrokinetics but also causes the fluid/particle velocity to be nonlinearly dependent on the strength of external electric field and/or the zeta potential. Shear-thinning nature of liquids tends to enhance electrokinetic phenomena, while shear-thickening nature of liquids leads to the reduction of electrokinetic effects. In addition, directions for the future studies are suggested and several theoretical issues in non-Newtonian electrokinetics are highlighted. 12. Non-Newtonian Properties of Relativistic Fluids CERN Document Server Koide, Tomoi 2010-01-01 We show that relativistic fluids behave as non-Newtonian fluids. First, we discuss the problem of acausal propagation in the diffusion equation and introduce the modified Maxwell-Cattaneo-Vernotte (MCV) equation. By using the modified MCV equation, we obtain the causal dissipative relativistic (CDR) fluid dynamics, where unphysical propagation with infinite velocity does not exist. We further show that the problems of the violation of causality and instability are intimately related, and the relativistic Navier-Stokes equation is inadequate as the theory of relativistic fluids. Finally, the new microscopic formula to calculate the transport coefficients of the CDR fluid dynamics is discussed. The result of the microscopic formula is consistent with that of the Boltzmann equation, i.e., Grad's moment method. 13. A Non-Newtonian Fluid Robot. Science.gov (United States) Hachmon, Guy; Mamet, Noam; Sasson, Sapir; Barkai, Tal; Hadar, Nomi; Abu-Horowitz, Almogit; Bachelet, Ido 2016-01-01 New types of robots inspired by biological principles of assembly, locomotion, and behavior have been recently described. In this work we explored the concept of robots that are based on more fundamental physical phenomena, such as fluid dynamics, and their potential capabilities. We report a robot made entirely of non-Newtonian fluid, driven by shear strains created by spatial patterns of audio waves. We demonstrate various robotic primitives such as locomotion and transport of metallic loads-up to 6-fold heavier than the robot itself-between points on a surface, splitting and merging, shapeshifting, percolation through gratings, and counting to 3. We also utilized interactions between multiple robots carrying chemical loads to drive a bulk chemical synthesis reaction. Free of constraints such as skin or obligatory structural integrity, fluid robots represent a radically different design that could adapt more easily to unfamiliar, hostile, or chaotic environments and carry out tasks that neither living organisms nor conventional machines are capable of. 14. Heat Transfer for Power Law Non-Newtonian Fluids Institute of Scientific and Technical Information of China (English) ZHENG Lian-Cun; ZHANG Xin-Xin; LU Chun-Qing 2006-01-01 We present a theoretical analysis for heat transfer in power law non-Newtonian fluid by assuming that the thermal diffusivity is a function of temperature gradient. The laminar boundary layer energy equation is considered as an example to illustrate the application. It is shown that the boundary layer energy equation subject to the corresponding boundary conditions can be transformed to a boundary value problem of a nonlinear ordinary differential equation when similarity variables are introduced. Numerical solutions of the similarity energy equation are presented. 15. Numerical Study of Non-Newtonian Boundary Layer Flow of Jeffreys Fluid Past a Vertical Porous Plate in a Non-Darcy Porous Medium Science.gov (United States) Ramachandra Prasad, V.; Gaffar, S. Abdul; Keshava Reddy, E.; Bég, O. Anwar 2014-07-01 Polymeric enrobing flows are important in industrial manufacturing technology and process systems. Such flows are non-Newtonian. Motivated by such applications, in this article we investigate the nonlinear steady state boundary layer flow, heat, and mass transfer of an incompressible Jefferys non-Newtonian fluid past a vertical porous plate in a non-Darcy porous medium. The transformed conservation equations are solved numerically subject to physically appropriate boundary conditions using a versatile, implicit, Keller-box finite-difference technique. The numerical code is validated with previous studies. The influence of a number of emerging non-dimensional parameters, namely Deborah number (De), Darcy number (Da), Prandtl number (Pr), ratio of relaxation to retardation times (λ), Schmidt number (Sc), Forchheimer parameter (Λ), and dimensionless tangential coordinate (ξ) on velocity, temperature, and concentration evolution in the boundary layer regime are examined in detail. Furthermore, the effects of these parameters on surface heat transfer rate, mass transfer rate, and local skin friction are also investigated. It is found that the boundary layer flow is decelerated with increasing De and Forchheimer parameter, whereas temperature and concentration are elevated. Increasing λ and Da enhances the velocity but reduces the temperature and concentration. The heat transfer rate and mass transfer rates are found to be depressed with increasing De and enhanced with increasing λ. Local skin friction is found to be decreased with a rise in De, whereas it is elevated with increasing λ. An increasing Sc decreases the velocity and concentration but increases temperature. 16. Non-Similar Computational Solution for Boundary Layer Flows of Non-Newtonian Fluid from an Inclined Plate with Thermal Slip Directory of Open Access Journals (Sweden) SUBBARAO ANNASAGARAM 2016-01-01 Full Text Available The laminar boundary layer flow and heat transfer of Casson non-Newtonian fluid from an inclined (solar collector plate in the presence of thermal and hydrodynamic slip conditions is analysed. The inclined plate surface is maintained at a constant temperature. The boundary layer conservation equations, which are parabolic in nature, are normalized into non-similar form and then solved numerically with the well-tested, efficient, implicit, stable Keller-box finite-difference scheme. Increasing velocity slip induces acceleration in the flow near the inclined plate surface. Increasing velocity slip consistently enhances temperatures throughout the boundary layer regime. An increase in thermal slip parameter strongly decelerates the flow and also reduces temperatures in the boundary layer regime. An increase in Casson rheological parameter acts to elevate considerably the velocity and this effect is pronounced at higher values of tangential coordinate. Temperatures are however very slightly decreased with increasing values of Casson rheological parameter. 17. Numerical Simulation of Bubble Evolution in Non-Newtonian Fluid Institute of Scientific and Technical Information of China (English) 唐亦农; 陈耀松; 陈文芳 1994-01-01 In this paper the bubble issuing from an orifice at the bottom of the boundary evolution in a finite Non-Newtonian fluid(such as Maxwell fluid,Carreu fluid)is numerically simulated The effects of the rheological behavior,physical parameters and circumstantial conditions are discussed in detail 18. The Inveterate Tinkerer: 5. Experiments with Non-Newtonian Fluids Chirag kalelkar 2017-07-01 In this series of articles, the authors discuss various phenomenain fluid dynamics, which may be investigated via tabletopexperiments using low-cost or home-made instruments.The fifth article in this series is about some fascinating experimentswith non-Newtonian fluids. 19. Flow Curve Determination for Non-Newtonian Fluids. Science.gov (United States) 1986-01-01 Describes an experimental program to examine flow curve determination for non-Newtonian fluids. Includes apparatus used (a modification of Walawender and Chen's set-up, but using a 50cc buret connected to a glass capillary through a Tygon tube), theoretical information, procedures, and typical results obtained. (JN) 20. Geophysical Aspects of Non-Newtonian Fluid Mechanics Science.gov (United States) Balmforth, N. J.; Craster, R. V. Non-Newtonian fluid mechanics is a vast subject that has several journals partly, or primarily, dedicated to its investigation (Journal of Non-Newtonian Fluid Mechanics, Rheologica Acta, Journal of Fluid Mechanics, Journal of Rheology, amongst others). It is an area of active research, both for industrial fluid problems and for applications elsewhere, notably geophysically motivated issues such as the flow of lava and ice, mud slides, snow avalanches and debris flows. The main motivati on for this research activity is that, apart from some annoyingly common fluids such as air and water, virtually no fluid is actually Newtonian (that is, having a simple linear relation between stress and strain-rate characterized by a constant viscosity). Several textbooks are useful sources of information; for example, [1-3] are standard texts giving mathematical and engineering perspectives upon the subject. In these lecture notes, Ancey's chapter on rheology (Chap. 3) gives further introduction. 1. Stagnation point flow of an non-Newtonian visco-elastic fluid Energy Technology Data Exchange (ETDEWEB) Teipel, I. [Univ. of Hannover, Inst. of Mechanics, Hannover (Germany) 1985-07-01 In this paper the flow near a two-dimensional stagnation point for a particular non-Newtonian fluid has been studied. Non-Newtonian fluids form a wide class of different materials, which will be used very often in chemical industries. From a practical point of view it is of great importance to obtain some results for example concerning the flow rate and the losses in a tube, the drag and the separation point of a boundary layer flow etc. for such fluids. Therefore it is necessary to assume a particular constitutive equation to calculate various aspects. (author) 2. The Rayleigh-Taylor instability of Newtonian and non-Newtonian fluids Science.gov (United States) Doludenko, A. N.; Fortova, S. V.; Son, E. E. 2016-10-01 Along with Newtonian fluids (for example, water), fluids with non-Newtonian rheology are widespread in nature and industry. The characteristic feature of a non-Newtonian fluid is the non-linear dependence between the shear stress and shear rate tensors. The form of this relation defines the types of non-Newtonian behavior: viscoplastic, pseudoplastic, dilatant and viscoelastic. The present work is devoted to the study of the Rayleigh-Taylor instability in pseudoplastic fluids. The main aim of the work is to undertake a direct three-dimensional numerical simulation of the mixing of two media with various rheologies and obtain the width of the mixing layer and the kinetic energy spectra, depending on the basic properties of the shear thinning liquids and the Atwood number. A theoretical study is carried out on the basis of the Navier-Stokes equation system for weakly compressible media. 3. Journal Bearings Lubrication Aspect Analysis Using Non-Newtonian Fluids Directory of Open Access Journals (Sweden) Abdessamed Nessil 2013-01-01 Full Text Available The aim of this work is related to an analysis of journal bearings lubrication using non-Newtonian fluids which are described by a power-law model. The performance characteristics of the journal bearings are determined for various values of the non-Newtonian power-law index “” which is equal to: 0.9, 1, and 1.1. Obtained numerical results show that for the dilatant fluids (, the load-carrying capacity, the pressure, the temperature, and the frictional force increased while for the pseudo-plastic fluids ( they decreased. The influence of the thermal effects on these characteristics is important at higher values of the flow behavior index “.” Obtained results are compared to those obtained by others. Good agreement is observed between the different results. 4. Verification of vertically rotating flume using non-newtonian fluids Science.gov (United States) Huizinga, R.J. 1996-01-01 Three tests on non-Newtonian fluids were used to verify the use of a vertically rotating flume (VRF) for the study of the rheological properties of debris flow. The VRF is described and a procedure for the analysis of results of tests made with the VRF is presented. The major advantages of the VRF are a flow field consistent with that found in nature, a large particle-diameter threshold, inexpensive operation, and verification using several different materials; the major limitations are a lack of temperature control and a certain error incurred from the use of the Bingham plastic model to describe a more complex phenomenon. Because the VRF has been verified with non-Newtonian fluids as well as Newtonian fluids, it can be used to measure the rheological properties of coarse-grained debris-flow materials. 5. Controlling and minimizing fingering instabilities in non-Newtonian fluids. Science.gov (United States) Fontana, João V; Dias, Eduardo O; Miranda, José A 2014-01-01 The development of the viscous fingering instability in Hele-Shaw cells has great practical and scientific importance. Recently, researchers have proposed different strategies to control the number of interfacial fingering structures, or to minimize as much as possible the amplitude of interfacial disturbances. Most existing studies address the situation in which an inviscid fluid displaces a viscous Newtonian fluid. In this work, we report on controlling and minimizing protocols considering the situation in which the displaced fluid is a non-Newtonian, power-law fluid. The necessary changes on the controlling schemes due to the shear-thinning and shear thickening nature of the displaced fluid are calculated analytically and discussed. 6. Electro-osmotic mobility of non-Newtonian fluids. Science.gov (United States) Zhao, Cunlu; Yang, Chun 2011-03-23 Electrokinetically driven microfluidic devices are usually used to analyze and process biofluids which can be classified as non-Newtonian fluids. Conventional electrokinetic theories resulting from Newtonian hydrodynamics then fail to describe the behaviors of these fluids. In this study, a theoretical analysis of electro-osmotic mobility of non-Newtonian fluids is reported. The general Cauchy momentum equation is simplified by incorporation of the Gouy-Chapman solution to the Poisson-Boltzmann equation and the Carreau fluid constitutive model. Then a nonlinear ordinary differential equation governing the electro-osmotic velocity of Carreau fluids is obtained and solved numerically. The effects of the Weissenberg number (Wi), the surface zeta potential (ψ¯s), the power-law exponent(n), and the transitional parameter (β) on electro-osmotic mobility are examined. It is shown that the results presented in this study for the electro-osmotic mobility of Carreau fluids are quite general so that the electro-osmotic mobility for the Newtonian fluids and the power-law fluids can be obtained as two limiting cases. 7. Memory Effects and Transport Coefficients for Non-Newtonian Fluids CERN Document Server Kodama, T 2008-01-01 We discuss the roles of viscosity in relativistic fluid dynamics from the point of view of memory effects. Depending on the type of quantity to which the memory effect is applied, different terms appear in higher order corrections. We show that when the memory effect applies on the extensive quantities, the hydrodynamic equations of motion become non-singular. We further discuss the question of memory effect in the derivation of transport coefficients from a microscopic theory. We generalize the application of the Green-Kubo-Nakano (GKN) to calculate transport coefficients in the framework of projection operator formalism, and derive the general formula when the fluid is non-Newtonian. 8. Static stability of collapsible tube conveying non-Newtonian fluid CERN Document Server Yushutin, V S 2014-01-01 The global static stability of a Starling Resistor conveying non-Newtonian fluid is considered. The Starling Resistor consists of two rigid circular tubes and axisymmetric collapsible tube mounted between them. Upstream and downstream pressures are the boundary condition as well as external to the collapsible tube pressure. Quasi one-dimensional model has been proposed and a boundary value problem in terms of nondimensional parameters obtained. Nonuniqueness of the boundary value problem is regarded as static instability. The analytical condition of instability which defines a surface in parameter space has been studied numerically. The influence of fluid rheology on stability of collapsible tube is established. 9. Random Attractors of Stochastic Non-Newtonian Fluids Institute of Scientific and Technical Information of China (English) Chun-xiao GUO; Bo-ling GUO; Yong-qian HAN 2012-01-01 The present paper investigates the asymptotic behavior of solutions for stochastic non-Newtonian fluids in a two-dimensional domain.Firstly,we prove the existence of random attractors AH(ω) in H; Secondly,we prove the existence of random attractors Av(ω) in V.Then we verify regularity of the random attractors by showing that AH(ω) =Av(ω),which implies the smoothing effect of the fluids in the sense that solution becomes eventually more regular than the initial data. 10. Dynamic wetting with viscous Newtonian and non-Newtonian fluids. Science.gov (United States) Wei, Y; Rame, E; Walker, L M; Garoff, S 2009-11-18 We examine various aspects of dynamic wetting with viscous Newtonian and non-Newtonian fluids. Rather than concentrating on the mechanisms that relieve the classic contact line stress singularity, we focus on the behavior in the wedge flow near the contact line which has the dominant influence on wetting with these fluids. Our experiments show that a Newtonian polymer melt composed of highly flexible molecules exhibits dynamic wetting behavior described very well by hydrodynamic models that capture the critical properties of the Newtonian wedge flow near the contact line. We find that shear thinning has a strong impact on dynamic wetting, by reducing the drag of the solid on the fluid near the contact line, while the elasticity of a Boger fluid has a weaker impact on dynamic wetting. Finally, we find that other polymeric fluids, nominally Newtonian in rheometric measurements, exhibit deviations from Newtonian dynamic wetting behavior. 11. Intermittent outgassing through a non-Newtonian fluid. Science.gov (United States) Divoux, Thibaut; Bertin, Eric; Vidal, Valérie; Géminard, Jean-Christophe 2009-05-01 We report an experimental study of the intermittent dynamics of a gas flowing through a column of a non-Newtonian fluid. In a given range of the imposed constant flow rate, the system spontaneously alternates between two regimes: bubbles emitted at the bottom either rise independently one from the other or merge to create a winding flue which then connects the bottom air entrance to the free surface. The observations are reminiscent of the spontaneous changes in the degassing regime observed on volcanoes and suggest that, in the nature, such a phenomenon is likely to be governed by the non-Newtonian properties of the magma. We focus on the statistical distribution of the lifespans of the bubbling and flue regimes in the intermittent steady state. The bubbling regime exhibits a characteristic time whereas, interestingly, the flue lifespan displays a decaying power-law distribution. The associated exponent, which is significantly smaller than the value 1.5 often reported experimentally and predicted in some standard intermittency scenarios, depends on the fluid properties and can be interpreted as the ratio of two characteristic times of the system. 12. Steady flow of a non-Newtonian fluid through a contraction Science.gov (United States) Gatski, T. B.; Lumley, J. L. 1978-01-01 A steady-state analysis is conducted to examine the basic flow structure of a non-Newtonian fluid in a domain including an inflow region, a contraction region, and an outflow region. A Cartesian grid system is used throughout the entire flow domain, including the contraction region, thus creating an irregular grid cell structure adjacent to the curved boundary. At node points adjacent to the curved boundary symmetry conditions are derived for the different flow variables in order to solve the governing difference equations. Attention is given to the motion and non-Newtonian constitutive equations, the boundary conditions, the numerical modeling of the non-Newtonian equations, the stream function contour lines for the non-Newtonian fluid, the vorticity contour lines for the non-Newtonian fluid, the velocity profile across the contraction, and the shear stress contour lines for the non-Newtonian fluid. 13. Mixed convection heat transfer from a vertical plate to non-Newtonian fluids Science.gov (United States) Wang, T.-Y. 1995-02-01 The nonsimilar boundary-layer analysis of steady laminar mixed-convection heat transfer between a vertical plate and non-Newtonian fluids is extended and unified. A mixed-convection parameter zeta is proposed to replace the conventional Richardson number, Gr/Re(exp 2/(2 - n)) and to serve as a controlling parameter that determines the relative importance of the forced and the free convection. The value of mixed-convection parameter lies between 0 and 1. In addition, the power-law model is used for non-Newtonian fluids with exponent n less than 1 for pseudoplastics; n = 1 for Newtonian fluids; and n greater than 1 for dilatant fluids. Furthermore, the coordinates and dependent variables are transformed to yield computationally efficient numerical solutions that are valid over the entire range of mixed convection, from the pure forced-convection limit to the pure free-convection limit, and the whole domain of non-Newtonian fluids, from pseudoplastics to dilatant fluids. The effects of the mixed-convection parameter, the power-law viscosity index, and the generalized Prandtl number on the velocity profiles, the temperature profiles, as well as on the wall skin friction and heat transfer rate are clearly illustrated for both cases of buoyancy assisting and opposing flow conditions. 14. A Critical Review of Dynamic Wetting by Complex Fluids: From Newtonian Fluids to Non-Newtonian Fluids and Nanofluids. Science.gov (United States) Lu, Gui; Wang, Xiao-Dong; Duan, Yuan-Yuan 2016-10-01 Dynamic wetting is an important interfacial phenomenon in many industrial applications. There have been many excellent reviews of dynamic wetting, especially on super-hydrophobic surfaces with physical or chemical coatings, porous layers, hybrid micro/nano structures and biomimetic structures. This review summarizes recent research on dynamic wetting from the viewpoint of the fluids rather than the solid surfaces. The reviewed fluids range from simple Newtonian fluids to non-Newtonian fluids and complex nanofluids. The fundamental physical concepts and principles involved in dynamic wetting phenomena are also reviewed. This review focus on recent investigations of dynamic wetting by non-Newtonian fluids, including the latest experimental studies with a thorough review of the best dynamic wetting models for non-Newtonian fluids, to illustrate their successes and limitations. This paper also reports on new results on the still fledgling field of nanofluid wetting kinetics. The challenges of research on nanofluid dynamic wetting is not only due to the lack of nanoscale experimental techniques to probe the complex nanoparticle random motion, but also the lack of multiscale experimental techniques or theories to describe the effects of nanoparticle motion at the nanometer scale (10(-9) m) on the dynamic wetting taking place at the macroscopic scale (10(-3) m). This paper describes the various types of nanofluid dynamic wetting behaviors. Two nanoparticle dissipation modes, the bulk dissipation mode and the local dissipation mode, are proposed to resolve the uncertainties related to the various types of dynamic wetting mechanisms reported in the literature. 15. UNIFORM ATTRACTOR FOR NONAUTONOMOUS INCOMPRESSIBLE NON-NEWTONIAN FLUID WITH A NEW CLASS OF EXTERNAL FORCES Institute of Scientific and Technical Information of China (English) Zhao Caidi; Jia Xiaolin; Yang Xinbo 2011-01-01 This paper is joint with [27].The authors prove in this article the existence and reveal its structure of uniform attractor for a two-dimensional nonautonomous incompressible non-Newtonian fluid with a new class of external forces. 16. H 2-regularity random attractors of stochastic non-Newtonian fluids with multiplicative noise Institute of Scientific and Technical Information of China (English) Chun-xiao GUO; Bo-ling GUO; Hui YANG 2014-01-01 In this paper, the authors study the long time behavior of solutions to stochastic non-Newtonian fluids in a two-dimensional bounded domain, and prove the existence of H 2-regularity random attractor. 17. Upper Semicontinuity of Attractors for a Non-Newtonian Fluid under Small Random Perturbations Directory of Open Access Journals (Sweden) Jianxin Luo 2014-01-01 Full Text Available This paper investigates the limiting behavior of attractors for a two-dimensional incompressible non-Newtonian fluid under small random perturbations. Under certain conditions, the upper semicontinuity of the attractors for diminishing perturbations is shown. 18. Decay of solutions to equations modelling incompressible bipolar non-newtonian fluids Directory of Open Access Journals (Sweden) Bo-Qing Dong 2005-11-01 Full Text Available This article concerns systems of equations that model incompressible bipolar non-Newtonian fluid motion in the whole space $mathbb{R}^n$. Using the improved Fourier splitting method, we prove that a weak solution decays in the $L^2$ norm at the same rate as $(1+t^{-n/4}$ as the time $t$ approaches infinity. Also we obtain optimal $L^2$ error-estimates for Newtonian and Non-Newtonian flows. 19. Applying Tiab’s direct synthesis technique to dilatant non-Newtonian/Newtonian fluids Directory of Open Access Journals (Sweden) Javier Andrés Martínez 2011-08-01 Full Text Available Non-Newtonian fluids, such as polymer solutions, have been used by the oil industry for many years as fracturing agents and drilling mud. These solutions, which normally include thickened water and jelled fluids, are injected into the formation to enhanced oil recovery by improving sweep efficiency. It is worth noting that some heavy oils behave non-Newtonianly. Non-Newtonian fluids do not have direct proportionality between applied shear stress and shear rate and viscosity varies with shear rate depending on whether the fluid is either pseudoplastic or dilatant. Viscosity decreases as shear rate increases for the former whilst the reverse takes place for dilatants. Mathematical models of conventional fluids thus fail when applied to non-Newtonian fluids. The pressure derivative curve is introduced in this descriptive work for a dilatant fluid and its pattern was observed. Tiab’s direct synthesis (TDS methodology was used as a tool for interpreting pressure transient data to estimate effective permeability, skin factors and non-Newtonian bank radius. The methodology was successfully verified by its application to synthetic examples. Also, comparing it to pseudoplastic behavior, it was found that the radial flow regime in the Newtonian zone of dilatant fluids took longer to form regarding both the flow behavior index and consistency factor. 20. Effect of non-Newtonian viscosity on the fluid-dynamic characteristics in stenotic vessels Science.gov (United States) Huh, Hyung Kyu; Ha, Hojin; Lee, Sang Joon 2015-08-01 Although blood is known to have shear-thinning and viscoelastic properties, the effects of such properties on the hemodynamic characteristics in various vascular environments are not fully understood yet. For a quantitative hemodynamic analysis, the refractive index of a transparent blood analogue needs to be matched with that of the flowing conduit in order to minimize the errors according to the distortion of the light. In this study, three refractive index-matched blood analogue fluids with different viscosities are prepared—one Newtonian and two non-Newtonian analogues—which correspond to healthy blood with 45 % hematocrit (i.e., normal non-Newtonian) and obese blood with higher viscosity (i.e., abnormal non-Newtonian). The effects of the non-Newtonian rheological properties of the blood analogues on the hemodynamic characteristics in the post-stenosis region of an axisymmetric stenosis model are experimentally investigated using particle image velocimetry velocity field measurement technique and pathline flow visualization. As a result, the centerline jet flow from the stenosis apex is suppressed by the shear-thinning feature of the blood analogues when the Reynolds number is smaller than 500. The lengths of the recirculation zone for abnormal and normal non-Newtonian blood analogues are 3.67 and 1.72 times shorter than that for the Newtonian analogue at Reynolds numbers smaller than 200. The Reynolds number of the transition from laminar to turbulent flow for all blood analogues increases as the shear-thinning feature increases, and the maximum wall shear stresses in non-Newtonian fluids are five times greater than those in Newtonian fluids. However, the shear-thinning effect on the hemodynamic characteristics is not significant at Reynolds numbers higher than 1000. The findings of this study on refractive index-matched non-Newtonian blood analogues can be utilized in other in vitro experiments, where non-Newtonian features dominantly affect the flow 1. Smart Fluids in Hydrology: Use of Non-Newtonian Fluids for Pore Structure Characterization Science.gov (United States) Abou Najm, M. R.; Atallah, N. M.; Selker, J. S.; Roques, C.; Stewart, R. D.; Rupp, D. E.; Saad, G.; El-Fadel, M. 2015-12-01 Classic porous media characterization relies on typical infiltration experiments with Newtonian fluids (i.e., water) to estimate hydraulic conductivity. However, such experiments are generally not able to discern important characteristics such as pore size distribution or pore structure. We show that introducing non-Newtonian fluids provides additional unique flow signatures that can be used for improved pore structure characterization while still representing the functional hydraulic behavior of real porous media. We present a new method for experimentally estimating the pore structure of porous media using a combination of Newtonian and non-Newtonian fluids. The proposed method transforms results of N infiltration experiments using water and N-1 non-Newtonian solutions into a system of equations that yields N representative radii (Ri) and their corresponding percent contribution to flow (wi). This method allows for estimating the soil retention curve using only saturated experiments. Experimental and numerical validation comparing the functional flow behavior of different soils to their modeled flow with N representative radii revealed the ability of the proposed method to represent the water retention and infiltration behavior of real soils. The experimental results showed the ability of such fluids to outsmart Newtonian fluids and infer pore size distribution and unsaturated behavior using simple saturated experiments. Specifically, we demonstrate using synthetic porous media that the use of different non-Newtonian fluids enables the definition of the radii and corresponding percent contribution to flow of multiple representative pores, thus improving the ability of pore-scale models to mimic the functional behavior of real porous media in terms of flow and porosity. The results advance the knowledge towards conceptualizing the complexity of porous media and can potentially impact applications in fields like irrigation efficiencies, vadose zone hydrology, soil 2. Introducing Non-Newtonian Fluid Mechanics Computations with Mathematica in the Undergraduate Curriculum Science.gov (United States) Binous, Housam 2007-01-01 We study four non-Newtonian fluid mechanics problems using Mathematica[R]. Constitutive equations describing the behavior of power-law, Bingham and Carreau models are recalled. The velocity profile is obtained for the horizontal flow of power-law fluids in pipes and annuli. For the vertical laminar film flow of a Bingham fluid we determine the… 3. Introducing Non-Newtonian Fluid Mechanics Computations with Mathematica in the Undergraduate Curriculum Science.gov (United States) Binous, Housam 2007-01-01 We study four non-Newtonian fluid mechanics problems using Mathematica[R]. Constitutive equations describing the behavior of power-law, Bingham and Carreau models are recalled. The velocity profile is obtained for the horizontal flow of power-law fluids in pipes and annuli. For the vertical laminar film flow of a Bingham fluid we determine the… 4. Free surface flow of a suspension of rigid particles in a non-Newtonian fluid DEFF Research Database (Denmark) Svec, Oldrich; Skocek, Jan; Stang, Henrik 2012-01-01 A numerical framework capable of predicting the free surface flow of a suspension of rigid particles in a non-Newtonian fluid is described. The framework is a combination of the lattice Boltzmann method for fluid flow, the mass tracking algorithm for free surface representation, the immersed... 5. Revisiting Newtonian and Non-Newtonian Fluid Mechanics Using Computer Algebra Science.gov (United States) Knight, D. G. 2006-01-01 This article illustrates how a computer algebra system, such as Maple[R], can assist in the study of theoretical fluid mechanics, for both Newtonian and non-Newtonian fluids. The continuity equation, the stress equations of motion, the Navier-Stokes equations, and various constitutive equations are treated, using a full, but straightforward,… 6. Effect of non-Newtonian fluid properties on bovine sperm motility. Science.gov (United States) Hyakutake, Toru; Suzuki, Hiroki; Yamamoto, Satoru 2015-09-18 The swimming process by which mammal spermatozoa progress towards an egg within the reproductive organs is important in achieving successful internal fertilization. The viscosity of oviductal mucus is more than two orders of magnitude greater than that of water, and oviductal mucus also has non-Newtonian properties. In this study, we experimentally observed sperm motion in fluids with various fluid rheological properties and investigated the influence of varying the viscosity and whether the fluid was Newtonian or non-Newtonian on the sperm motility. We selected polyvinylpyrrolidone and methylcellulose as solutes to create solutions with different rheological properties. We used the semen of Japanese cattle and investigated the following parameters: the sperm velocity, the straight-line velocity and the amplitude from the trajectory, and the beat frequency from the fragellar movement. In a Newtonian fluid environment, as the viscosity increased, the motility of the sperm decreased. However, in a non-Newtonian fluid, the straight-line velocity and beat frequency were significantly higher than in a Newtonian fluid with comparable viscosity. As a result, the linearity of the sperm movement increased. Additionally, increasing the viscosity brought about large changes in the sperm flagellar shape. At low viscosities, the entire flagellum moved in a curved flapping motion, whereas in the high-viscosity, only the tip of the flagellum flapped. These results suggest that the bovine sperm has evolved to swim toward the egg as quickly as possible in the actual oviduct fluid, which is a high-viscosity non-Newtonian fluid. 7. Learning about Non-Newtonian Fluids in a Student-Driven Classroom Science.gov (United States) Dounas-Frazer, D. R.; Lynn, J.; Zaniewski, A. M.; Roth, N. 2013-01-01 We describe a simple, low-cost experiment and corresponding pedagogical strategies for studying fluids whose viscosities depend on shear rate, referred to as "non-Newtonian fluids." We developed these materials teaching for the Compass Project, an organization that fosters a creative, diverse, and collaborative community of science… 8. Revisiting Newtonian and Non-Newtonian Fluid Mechanics Using Computer Algebra Science.gov (United States) Knight, D. G. 2006-01-01 This article illustrates how a computer algebra system, such as Maple[R], can assist in the study of theoretical fluid mechanics, for both Newtonian and non-Newtonian fluids. The continuity equation, the stress equations of motion, the Navier-Stokes equations, and various constitutive equations are treated, using a full, but straightforward,… 9. Stretch flow of confined non-Newtonian fluids: nonlinear fingering dynamics. Science.gov (United States) Brandão, Rodolfo; Fontana, João V; Miranda, José A 2013-12-01 We employ a weakly nonlinear perturbative scheme to investigate the stretch flow of a non-Newtonian fluid confined in Hele-Shaw cell for which the upper plate is lifted. A generalized Darcy's law is utilized to model interfacial fingering formation in both the weak shear-thinning and weak shear-thickening limits. Within this context, we analyze how the interfacial finger shapes and the nonlinear competition dynamics among fingers are affected by the non-Newtonian nature of the stretched fluid. 10. Similarity solutions for non-Newtonian power-law fluid flow Institute of Scientific and Technical Information of China (English) D.M.WEI; S.AL-ASHHAB 2014-01-01 The problem of the boundary layer flow of power law non-Newtonian fluids with a novel boundary condition is studied. The existence and uniqueness of the solutions are examined, which are found to depend on the curvature of the solutions for different values of the power law index n. It is established with the aid of the Picard-Lindel¨of theorem that the nonlinear boundary value problem has a unique solution in the global domain for all values of the power law index n but with certain conditions on the curva-ture of the solutions. This is done after a suitable transformation of the dependent and independent variables. For 0 1, the solution has a negative or zero curvature on some part of the global domain. Some solutions are presented graphically to illustrate the results and the behaviors of the solutions. 11. Interfacial instabilities affect microfluidic extraction of small molecules from non-Newtonian fluids. Science.gov (United States) Helton, Kristen L; Yager, Paul 2007-11-01 As part of a project to develop an integrated microfluidic biosensor for the detection of small molecules in saliva, practical issues of extraction of analytes from non-Newtonian samples using an H-filter were explored. The H-filter can be used to rapidly and efficiently extract small molecules from a complex sample into a simpler buffer. The location of the interface between the sample and buffer streams is a critical parameter in the function of the H-filter, so fluorescence microscopy was employed to monitor the interface position; this revealed apparently anomalous fluorophore diffusion from the samples into the buffer solutions. Using confocal microscopy to understand the three-dimensional distribution of the fluorophore, it was found that the interface between the non-Newtonian sample and Newtonian buffer was both curved and unstable. The core of the non-Newtonian sample extended into the Newtonian buffer and its position was unstable, producing a fluorescence intensity profile that gave rise to the apparently anomalously fast fluorophore transport. These instabilities resulted from the pairing of rheologically dissimilar fluid streams and were flowrate dependent. We conclude that use of non-Newtonian fluids, such as saliva, in the H-filter necessitates pretreatment to reduce viscoelasticity. The interfacial variation in position, stability and shape caused by the non-Newtonian samples has substantial implications for the use of biological samples for quantitative analysis and analyte extraction in concurrent flow extraction devices. 12. Smart Fluids in Hydrology: Use of Non-Newtonian Fluids for Pore Structure Characterization Science.gov (United States) Abou Najm, Majdi; Atallah, Nabil; Selker, John; Roques, Clément; Stewart, Ryan; Rupp, David; Saad, George; El-Fadel, Mutasem 2016-04-01 Classic porous media characterization relies on typical infiltration experiments with Newtonian fluids (i.e., water) to estimate hydraulic conductivity. However, such experiments are generally not able to discern important characteristics such as pore size distribution or pore structure. We show that introducing non-Newtonian fluids provides additional unique flow signatures that can be used for improved pore structure characterization. We present a new method that transforms results of N infiltration experiments using water and N-1 non-Newtonian solutions into a system of equations that yields N representative radii (Ri) and their corresponding percent contribution to flow (wi). Those radii and weights are optimized in terms of flow and porosity to represent the functional hydraulic behavior of real porous media. The method also allows for estimating the soil retention curve using only saturated experiments. Experimental and numerical validation revealed the ability of the proposed method to represent the water retention and functional infiltration behavior of real soils. The experimental results showed the ability of such fluids to outsmart Newtonian fluids and infer pore size distribution and unsaturated behavior using simple saturated experiments. Specifically, we demonstrate using synthetic porous media composed of different combinations of sizes and numbers of capillary tubes that the use of different non-Newtonian fluids enables the prediction of the pore structure. The results advance the knowledge towards conceptualizing the complexity of porous media and can potentially impact applications in fields like irrigation efficiencies, vadose zone hydrology, soil-root-plant continuum, carbon sequestration into geologic formations, soil remediation, petroleum reservoir engineering, oil exploration and groundwater modeling. 13. Applied holography for drop formation of non-Newtonian fluids in centrifugal atomizers Science.gov (United States) Timko, J. J. Holography made possible the analysis of drop formation in Newtonian and non-Newtonian fluids. The drops were illuminated at the moment of their formation with an impulse ruby laser, and from the holograms the whole spray was reconstructed with a closed-circuit TV loop. From the pictures taken from different planes of the spray, the size and the spatial distribution of the drops were determined with an electrooptical analyzer. The holographic measuring method provided quantitative data phenomena which were qualitatively observable on high-speed films. The experiments also verified an equation involving dimensionless criteria, deduced fo the atomization of non-Newtonian substances. 14. Numerical Modelling of Non-Newtonian Fluid in a Rotational Cross-Flow MBR DEFF Research Database (Denmark) Bentzen, Thomas Ruby; Ratkovich, Nicolas Rios; Rasmussen, Michael R. 2011-01-01 . Validation of the CFD model was made against LDA tangential velocity measurements (error less than 8 %) using water a fluid. The shear stress over the membrane surface was inferred from the CFD simulations for water. However, activated sludge is a non-Newtonian liquid, for which the CFD model was modified...... incorporating the non-Newtonian behaviour of activated sludge. Shear stress and area-weighted average shear stress relationships were made giving error less that 8 % compared to the CFD results. An empirical relationship for the area-weighted average shear stress was developed for water and activated sludge... 15. Validation of computational non-Newtonian fluid model for membrane bioreactor DEFF Research Database (Denmark) Sørensen, Lasse; Bentzen, Thomas Ruby; Skov, Kristian 2015-01-01 for optimizing MBR-systems is computational fluid dynamics (CFD) modelling, giving the ability to describe the flow in the systems. A parameter which is often neglected in such models is the non-Newtonian properties of active sludge, which is of great importance for MBR systems since they operate at sludge...... concentrations up to a factor 10 compared to conventional activated sludge (CAS) systems, resulting in strongly shear thinning liquids. A CFD-model is validated against measurements conducted in a system with rotating cross flow membranes submerged in non-Newtonian liquids, where tangential velocities... 16. RANDOM ATTRACTOR FOR A TWO-DIMENSIONAL INCOMPRESSIBLE NON-NEWTONIAN FLUID WITH MULTIPLICATIVE NOISE Institute of Scientific and Technical Information of China (English) Zhao Caidi; Li Yongsheng; Zhou Shengfan 2011-01-01 This article proves that the random dynamical system generated by a two- dimensional incompressible non-Newtonian fluid with multiplicative noise has a global random attractor, which is a random compact set absorbing any bounded nonrandom subset of the phase space. 17. Modeling of flow of particles in a non-Newtonian fluid using lattice Boltzmann method DEFF Research Database (Denmark) Skocek, Jan; Svec, Oldrich; Spangenberg, Jon 2011-01-01 is necessary. In this contribution, the model at the scale of aggregates is introduced. The conventional lattice Boltzmann method for fluid flow is enriched with the immersed boundary method with direct forcing to simulate the flow of rigid particles in a non- Newtonian liquid. Basic ingredients of the model... 18. Heat Transfer of Non-Newtonian Dilatant Power Law Fluids in Square and Rectangular Cavities Directory of Open Access Journals (Sweden) 2011-01-01 Full Text Available Steady two-dimensional natural convection in fluid filled cavities is numerically investigated for the case of non- Newtonian shear thickening power law liquids. The conservation equations of mass, momentum and energy under the assumption of a Newtonian Boussinesq fluid have been solved using the finite volume method for Newtonian and non-Newtonian fluids. The computations were performed for a Rayleigh number, based on cavity height, of 105 and a Prandtl number of 100. In all of the numerical experiments, the channel is heated from below and cooled from the top with insulated side-walls and the inclination angle is varied. The simulations have been carried out for aspect ratios of 1 and 4. Comparison between the Newtonian and the non-Newtonian cases is conducted based on the dependence of the average Nusselt number on angle of inclination. It is shown that despite significant variation in heat transfer rate both Newtonian and non-Newtonian fluids exhibit similar behavior with the transition from multi-cell flow structure to a single-cell regime. 19. Oscillatory Spreading and Surface Instability of a Non-Newtonian Fluid under Compression OpenAIRE Choudhury, Moutushi Dutta; Chandra, Subrata; Nag, Soma; Das, Shantanu; Tarafdar, Sujata 2010-01-01 Starch solutions, which are strongly non-Newtonian, show a surface instability, when subjected to a load. A droplet of the fluid is sandwiched between two glass plates and a weight varying from 1 to 5 kgs. is placed on the top plate. The area of contact between the fluid and plate increases in an oscillatory manner, unlike Newtonian fluids in a similar situation. The periphery moreover, develops a viscous fingering like instability, which is not expected under compression. We attempt to model... 20. Spreading of Non-Newtonian and Newtonian Fluids on a Solid Substrate under Pressure Science.gov (United States) Dutta Choudhury, Moutushi; Chandra, Subrata; Nag, Soma; Das, Shantanu; Tarafdar, Sujata 2011-09-01 Strongly non-Newtonian fluids namely, aqueous gels of starch, are shown to exhibit visco-elastic behavior, when subjected to a load. We study arrowroot and potato starch gels. When a droplet of the fluid is sandwiched between two glass plates and compressed, the area of contact between the fluid and plates increases in an oscillatory manner. This is unlike Newtonian fluids, where the area increases monotonically in a similar situation. The periphery moreover, develops an instability, which looks similar to Saffman Taylor fingers. This is not normally seen under compression. The loading history is also found to affect the manner of spreading. We attempt to describe the non-Newtonian nature of the fluid through a visco-elastic model incorporating generalized calculus. This is shown to reproduce qualitatively the oscillatory variation in the surface strain. 1. Spreading of Non-Newtonian and Newtonian Fluids on a Solid Substrate under Pressure Energy Technology Data Exchange (ETDEWEB) Choudhury, Moutushi Dutta; Chandra, Subrata; Nag, Soma; Tarafdar, Sujata [Condensed Matter Physics Research Centre, Physics Department, Jadavpur University, Kolkata 700032 (India); Das, Shantanu, E-mail: [email protected] [Reactor Control Division, Bhabha Atomic Research Center, Trombay, Mumbai 400085 (India) 2011-09-15 Strongly non-Newtonian fluids namely, aqueous gels of starch, are shown to exhibit visco-elastic behavior, when subjected to a load. We study arrowroot and potato starch gels. When a droplet of the fluid is sandwiched between two glass plates and compressed, the area of contact between the fluid and plates increases in an oscillatory manner. This is unlike Newtonian fluids, where the area increases monotonically in a similar situation. The periphery moreover, develops an instability, which looks similar to Saffman Taylor fingers. This is not normally seen under compression. The loading history is also found to affect the manner of spreading. We attempt to describe the non-Newtonian nature of the fluid through a visco-elastic model incorporating generalized calculus. This is shown to reproduce qualitatively the oscillatory variation in the surface strain. 2. Axial dispersion in packed bed reactors involving viscoinelastic and viscoelastic non-Newtonian fluids. Science.gov (United States) Gupta, Renu; Bansal, Ajay 2013-08-01 Axial dispersion is an important parameter in the performance of packed bed reactors. A lot of fluids exhibit non-Newtonian behaviour but the effect of rheological parameters on axial dispersion is not available in literature. The effect of rheology on axial dispersion has been analysed for viscoinelastic and viscoelastic non-Newtonian fluids. Aqueous solutions of carboxymethyl cellulose and polyacrylamide have been chosen to represent viscoinelastic and viscoelastic liquid-phases. Axial dispersion has been measured in terms of BoL number. The single parameter axial dispersion model has been applied to analyse RTD response curve. The BoL numbers were observed to increase with increase in liquid flow rate and consistency index 'K' for viscoinelastic as well as viscoelastic fluids. Bodenstein correlation for Newtonian fluids proposed has been modified to account for the effect of fluid rheology. Further, Weissenberg number is introduced to quantify the effect of viscoelasticity. 3. Gass-Assisted Displacement of Non-Newtonian Fluids DEFF Research Database (Denmark) Rasmussen, Henrik Koblitz; Eriksson, Torbjörn Gerhard 2003-01-01 During the resent years several publications (for instance Hyzyak and Koelling, J. Non-Newt. Fluid Mech. 71,73-88 (1997) and Gauri and Koelling, Rheol. Acta, 38, 458-470 (1999)) have concerned gas assisted displacement of viscoelastic fluids (polymer melts and polymeric solutions) contained...... in a circular cylinder. This is a simple model system used to investigate the gas-fluid displacement, as the problem is reduced to an axis-symmetric flow problem. The understanding of this process is relevant for the geometrically much more complex polymer processing operation Gas-assisted injection moulding... 4. Classical XY model with conserved angular momentum is an archetypal non-Newtonian fluid. Science.gov (United States) Evans, R M L; Hall, Craig A; Simha, R Aditi; Welsh, Tom S 2015-04-03 We find that the classical one-dimensional XY model, with angular-momentum-conserving Langevin dynamics, mimics the non-Newtonian flow regimes characteristic of soft matter when subjected to counterrotating boundaries. An elaborate steady-state phase diagram has continuous and first-order transitions between states of uniform flow, shear-banding, solid-fluid coexistence and slip planes. Results of numerical studies and a concise mean-field constitutive relation offer a paradigm for diverse nonequilibrium complex fluids. 5. Gaseous bubble oscillations in anisotropic non-Newtonian fluids under influence of high-frequency acoustic field Science.gov (United States) Golykh, R. N. 2016-06-01 Progress of technology and medicine dictates the ever-increasing requirements (heat resistance, corrosion resistance, strength properties, impregnating ability, etc.) for non-Newtonian fluids and materials produced on their basis (epoxy resin, coating materials, liquid crystals, etc.). Materials with improved properties obtaining is possible by modification of their physicochemical structure. One of the most promising approaches to the restructuring of non-Newtonian fluids is cavitation generated by high-frequency acoustic vibrations. The efficiency of cavitation in non-Newtonian fluid is determined by dynamics of gaseous bubble. Today, bubble dynamics in isotropic non-Newtonian fluids, in which cavitation bubble shape remains spherical, is most full investigated, because the problem reduces to ordinary differential equation for spherical bubble radius. However, gaseous bubble in anisotropic fluids which are most wide kind of non-Newtonian fluids (due to orientation of macromolecules) deviates from spherical shape due to viscosity dependence on shear rate direction. Therefore, the paper presents the mathematical model of gaseous bubble dynamics in anisotropic non-Newtonian fluids. The model is based on general equations for anisotropic non-Newtonian fluid flow. The equations are solved by asymptotic decomposition of fluid flow parameters. It allowed evaluating bubble size and shape evolution depending on rheological properties of liquid and acoustic field characteristics. 6. Turbulent Characteristic of Liquid Around a Chain of Bubbles in Non-Newtonian Fluid Institute of Scientific and Technical Information of China (English) 李少白; 马友光; 朱春英; 付涛涛; 李怀志 2012-01-01 The turbulence behavior of gas-liquid two-phase flow plays an important role in heat transfer and mass transfer in many chemical processes. In this work, a 2D particle image velocimetry (PIV) was used to investigate the turbulent characteristic of fluid induced by a chain of bubbles rising in Newtonian and non-Newtonian fluids. The instantaneous flow field, turbulent kinetic energy (TKE) and TKE dissipation rate were measured. The results demonstrated that the TKE profiles were almost symmetrical along the column center and showed higher values in the central region of the column. The TKE was enhanced with the increase of gas flow and decrease of liquid viscosity. The maximum TKE dissipation rate appeared on both sides of the bubble chain, and increased with the increase of gas flow rate or liquid viscosity. These results provide an understanding for gas-liquid mass transfer in non-Newtonian fluids. 7. A comparison of numerical methods for non-Newtonian fluid flows in a sudden expansion Science.gov (United States) Ilio, G. Di; Chiappini, D.; Bella, G. 2016-06-01 A numerical study on incompressible laminar flow in symmetric channel with sudden expansion is conducted. In this work, Newtonian and non-Newtonian fluids are considered, where non-Newtonian fluids are described by the power-law model. Three different computational methods are employed, namely a semi-implicit Chorin projection method (SICPM), an explicit algorithm based on fourth-order Runge-Kutta method (ERKM) and a Lattice Boltzmann method (LBM). The aim of the work is to investigate on the capabilities of the LBM for the solution of complex flows through the comparison with traditional computational methods. In the range of Reynolds number investigated, excellent agreement with the literature results is found. In particular, the LBM is found to be accurate in the prediction of the fluid flow behavior for the problem under consideration. 8. Gass-Assisted Displacement of Non-Newtonian Fluids DEFF Research Database (Denmark) Rasmussen, Henrik Koblitz; Eriksson, Torbjörn Gerhard 2003-01-01 (GAIM). This is a process, where a mould is filled partly with a polymer melt followed by the injection of inert gas into the core of the polymer melt. The numerical analysis of the fluid flow concerning the experimental observations data in these publications is all based on Newtonian or general......-B constitutive model will be used throughout this paper. A numerical method is needed in order to calculate the flow of the viscoelastic fluid during the displacement. To model the displacement numerically, the time-dependent finite element method from Rasmussen [1] is used. This method has second order...... convergence both in the time and the spatial discretization. The non-dimensional geometrical groups in this displacement are the Deborah and the surface elasticity number. The Deborah number is in a general definition (e.g. independent of constitutive equation) given as De=(2·U/R)·Ø1(2·U/R)/(2·çp(2·U... 9. Numerical Solution of Hydrodynamics Lubrications with Non-Newtonian Fluid Flow Science.gov (United States) Osman, Kahar; Sheriff, Jamaluddin Md; Bahak, Mohd. Zubil; Bahari, Adli; Asral 2010-06-01 This paper focuses on solution of numerical model for fluid film lubrication problem related to hydrodynamics with non-Newtonian fluid. A programming code is developed to investigate the effect of bearing design parameter such as pressure. A physical problem is modeled by a contact point of sphere on a disc with certain assumption. A finite difference method with staggered grid is used to improve the accuracy. The results show that the fluid characteristics as defined by power law fluid have led to a difference in the fluid pressure profile. Therefore a lubricant with special viscosity can reduced the pressure near the contact area of bearing. 10. Transfer of Microparticles across Laminar Streams from Non-Newtonian to Newtonian Fluid. Science.gov (United States) Ha, Byunghang; Park, Jinsoo; Destgeer, Ghulam; Jung, Jin Ho; Sung, Hyung Jin 2016-04-19 Engineering inertial lift forces and elastic lift forces is explored to transfer microparticles across laminar streams from non-Newtonian to Newtonian fluid. A co-stream of non-Newtonian flow loaded with microparticles (9.9 and 2.0 μm in diameter) and a Newtonian carrier medium flow in a straight rectangular conduit is devised. The elastic lift forces present in the non-Newtonian fluid, undeterred by particle-particle interaction, successfully pass most of the larger (9.9 μm) particles over to the Newtonian fluid. The Newtonian fluid takes over the larger particles and focus them on the equilibrium position, separating the larger particles from the smaller particles. This mechanism enabled processing of densely suspended particle samples. The method offers dilution-free (for number densities up to 10,000 μL(-1)), high throughput (6700 beads/s), and highly efficient (>99% recovery rate, >97% purity) particle separation operated over a wide range of flow rate (2 orders of magnitude). 11. FDA's nozzle numerical simulation challenge: non-Newtonian fluid effects and blood damage. Science.gov (United States) Trias, Miquel; Arbona, Antonio; Massó, Joan; Miñano, Borja; Bona, Carles 2014-01-01 Data from FDA's nozzle challenge-a study to assess the suitability of simulating fluid flow in an idealized medical device-is used to validate the simulations obtained from a numerical, finite-differences code. Various physiological indicators are computed and compared with experimental data from three different laboratories, getting a very good agreement. Special care is taken with the derivation of blood damage (hemolysis). The paper is focused on the laminar regime, in order to investigate non-Newtonian effects (non-constant fluid viscosity). The code can deal with these effects with just a small extra computational cost, improving Newtonian estimations up to a ten percent. The relevance of non-Newtonian effects for hemolysis parameters is discussed. 12. Experimental and modeling study of Newtonian and non-Newtonian fluid flow in pore network micromodels. Science.gov (United States) Perrin, Christian L; Tardy, Philippe M J; Sorbie, Ken S; Crawshaw, John C 2006-03-15 The in situ rheology of polymeric solutions has been studied experimentally in etched silicon micromodels which are idealizations of porous media. The rectangular channels in these etched networks have dimensions typical of pore sizes in sandstone rocks. Pressure drop/flow rate relations have been measured for water and non-Newtonian hydrolyzed-polyacrylamide (HPAM) solutions in both individual straight rectangular capillaries and in networks of such capillaries. Results from these experiments have been analyzed using pore-scale network modeling incorporating the non-Newtonian fluid mechanics of a Carreau fluid. Quantitative agreement is seen between the experiments and the network calculations in the Newtonian and shear-thinning flow regions demonstrating that the 'shift factor,'alpha, can be calculated a priori. Shear-thickening behavior was observed at higher flow rates in the micromodel experiments as a result of elastic effects becoming important and this remains to be incorporated in the network model. 13. The effect of non-Newtonian viscosity on the stability of the Blasius boundary layer Science.gov (United States) Griffiths, P. T.; Gallagher, M. T.; Stephen, S. O. 2016-07-01 We consider, for the first time, the stability of the non-Newtonian boundary layer flow over a flat plate. Shear-thinning and shear-thickening flows are modelled using a Carreau constitutive viscosity relationship. The boundary layer equations are solved in a self-similar fashion. A linear asymptotic stability analysis, that concerns the lower-branch structure of the neutral curve, is presented in the limit of large Reynolds number. It is shown that the lower-branch mode is destabilised and stabilised for shear-thinning and shear-thickening fluids, respectively. Favourable agreement is obtained between these asymptotic predictions and numerical results obtained from an equivalent Orr-Sommerfeld type analysis. Our results indicate that an increase in shear-thinning has the effect of significantly reducing the value of the critical Reynolds number, this suggests that the onset of instability will be significantly advanced in this case. This postulation, that shear-thinning destabilises the boundary layer flow, is further supported by our calculations regarding the development of the streamwise eigenfunctions and the relative magnitude of the temporal growth rates. 14. ANALYSIS OF MARANGONI CONVECTION OF NON-NEWTONIAN POWER LAW FLUIDS WITH LINEAR TEMPERATURE DISTRIBUTION Directory of Open Access Journals (Sweden) Yan Zhang 2011-01-01 Full Text Available The problem of steady, laminar, thermal Marangoni convection flow of non-Newtonian power law fluid along a horizontal surface with variable surface temperature is studied. The partial differential equations are transformed into ordinary differential equations by using a suitable similarity transformation and analytical approximate solutions are obtained by an efficient transformation, asymptotic expansion and Padé approximants technique. The effects of power law index and Marangoni number on velocity and temperature profiles are examined and discussed. 15. Studies on heat transfer to Newtonian and non-Newtonian fluids in agitated vessel Science.gov (United States) Triveni, B.; Vishwanadham, B.; Venkateshwar, S. 2008-09-01 Heat transfer studies to Newtonian and non-Newtonian fluids are carried out in a stirred vessel fitted with anchor/turbine impeller and a coil for heating/cooling with an objective of determining experimentally the heat transfer coefficient of few industrially important systems namely castor oil and its methyl esters, soap solution, CMC and chalk slurries. The effect of impeller geometry, speed and aeration is investigated. Generalized Reynolds and Prandtl numbers are calculated using an apparent viscosity for non-Newtonian fluids. The data is correlated using a Sieder-Tate type equation. A trend of increase in heat transfer coefficient with RPM in presence and absence of solids has been observed. Relatively high values of Nusselt numbers are obtained for non-Newtonian fluids when aeration is coupled with agitation. The contribution of natural convection to heat transfer has been accounted for by incorporating the Grashof number. The correlations developed based on these studies are applied for design of commercial scale soponification reactor. Power per unit volume resulted in reliable design of a reactor. 16. CFD simulation of non-Newtonian fluid flow in anaerobic digesters. Science.gov (United States) Wu, Binxin; Chen, Shulin 2008-02-15 A general mathematical model that predicts the flow fields in a mixed-flow anaerobic digester was developed. In this model, the liquid manure was assumed to be a non-Newtonian fluid, and the flow governed by the continuity, momentum, and k-epsilon standard turbulence equations, and non-Newtonian power law model. The commercial computational fluid dynamics (CFD) software, Fluent, was applied to simulate the flow fields of lab-scale, scale-up, and pilot-scale anaerobic digesters. The simulation results were validated against the experimental data from literature. The flow patterns were qualitatively compared for Newtonian and non-Newtonian fluids flow in a lab-scale digester. Numerical simulations were performed to predict the flow fields in scale-up and pilot-scale anaerobic digesters with different water pump power inputs and different total solid concentration (TS) in the liquid manure. The optimal power inputs were determined for the pilot-scale anaerobic digester. Some measures for reducing dead and low velocity zones were proposed based upon the CFD simulation results. 17. Theoretical aspects of non-newtonian fluids flow simulation in food technologies Directory of Open Access Journals (Sweden) E. Biletskii 2015-05-01 Full Text Available Introduction. The problems of simulating viscoplastic longitudinal and cross-sectional flow of non-Newtonian fluids are overviewed. Materials and methods. For the first time the superposition method by expressing the components of the stress tensor for building flow fields with higher dimension from flow fields with lower dimension with various boundary conditions when rheological parameters change depending on pressure was used. The flows in the channel are categorized by velocity and pressure values in each point of the section. Results.The theoretical methods for simulating flows of non-Newtonian fluids in channels of different geometry with moving bounds and pressure drop on channel edges with respect to functional connections between main process parameters are described using the superposition method. It is shown that longitudinal and cross-sectional are reduced to the collection of one-dimensional longitudinal flows of the same type which allow to describe three-dimensional isothermal in rectangular channel and two-dimensional flows in flat channels with different channel aspect ratio. The received theoretical two- and three-dimensional model of viscous flows in channels with basic geometry allow to research main regularities of the process and to establish optimal macro-kinetic and macro-dynamic flow characteristics of non-Newtonian materials which are aimed at reducing energy costs and material consumption of food processing equipment. Conclusion.The developed and theoretically reasonable three-dimensional models flows of non-Newtonian fluids in channels allow to perform qualitatively new design of food processing equipment which allows to reduce energy costs and material consumption. 18. Mass transport in a porous microchannel for non-Newtonian fluid with electrokinetic effects. Science.gov (United States) Mondal, Sourav; De, Sirshendu 2013-03-01 Quantification of mass transfer in porous microchannel is of paramount importance in several applications. Transport of neutral solute in presence of convective-diffusive EOF having non-Newtonian rheology, in a porous microchannel was presented in this article. The governing mass transfer equation coupled with velocity field was solved along with associated boundary conditions using a similarity solution method. An analytical solution of mass transfer coefficient and hence, Sherwood number were derived from first principles. The corresponding effects of assisting and opposing pressure-driven flow and EOF were also analyzed. The influence of wall permeation, double-layer thickness, rheology, etc. on the mass transfer was also investigated. Permeation at the wall enhanced the mass transfer coefficient more than five times compared to impervious conduit in case of pressure-driven flow assisting the EOF at higher values of κh. Shear thinning fluid exhibited more enhancement of Sherwood number in presence of permeation compared to shear thickening one. The phenomenon of stagnation was observed at a particular κh (∼2.5) in case of EOF opposing the pressure-driven flow. This study provided a direct quantification of transport of a neutral solute in case of transdermal drug delivery, transport of drugs from blood to target region, etc. 19. Wall effects on the terminal velocity of spherical particles in Newtonian and non-Newtonian fluids Directory of Open Access Journals (Sweden) ATAÍDE C. H. 1999-01-01 Full Text Available The objective of this work is to study the effect of confining walls on the free settling of spherical particles along the axes of five vertical cylindrical tubes in Newtonian and non-Newtonian liquids. Experimental results were predominantly obtained in the particle flow region between the Stokes and the Newton regimes (intermediate region and displayed Reynolds numbers in the ranges 0.7non-Newtonian fluids. 20. Mixed convection effects on heat and mass transfer in a non Newtonian fluid with chemical reaction over a vertical plate Institute of Scientific and Technical Information of China (English) 2011-01-01 This paper studies mixed convection,double dispersion and chemical reaction effects on heat and mass transfer in a non-Darcy non-Newtonian fluid over a vertical surface in a porous medium under the constant temperature and concentration.The governing boundary layer equations,namely,momentum,energy and concentration,are converted to ordinary differential equations by introducing similarity variables and then are solved numerically by means of fourth-order Runge-Kutta method coupled with double-shooting techn... 1. Hodographic study of non-Newtonian MHD aligned steady plane fluid flows Directory of Open Access Journals (Sweden) P. V. Nguyen 1990-01-01 Full Text Available A study is made of non-Newtonian HHD aligned steady plane fluid flows to find exact solutions for various flow configurations. The equations of motion have been transformed to the hodograph plane. A Legendre-transform function is used to recast the equations in the hodograph plane in terms of this transform function. Solutions for various flow configurations are obtained. Applications are investigated for the fluids of finite and infinite electrical conductivity bringing out the similarities and contrasts in the solutions of these types of fluids. 2. Viscoelastic fluid-structure interaction between a non-Newtonian fluid flow and flexible cylinder Science.gov (United States) Dey, Anita; Modarres-Sadeghi, Yahya; Rothstein, Jonathan 2016-11-01 It is well known that when a flexible or flexibly-mounted structure is placed perpendicular to the flow of a Newtonian fluid, it can oscillate due to the shedding of separated vortices at high Reynolds numbers. If the same flexible object is placed in non-Newtonian flows, however, the structure's response is still unknown. Unlike Newtonian fluids, the flow of viscoelastic fluids can become unstable at infinitesimal Reynolds numbers due to a purely elastic flow instability. In this talk, we will present a series of experiments investigating the response of a flexible cylinder placed in the cross flow of a viscoelastic fluid. The elastic flow instabilities occurring at high Weissenberg numbers can exert fluctuating forces on the flexible cylinder thus leading to nonlinear periodic oscillations of the flexible structure. These oscillations are found to be coupled to the time-dependent state of viscoelastic stresses in the wake of the flexible cylinder. The static and dynamic responses of the flexible cylinder will be presented over a range of flow velocities, along with measurements of velocity profiles and flow-induced birefringence, in order to quantify the time variation of the flow field and the state of stress in the fluid. 3. Squeeze film lubrication for non-Newtonian fluids with application to manual medicine. Science.gov (United States) Chaudhry, Hans; Bukiet, Bruce; Roman, Max; Stecco, Antonio; Findley, Thomas 2013-01-01 In this paper, we computed fluid pressure and force on fascia sheets during manual therapy treatments using Squeeze Film Lubrication theory for non-Newtonian fluids. For this purpose, we developed a model valid for three dimensional fluid flow of a non-Newtonian liquid. Previous models considered only one-dimensional flows in two dimensions. We applied this model to compare the one-dimensional flow of HA, considered as a lubricating fluid, around or within the fascia during sliding, vibration, and back-and-forth sliding manipulation treatment techniques. The fluid pressure of HA increases dramatically as fascia is deformed during manual therapies. The fluid force increases more during vertical vibratory manipulation treatment than in constant sliding, and back and forth motion. The variation of fluid pressure/force causes HA to flow near the edges of the fascial area under manipulation in sliding and back and forth motion which may result in greater lubrication. The fluid pressure generated in manual therapy techniques may improve sliding and permit muscles to work more efficiently. 4. Analysis of HD Journal Bearings Considering Elastic Deformation and Non-Newtonian Rabinowitsch Fluid Model Directory of Open Access Journals (Sweden) J. Javorova 2016-06-01 Full Text Available The purpose of this paper is to study the performance of a finite length journal bearing, taking into account effects of non-Newtonian Rabinowitsch flow rheology and elastic deformations of the bearing liner. According to the Rabinowitsch fluid model, the cubic-stress constitutive equation is used to account for the non-Newtonian effects of pseudoplastic and dilatant lubricants. Integrating the continuity equation across the film, the nonlinear non-Newtonian Reynolds-type equation is derived. The elasticity part of the problem is solved on the base of Vlassov model of an elastic foundation. The numerical solution of the modified Reynolds equation is carried out by using FDM with over-relaxation technique. The results for steady state bearing performance characteristics have been calculated for various values of nonlinear factor and elasticity parameters. It was concluded that in comparison with the Newtonian lubricants, higher values of film pressure and load carrying capacity have been obtained for dilatant lubricants, while the case was reversed for pseudoplastic lubricants. 5. Learning about non-Newtonian fluids in a student-driven classroom CERN Document Server Dounas-Frazer, D R; Zaniewski, A M; Roth, N 2012-01-01 We describe a simple, low-cost experiment and corresponding pedagogical strategies for studying fluids whose viscosities depend on shear rate, referred to as non-Newtonian fluids. We developed these materials teaching for the Compass Project, an organization that fosters a creative, diverse, and collaborative community of science students at UC Berkeley. Incoming freshmen worked together in a week-long, residential program to explore physical phenomena through a combination of conceptual model-building and hands-on experimentation. During the program, students were exposed to three major aspects of scientific discovery: developing a model, testing the model, and investigating deviations from the model. 6. Experimental study on the special shear thinning process of a kind of non-Newtonian fluid Institute of Scientific and Technical Information of China (English) CHEN HaoSheng; CHEN DaRong; WANG JiaDao; LI YongJian 2007-01-01 To study the effect of long chain molecule and surface active agent on non-Newtonian fluid properties, rheological experiments on two different fluids have been done. The first group of the fluid is the hydroxyethyl cellulose water solution, and the second is the water solution containing the mixture of dodecyltriethyl ammonium bromide and lauryl sodium sulfate. With the increasing shear rate, shear thinning phenomenon appears in the first group of solution, and a special shear thickening-shear thinning phenomenon appears in the second group. It is considered that the special rheological phenomenon is caused by the difference between the aggregating and the departing speed of the colloidal particles formed in the fluid. The difference between the two speeds relates with the shear rate. The experiment results indicate that the rheological properties can be designed by choosing proper additives at a certain shear rate, and such a fluid with special viscosity variation should be included in the classification of the non-Newtonian fluid. 7. Experimental study on the special shear thinning process of a kind of non-Newtonian fluid Institute of Scientific and Technical Information of China (English) 2007-01-01 To study the effect of long chain molecule and surface active agent on non-Newtonian fluid properties, rheological experiments on two different fluids have been done. The first group of the fluid is the hydroxyethyl cellulose water solution, and the second is the water solution containing the mixture of dodecyl- triethyl ammonium bromide and lauryl sodium sulfate. With the increasing shear rate, shear thinning phenomenon appears in the first group of solution, and a spe- cial shear thickening-shear thinning phenomenon appears in the second group. It is considered that the special rheological phenomenon is caused by the difference between the aggregating and the departing speed of the colloidal particles formed in the fluid. The difference between the two speeds relates with the shear rate. The experiment results indicate that the rheological properties can be designed by choosing proper additives at a certain shear rate, and such a fluid with special vis- cosity variation should be included in the classification of the non-Newtonian fluid. 8. Study on local resistance of non-Newtonian power law fluid in elbow pipes Science.gov (United States) Zhang, Hao; Xu, Tiantian; Zhang, Xinxin; Wang, Yuxiang; Wang, Yuancheng; Liu, Xueting 2016-06-01 This paper focuses on the flow characteristic and local resistance of non-Newtonian power law fluid in a curved 90° bend pipe with circular cross-sections, which are widely used in industrial applications. By employing numerical simulation and theoretical analysis the properties of the flow and local resistance of power law fluid under different working conditions are obtained. To explore the change rule the experiment is carried out by changing the Reynolds number, the wall roughness and different diameter ratio of elbow pipe. The variation of the local resistance coefficient with the Reynolds number, the diameter ratio and the wall roughness is presented comprehensively in the paper. The results show that the local resistance force coefficient hardly changes with Reynolds number of the power law fluid; the wall roughness has a significant impact on the local resistance coefficient. As the pipe wall roughness increasing, the coefficient of local resistance force will increase. The main reason of the influence of the roughness on the local resistance coefficient is the increase of the eddy current region in the power law fluid flow, which increases the kinetic energy dissipation of the main flow. This paper provides theoretical and numerical methods to understand the local resistance property of non-Newtonian power law fluid in elbow pipes. 9. Numerical analysis of dynamic electro-osmotic flows of non-Newtonian fluids in rectangular microchannels CERN Document Server Zhao, Cunlu 2010-01-01 Numerical analyses of transient electro-osmosis of a typical non-Newtonian liquid induced by DC and AC electric fields in a rectangular microchannel are conducted in the framework of continuum fluid mechanics. The famous power-law constitutive model is used to express the fluid dynamic viscosity in terms of the velocity gradient. Transient start-up characteristics of electro-osmotic power-law liquid flow in rectangular microchannels are simulated by using finite element method. Under a DC electric field, it is found out and the fluid is more inert to the external electric field and the steady-state velocity profile becomes more plug-like with decrease of the flow behavior index of the power-law liquids. The numerical calculations also confirm the validity of the generalized Smoluchowski slip velocity which can serve as the counterpart for the classic Smoluchowski slip velocity when dealing with electrokinetic flow of non-Newtonian power-law fluids. Under AC electric fields, the fluid is more obviously acceler... 10. Spreading of completely wetting, non-Newtonian fluids with non-power-law rheology. Science.gov (United States) Min, Qi; Duan, Yuan-Yuan; Wang, Xiao-Dong; Liang, Zhan-Peng; Lee, Duu-Jong; Su, Ay 2010-08-01 Spreading non-Newtonian liquids with non-power-law rheology on completely wetting surfaces are seldom investigated. This study assessed the wetting behavior of polydimethylsiloxane (PDMS), a Newtonian fluid, two carboxymethylcellulose (CMC) sodium solutions, a PDMS+2%w/w silica nanoparticle suspension and three polyethylene glycol (PEG400)+5-10%w/w silica nanoparticle suspensions (non-power-law fluids) on a mica surface. The theta(D)-U and R-t data for spreading drops of the six tested, non-power-law fluids can be described by power-law wetting models. We propose that this behavior is attributable to a uniform shear rate (a few tens to a few hundreds of s(-1)) distributed over the thin-film regime that controls spreading dynamics. Estimated film thickness was below the resolution of an optical microscope for direct observation. Approximating a general non-Newtonian fluid spreading as a power-law fluid greatly simplifies theoretical analysis and data interpretation. 11. Unsteady MHD Slip Flow of a Non-Newtonian Casson Fluid due to Stretching Sheet with Suction or Blowing Effect Directory of Open Access Journals (Sweden) A Mahdy 2016-01-01 Full Text Available In this contribution a numerical study is carried out to analyze the effect of slip at the boundary of unsteady two-dimensional MHD flow of a non-Newtonian fluid over a stretching surface having a prescribed surface temperature in the presence of suction or blowing at the surface. Casson fluid model is used to characterize the non-Newtonian fluid behavior. With the help of similarity transformations, the governing partial differential equations corresponding to the momentum and heat transfer are reduced to a set of non-linear ordinary differential equations, which are then solved for local similar solutions using the very robust computer algebra software MATLAB. The flow features and heat transfer characteristics for different values of the governing parameters are graphically presented and discussed in detail. Comparison with available results for certain cases is excellent. The effect of increasing values of the Casson parameter is seen to suppress the velocity field. But the temperature is enhanced with increasing Casson parameter. For increasing slip parameter, velocity increases and thermal boundary layer becomes thinner in the case of suction or blowing. 12. Computational fluid dynamics investigation of turbulence models for non-newtonian fluid flow in anaerobic digesters. Science.gov (United States) Wu, Binxin 2010-12-01 In this paper, 12 turbulence models for single-phase non-newtonian fluid flow in a pipe are evaluated by comparing the frictional pressure drops obtained from computational fluid dynamics (CFD) with those from three friction factor correlations. The turbulence models studied are (1) three high-Reynolds-number k-ε models, (2) six low-Reynolds-number k-ε models, (3) two k-ω models, and (4) the Reynolds stress model. The simulation results indicate that the Chang-Hsieh-Chen version of the low-Reynolds-number k-ε model performs better than the other models in predicting the frictional pressure drops while the standard k-ω model has an acceptable accuracy and a low computing cost. In the model applications, CFD simulation of mixing in a full-scale anaerobic digester with pumped circulation is performed to propose an improvement in the effective mixing standards recommended by the U.S. EPA based on the effect of rheology on the flow fields. Characterization of the velocity gradient is conducted to quantify the growth or breakage of an assumed floc size. Placement of two discharge nozzles in the digester is analyzed to show that spacing two nozzles 180° apart with each one discharging at an angle of 45° off the wall is the most efficient. Moreover, the similarity rules of geometry and mixing energy are checked for scaling up the digester. 13. Experimental Investigation and Pore-Scale Modeling of Non-Newtonian Fluid Flow in Porous Media Science.gov (United States) Hauswirth, S.; Dye, A. L.; Miller, C. T.; Tapscott, C.; Schultz, P. B. 2015-12-01 Systems involving the flow of non-Newtonian fluids in porous media arise in a number of settings, including hydraulic fracturing, enhanced oil recovery, contaminant remediation, and biological systems. Development of accurate macroscale models of such systems requires an understanding of the relationship between the fluid and medium properties at the microscale and averaged macroscale properties. This study investigates the flow of aqueous solutions of guar gum, a major component of hydraulic fracturing fluids that exhibits Cross model rheological behavior. The rheological properties of solutions containing varying concentrations of guar gum were characterized using a rotational rheometer and the data were fit to a model relating viscosity to shear rate and concentration. Flow experiments were conducted in a porous medium-packed column to measure the pressure response during the flow of guar gum solutions at a wide range of flow rates and determine apparent macroscale viscosities and shear rates. To investigate the relationship between the fluid rheology, microscale physics, and the observed macroscale properties, a lattice Boltzmann pore scale simulator incorporating non-Newtonian behavior was developed. The model was validated, then used to simulate systems representative of the column experiments, allowing direct correlation of detailed microscale physics to the macroscale observations. 14. Pullback Asymptotic Behavior of Solutions for a 2D Non-autonomous Non-Newtonian Fluid Science.gov (United States) Liu, Guowei 2016-10-01 This paper studies the pullback asymptotic behavior of solutions for the non-autonomous incompressible non-Newtonian fluid in 2D bounded domains. Firstly, with a little high regularity of the force, the semigroup method and ɛ -regularity method are used to establish the existence of compact pullback absorbing sets. Then, with a minimal regularity of the force, by verifying the flattening property also known as the "Condition (C)", the author proves the existence of pullback attractors for the universe of fixed bounded sets and for the another universe given by a tempered condition. Furthermore, the regularity of pullback attractors is given. 15. Acoustic waveform of continuous bubbling in a non-Newtonian fluid. Science.gov (United States) Vidal, Valérie; Ichihara, Mie; Ripepe, Maurizio; Kurita, Kei 2009-12-01 We study experimentally the acoustic signal associated with a continuous bubble bursting at the free surface of a non-Newtonian fluid. Due to the fluid rheological properties, the bubble shape is elongated, and, when bursting at the free surface, acts as a resonator. For a given fluid concentration, at constant flow rate, repetitive bubble bursting occurs at the surface. We report a modulation pattern of the acoustic waveform through time. Moreover, we point out the existence of a precursor acoustic signal, recorded on the microphone array, previous to each bursting. The time delay between this precursor and the bursting signal is well correlated with the bursting signal frequency content. Their joint modulation through time is driven by the fluid rheology, which strongly depends on the presence of small satellite bubbles trapped in the fluid due to the yield stress. 16. Force effects on rotor of squeeze film damper using Newtonian and non-Newtonian fluid Science.gov (United States) Dominik, Šedivý; Petr, Ferfecki; Simona, Fialová 2017-09-01 This article presents the evaluation of force effects on rotor of squeeze film damper. Rotor is eccentric placed and its motion is translate-circular. The amplitude of rotor motion is smaller than its initial eccentricity. The force effects are calculated from pressure and viscous forces which were gained by using computational modeling. Two types of fluid were considered as filling of damper. First type of fluid is Newtonian (has constant viscosity) and second type is magnetorheological fluid (does not have constant viscosity). Viscosity of non-Newtonian fluid is given using Bingham rheology model. Yield stress is a function of magnetic induction which is described by many variables. The most important variables of magnetic induction are electric current and gap width which is between rotor and stator. Comparison of application two given types of fluids is shown in results. 17. Hydromagnetic Non-Darcian Free-Convective Flow of a Non-Newtonian Fluid with Temperature Jump Directory of Open Access Journals (Sweden) Ahmed M. Salem 2013-01-01 Full Text Available In the present study, the effect of viscous dissipation on magnetohydrodynamic (MHD non-Darcian free-convection flow of a non-Newtonian power-law fluid past a vertical flat plate in a saturated porous medium with variable viscosity and temperature jump is considered. The fluid is permeated by a transverse magnetic field imposed perpendicularly to the plate on the assumption of a small magnetic Reynolds number. The fluid viscosity is assumed to vary as a reciprocal of linear function of temperature. The governing boundary layer equations and boundary conditions are cast into a dimensionless form and simplified by using a similarity transformation into a system of nonlinear ordinary differential equations and solved numerically. The effects of the governing parameters on the flow fields and heat transfer are shown in graphs and tabular form. 18. Single-Phase Flow of Non-Newtonian Fluids in Porous Media CERN Document Server Sochi, Taha 2009-01-01 The study of flow of non-Newtonian fluids in porous media is very important and serves a wide variety of practical applications in processes such as enhanced oil recovery from underground reservoirs, filtration of polymer solutions and soil remediation through the removal of liquid pollutants. These fluids occur in diverse natural and synthetic forms and can be regarded as the rule rather than the exception. They show very complex strain and time dependent behavior and may have initial yield-stress. Their common feature is that they do not obey the simple Newtonian relation of proportionality between stress and rate of deformation. Non-Newtonian fluids are generally classified into three main categories: time-independent whose strain rate solely depends on the instantaneous stress, time-dependent whose strain rate is a function of both magnitude and duration of the applied stress and viscoelastic which shows partial elastic recovery on removal of the deforming stress and usually demonstrates both time and str... 19. Non-Newtonian fluid model incorporated into elastohydrodynamic lubrication of rectangular contacts Science.gov (United States) Jacobson, B. O.; Hamrock, B. J. 1984-01-01 A procedure is outlined for the numerical solution of the complete elastohydrodynamic lubrication of rectangular contacts incorporating a non-Newtonian fluid model. The approach uses a Newtonian model as long as the shear stress is less than a limiting shear stress. If the shear stress exceeds the limiting value, the shear stress is set equal to the limiting value. The numerical solution requires the coupled solution of the pressure, film shape, and fluid rheology equations from the inlet to the outlet. Isothermal and no-side-leakage assumptions were imposed in the analysis. The influence of dimensionless speed, load, materials, and sliding velocity and limiting-shear-strength proportionality constant on dimensionless minimum film thickness was investigated. Fourteen cases were used in obtaining the minimum-film-thickness equation for an elastohydrodynamically lubricated rectangular contact incorporating a non-Newtonian fluid model. Computer plots are also presented that indicate in detail pressure distribution, film shape, shear stress at the surfaces, and flow throughout the conjunction. 20. CFD simulation of gas and non-Newtonian fluid two-phase flow in anaerobic digesters. Science.gov (United States) Wu, Binxin 2010-07-01 This paper presents an Eulerian multiphase flow model that characterizes gas mixing in anaerobic digesters. In the model development, liquid manure is assumed to be water or a non-Newtonian fluid that is dependent on total solids (TS) concentration. To establish the appropriate models for different TS levels, twelve turbulence models are evaluated by comparing the frictional pressure drops of gas and non-Newtonian fluid two-phase flow in a horizontal pipe obtained from computational fluid dynamics (CFD) with those from a correlation analysis. The commercial CFD software, Fluent12.0, is employed to simulate the multiphase flow in the digesters. The simulation results in a small-sized digester are validated against the experimental data from literature. Comparison of two gas mixing designs in a medium-sized digester demonstrates that mixing intensity is insensitive to the TS in confined gas mixing, whereas there are significant decreases with increases of TS in unconfined gas mixing. Moreover, comparison of three mixing methods indicates that gas mixing is more efficient than mixing by pumped circulation while it is less efficient than mechanical mixing. 1. Pressure falloff behavior in vertically fractured wells: Non-Newtonian power-law fluids Energy Technology Data Exchange (ETDEWEB) Vongvuthipornchai, S.; Raghauan, R.; Reynolds, A.C. 1984-09-01 This paper examines pressure falloff behavior in fractured wells following the injection of a non-Newtonian power-law fluid. Results are presented in a form suitable for field application. Responses at wells intercepting infinite-conductivity and uniformflux fractures are considered. Procedures to identify flow regimes are discussed. The solutions presented here are new and to our knowledge not available in the literature. The consequences of neglecting the non-Newtonian characteristics of the injected fluid are examined. The results of this work were obtained by a finite difference model. Procedures to compute the apparent viscosity of power-law fluids for twodimensional flow through porous media are discussed. The formulation given here avoids numerical problems (multiple solutions, cross over, etc.) reported in other studies. Although, the main objective of the work is to examine pressure falloff behavior at fractured wells, the authors also examine responses at unfractured wells. The main objective of this part of a study is to examine the validity of using the superposition principle to analyze pressure falloff data. (The pressure distribution for this problem is governed by a nonlinear partial differential equation.) If the solutions given in the literature are used, then correction factors are needed to analyze pressure falloff data. The results of this phase of the work can also be used to analyze data in fractured wells provided that pseudoradial flow conditions exist. 2. Non-Newtonian Effects of Second-Order Fluids on the Hydrodynamic Lubrication of Inclined Slider Bearings. Science.gov (United States) 2014-01-01 Theoretical study of non-Newtonian effects of second-order fluids on the performance characteristics of inclined slider bearings is presented. An approximate method is used for the solution of the highly nonlinear momentum equations for the second-order fluids. The closed form expressions for the fluid film pressure, load carrying capacity, frictional force, coefficient of friction, and centre of pressure are obtained. The non-Newtonian second order fluid model increases the film pressure, load carrying capacity, and frictional force whereas the center of pressure slightly shifts towards exit region. Further, the frictional coefficient decreases with an increase in the bearing velocity as expected for an ideal fluid. 3. Development and Implementation of Non-Newtonian Rheology Into the Generalized Fluid System Simulation Program (GFSSP) Science.gov (United States) DiSalvo, Roberto; Deaconu, Stelu; Majumdar, Alok 2006-01-01 One of the goals of this program was to develop the experimental and analytical/computational tools required to predict the flow of non-Newtonian fluids through the various system components of a propulsion system: pipes, valves, pumps etc. To achieve this goal we selected to augment the capabilities of NASA's Generalized Fluid System Simulation Program (GFSSP) software. GFSSP is a general-purpose computer program designed to calculate steady state and transient pressure and flow distributions in a complex fluid network. While the current version of the GFSSP code is able to handle various systems components the implicit assumption in the code is that the fluids in the system are Newtonian. To extend the capability of the code to non-Newtonian fluids, such as silica gelled fuels and oxidizers, modifications to the momentum equations of the code have been performed. We have successfully implemented in GFSSP flow equations for fluids with power law behavior. The implementation of the power law fluid behavior into the GFSSP code depends on knowledge of the two fluid coefficients, n and K. The determination of these parameters for the silica gels used in this program was performed experimentally. The n and K parameters for silica water gels were determined experimentally at CFDRC's Special Projects Laboratory, with a constant shear rate capillary viscometer. Batches of 8:1 (by weight) water-silica gel were mixed using CFDRC s 10-gallon gelled propellant mixer. Prior to testing the gel was allowed to rest in the rheometer tank for at least twelve hours to ensure that the delicate structure of the gel had sufficient time to reform. During the tests silica gel was pressure fed and discharged through stainless steel pipes ranging from 1", to 36", in length and three diameters; 0.0237", 0.032", and 0.047". The data collected in these tests included pressure at tube entrance and volumetric flowrate. From these data the uncorrected shear rate, shear stress, residence time 4. Non-newtonian fluid flow through three-dimensional disordered porous media. Science.gov (United States) Morais, Apiano F; Seybold, Hansjoerg; Herrmann, Hans J; Andrade, José S 2009-11-06 We investigate the flow of various non-newtonian fluids through three-dimensional disordered porous media by direct numerical simulation of momentum transport and continuity equations. Remarkably, our results for power-law (PL) fluids indicate that the flow, when quantified in terms of a properly modified permeability-like index and Reynolds number, can be successfully described by a single (universal) curve over a broad range of Reynolds conditions and power-law exponents. We also study the flow behavior of Bingham fluids described in terms of the Herschel-Bulkley model. In this case, our simulations reveal that the interplay of (i) the disordered geometry of the pore space, (ii) the fluid rheological properties, and (iii) the inertial effects on the flow is responsible for a substantial enhancement of the macroscopic hydraulic conductance of the system at intermediate Reynolds conditions. 5. ON THE FILTRATION OF NON-NEWTONIAN FLUID IN POROUS MEDIA WITH A MULTIPLE PARAMETER MODEL Institute of Scientific and Technical Information of China (English) 2001-01-01 A multiple parameter model to describe the Non-Newtonianproperties of fluid filtration in porous media is presented with regard to the pressure gradient expression in terms of the velocity of filtration, where the multiple parameters should be determined by measurements. Based on such a model, an analysis was furnished to deduce the formula for the rate of production of a oil well, and the governing equations for single phase Non-Newtonian fluid fritration. In order to examine the effects of model parameters, the governing equations were numerically solved with the method of cross-diagonal decomposition ZG method. It is found that, for constant rate of production, the power index n of the model influences the pressure distribution considerably, particularly in the vicinity of a single well. The well-bore pressure of Leibenzonian fluid is lower than that of the power-law fluid in the case of the same parameter B and the power index n = 0.5. 6. Coalescence of drops and bubbles rising through a non-Newtonian fluid in a tube. Science.gov (United States) Al-Matroushi, Eisa; Borhan, Ali 2009-04-01 We conducted an experimental study of the interaction and coalescence of two drops (of the same fluid) or bubbles translating under the action of buoyancy in a cylindrical tube. The close approach of two Newtonian fluid particles of different size in a non-Newtonian continuous phase was examined using image analysis, and measurements of the coalescence time are reported for various particle size ratios, Bond numbers, and particle-to-suspending-fluid viscosity ratios. The flow disturbance behind the leading bubble and the viscoelastic nature of the continuous phase seemed to retard bubble coalescence. The time scale for coalescence of liquid drops in highly elastic continuous phase was influenced by the relative motion of the drops and their coalescence behavior. 7. Geometry of elastic hydrofracturing by injection of an over pressured non-Newtonian Fluid CERN Document Server Cerca, Mariano; Barrientos, Bernardino; Soto, Enrique; Mares, Carlos 2009-01-01 The nucleation and propagation of hydrofractures by injection of over pressured fluids in an elastic and isotropic medium are studied experimentally. Non-Newtonian fluids are injected inside a gelatine whose mechanical properties are assumed isotropic at the experimental strain rates. Linear elastic theory predicts that plastic deformation associated to breakage of gelatin bonds is limited to a small zone ahead of the tip of the propagating fracture and that propagation will be maintained while the fluid pressure exceeds the normal stress to the fracture walls (Ch\\'avez-\\'Alvarez,2008) (i.e., the minimum compressive stress), resulting in a single mode I fracture geometry. However, we observed the propagation of fractures type II and III as well as nucleation of secondary fractures, with oblique to perpendicular trajectories with respect to the initial fracture. In the Video (http://hdl.handle.net/1813/14122) experimental evidence shows that the fracture shape depends on the viscoelastic properties of gelatine... 8. Free surface flow of a suspension of rigid particles in a non-Newtonian fluid DEFF Research Database (Denmark) Svec, Oldrich; Skocek, Jan; Stang, Henrik 2012-01-01 efficient, allowing simulations of tens of thousands of rigid particles within a reasonable computational time. Furthermore, the framework does not require any fitting constants or parameters devoid of a clear physical meaning and it is stable, robust and can be easily generalized to a variety of problems......A numerical framework capable of predicting the free surface flow of a suspension of rigid particles in a non-Newtonian fluid is described. The framework is a combination of the lattice Boltzmann method for fluid flow, the mass tracking algorithm for free surface representation, the immersed...... boundary method for two-way coupled interactions between fluid and rigid particles and an algorithm for the dynamics and mutual interactions of rigid particles. The framework is able to simulate the flow of suspensions at the level of the largest suspended particles and, at the same time, the model is very... 9. Validation of computational non-Newtonian fluid model for membrane bioreactor. Science.gov (United States) Sørensen, Lasse; Bentzen, Thomas Ruby; Skov, Kristian 2015-01-01 Membrane bioreactor (MBR) systems are often considered as the wastewater treatment method of the future due to their high effluent quality. One of the main problems with such systems is a relative large energy consumption, compared to conventional activated sludge (CAS) systems, which has led to further research in this specific area. A powerful tool for optimizing MBR-systems is computational fluid dynamics (CFD) modelling, which gives researchers the ability to describe the flow in the systems. A parameter which is often neglected in such models is the non-Newtonian properties of active sludge, which is of great importance for MBR systems since they operate at sludge concentrations up to a factor of 10 compared to CAS systems, resulting in strongly shear thinning liquids. A CFD-model is validated against measurements conducted in a system with rotating cross-flow membranes submerged in non-Newtonian liquids, where tangential velocities are measured with a Laser Doppler Anemometer (LDA). The CFD model is found to be capable of modelling the correct velocities in a range of setups, making CFD models a powerful tool for optimization of MBR systems. 10. Generalized multiscale finite element method for non-Newtonian fluid flow in perforated domain Science.gov (United States) Chung, E. T.; Iliev, O.; Vasilyeva, M. V. 2016-10-01 In this work, we consider a non-Newtonian fluid flow in perforated domains. Fluid flow in perforated domains have a multiscale nature and solution techniques for such problems require high resolution. In particular, the discretization needs to honor the irregular boundaries of perforations. This gives rise to a fine-scale problems with many degrees of freedom which can be very expensive to solve. In this work, we develop a multiscale approach that attempt to solve such problems on a coarse grid by constructing multiscale basis functions. We follow Generalized Multiscale Finite Element Method (GMsFEM) [1, 2] and develop a multiscale procedure where we identify multiscale basis functions in each coarse block using snapshot space and local spectral problems [3, 4]. We show that with a few basis functions in each coarse block, one can accurately approximate the solution, where each coarse block can contain many small inclusions. 11. Microconfined shear deformation of a droplet in an equiviscous non-newtonian immiscible fluid: experiments and modeling. Science.gov (United States) Minale, Mario; Caserta, Sergio; Guido, Stefano 2010-01-05 In this work, the microconfined shear deformation of a droplet in an equiviscous non-Newtonian immiscible fluid is investigated by modeling and experiments. A phenomenological model based on the assumption of ellipsoidal shape and taking into account wall effects is proposed for systems made of non-Newtonian second-order fluids. The model, without any adjustable parameters, is tested by comparison with experiments under simple shear flow performed in a sliding plate apparatus, where the ratio between the distance between the confining walls and the droplet radius can be varied. The agreement between model predictions and experimental data is good both in steady state shear and in transient drop retraction upon cessation of flow. The results obtained in this work are relevant for microfluidics applications where non-Newtonian fluids are used. 12. A Conditionally Stable Scheme for a Transient Flow of a Non-Newtonian Fluid Saturating a Porous Medium KAUST Repository El-Amin, Mohamed 2012-06-02 The problem of thermal dispersion effects on unsteady free convection from an isothermal horizontal circular cylinder to a non-Newtonian fluid saturating a porous medium is examined numerically. The Darcy-Brinkman-Forchheimer model is employed to describe the flow field. The thermal diffusivity coefficient has been assumed to be the sum of the molecular diffusivity and the dynamic diffusivity due to mechanical dispersion. The simultaneous development of the momentum and thermal boundary layers are obtained by using finite difference method. The stability conditions are determined for each difference equation. Using an explicit finite difference scheme, solutions at each time-step have been found and then stepped forward in time until reaching steady state solution. Velocity and temperature profiles are shown graphically. It is found that as time approaches infinity, the values of friction factor and heat transfer coefficient approach the steady state values. 13. Mathematical simulation of nonisothermal filling of plane channel with non-Newtonian fluid Science.gov (United States) Borzenko, E.; Ryltseva, K.; Frolov, O.; Shrager, G. 2016-10-01 In this paper, the fountain flow of a non-Newtonian fluid during the filling of a plane vertical channel with due account of dissipative heating is investigated. The rheological features of the medium are defined by Ostwald de Waele power-law with exponential temperature dependence of viscosity. The numerical solution of the problem is obtained using a finite-difference method, based on the SIMPLE algorithm, and the method of invariants for compliance with the natural boundary conditions on free surface. It was shown that the flow separates into a two-dimensional flow zone in the vicinity of the free surface and a onedimensional flow zone away from it. The parametrical investigations of kinematic and thermophysical properties of the flow and the dependence of the free surface behavior on the basic criteria and rheological parameters are implemented. 14. Existence for a Class of Non-Newtonian Fluids with a Nonlocal Friction Boundary Condition Institute of Scientific and Technical Information of China (English) L.CONSIGLIERI 2006-01-01 We deal with a variational inequality describing the motion of incompressible fluids, whose viscous stress tensors belong to the subdifferential of a functional at the point given by the symmetric part of the velocity gradient, with a nonlocal friction condition on a part of the boundary obtained by a generalized mollification of the stresses. We establish an existence result of a solution to the nonlocal friction problem for this class of non-Newtonian flows. The result is based on the Faedo-Galerkin and Moreau-Yosida methods, the duality theory of convex analysis and the Tychonov-Kakutani-Glicksberg fixed point theorem for multi-valued mappings in an appropriate functional space framework. 15. Modeling of flow of particles in a non-Newtonian fluid using lattice Boltzmann method DEFF Research Database (Denmark) Skocek, Jan; Svec, Oldrich; Spangenberg, Jon 2011-01-01 To predict correctly the castings process of self compacting concrete a numerical model capable of simulating flow patterns at the structural scale and at the same time the impact of the varying volume fraction of aggregates and other phenomena at the scale of aggregates on the flow evolution...... is necessary. In this contribution, the model at the scale of aggregates is introduced. The conventional lattice Boltzmann method for fluid flow is enriched with the immersed boundary method with direct forcing to simulate the flow of rigid particles in a non- Newtonian liquid. Basic ingredients of the model...... are presented and discussed with the emphasis on a newly developed algorithm for the dynamics of particles whose interactions strongly depend on velocities of particles. The application of the model is demonstrated by a parametric study with varying volume fractions of aggregates and speed of shearing used... 16. On the rheology of refractive-index-matched, non-Newtonian blood-analog fluids for PIV experiments Science.gov (United States) Najjari, Mohammad Reza; Hinke, Jessica A.; Bulusu, Kartik V.; Plesniak, Michael W. 2016-06-01 Four commonly used refractive-index (RI)-matched Newtonian blood-analog fluids are reviewed, and different non-Newtonian blood-analogs, with RI of 1.372-1.495, are investigated. Sodium iodide (NaI), sodium thiocyanate (NaSCN) and potassium thiocyanate are used to adjust the RI of blood-analogs to that of test sections for minimizing optical distortions in particle image velocimetry data, and xanthan gum (XG) is added to the fluids to give them non-Newtonian properties (shear thinning and viscoelasticity). Our results support the general belief that adding NaI to Newtonian fluids matches the RI without changing the kinematic viscosity. However, in contrast to claims made in a few studies that did not measure rheology, our investigation revealed that adding NaI or NaSCN to XG-based non-Newtonian fluids changes the viscosity of the fluids considerably and reduces the shear-thinning property. Therefore, the RI of non-Newtonian blood-analog fluids with XG cannot be adjusted easily by varying the concentration of NaI or NaSCN and needs more careful rheological study. 17. A Finite Difference Scheme for Double-Diffusive Unsteady Free Convection from a Curved Surface to a Saturated Porous Medium with a Non-Newtonian Fluid KAUST Repository El-Amin, Mohamed 2011-05-14 In this paper, a finite difference scheme is developed to solve the unsteady problem of combined heat and mass transfer from an isothermal curved surface to a porous medium saturated by a non-Newtonian fluid. The curved surface is kept at constant temperature and the power-law model is used to model the non-Newtonian fluid. The explicit finite difference method is used to solve simultaneously the equations of momentum, energy and concentration. The consistency of the explicit scheme is examined and the stability conditions are determined for each equation. Boundary layer and Boussinesq approximations have been incorporated. Numerical calculations are carried out for the various parameters entering into the problem. Velocity, temperature and concentration profiles are shown graphically. It is found that as time approaches infinity, the values of wall shear, heat transfer coefficient and concentration gradient at the wall, which are entered in tables, approach the steady state values. 18. Non-Newtonian effects of blood on LDL transport inside the arterial lumen and across multi-layered arterial wall with and without stenosis Science.gov (United States) Deyranlou, Amin; Niazmand, Hamid; Sadeghi, Mahmood-Reza; Mesri, Yaser 2016-06-01 Blood non-Newtonian behavior on low-density lipoproteins (LDL) accumulation is analyzed numerically, while fluid-multilayered arteries are adopted for nonstenotic and 30%-60% symmetrical stenosed models. Present model considers non-Newtonian effects inside the lumen and within arterial layers simultaneously, which has not been examined in previous studies. Navier-Stokes equations are solved along with the mass transport convection-diffusion equations and Darcy’s model for species transport inside the luminal flow and across wall layers, respectively. Carreau model for the luminal flow and the modified Darcy equation for the power-law fluid within arterial layers are employed to model blood rheological characteristics, appropriately. Results indicate that in large arteries with relatively high Reynolds number Newtonian model estimates LDL concentration patterns well enough, however, this model seriously incompetent for regions with low WSS. Moreover, Newtonian model for plasma underestimates LDL concentration especially on luminal surface and across arterial wall. Therefore, applying non-Newtonian model seems essential for reaching to a more accurate estimation of LDL distribution in the artery. Finally, blood flow inside constricted arteries demonstrates that LDL concentration patterns along the stenoses inside the luminal flow and across arterial layers are strongly influenced as compared to the nonstenotic arteries. Additionally, among four stenosis severity grades, 40% stenosis is prone to more LDL accumulation along the post-stenotic regions. 19. Drag Force of Non-newtonian Fluid on a Continuous Moving Surface with Strong Suction/Blowing Institute of Scientific and Technical Information of China (English) 郑连存; 张欣欣; 赫冀成 2003-01-01 A theoretical analysis for the laminar boundary layer flow of a non-Newtonian fluid on a continuous moving flat plate with surface strong suction/blowing is made. The types of potential flows necessary for similar solutions to the boundary layer are determined and both analytical and numerical solutions are presented. It is shown that the solution of the boundary layer problem depends not only on the ratio of the velocity of the plate to the velocity of the free stream, but also on the suction/blowing parameter. The skin friction decreases with increasing the parameters of power law and blowing. In the case of existing suction, the shear force decreases with the increases of tangential velocity, the largest shear force occurs at wall and the smallest shear force occurs at the edge of the boundary layer. However, in the case of existing surface blowing, the shear force initially increases with tangentialvelocity and the biggest shear force occurs at the interior of the boundary layer, the skin friction approaches to zero as the blowing rate approaches the critical value. 20. Non-Newtonian fluids: Frictional pressure loss prediction for fully-developed flow in straight pipes Science.gov (United States) 1991-10-01 ESDU 91025 discusses models used to describe the rheology of time independent pseudohomogeneous non-Newtonian fluids (power-law, Bingham, Herschel-Bulkley and a generalized model due to Metzner and Reed); they are used to calculate the laminar flow pressure drop (which is independent of pipe roughness in this regime). Values of a generalized Reynolds number are suggested to define transitional and turbulent flow. For turbulent flow in smooth pipes, pressure loss is estimated on the basis of an experimentally determined rheogram using either the Dodge-Metzner or Bowen approach depending on the available measurements. Bowen requires results for at least two pipe diameters. The choice of Dodge-Metzner when data are limited is discussed; seven possible methods are assessed against five sets of experimental results drawn from the literature. No method is given for transitional flow, which it is suggested should be avoided, but the turbulent correlation is recommended because it will yield an overestimate. Suggestions are made for the treatment of roughness effects. Several worked examples illustrate the use of the methods and a flowchart guides the user through the process from experimentally characterizing the behavior of the fluid to determining the pressure drop. A computer program, ESDUpac A9125, is also provided. 1. Non-Newtonian flow effects on the coalescence and mixing of initially stationary droplets of shear-thinning fluids. Science.gov (United States) Sun, Kai; Wang, Tianyou; Zhang, Peng; Law, Chung K 2015-02-01 The coalescence of two initially stationary droplets of shear-thinning fluids in a gaseous environment is investigated numerically using the lattice Boltzmann method, with particular interest in non-Newtonian flow effects on the internal mixing subsequent to coalescence. Coalescence of equal-sized droplets, with one being Newtonian while the other is non-Newtonian, leads to the non-Newtonian droplet wrapping around the Newtonian one and hence minimal fine-scale mixing. For unequal-sized droplets, mixing is greatly promoted if both droplets are shear-thinning. When only one of the droplets is shear-thinning, the non-Newtonian effect from the smaller droplet is found to be significantly more effective than that from the larger droplet in facilitating internal jetlike mixing. Parametric study with the Carreau-Yasuda model indicates that the phenomena are universal to a wide range of shear-thinning fluids, given that the extent of shear thinning reaches a certain level, and the internal jet tends to be thicker and develops more rapidly with increasing extent of the shear-thinning effect. 2. On Laminar Flow of Non-Newtonian Fluids in Porous Media KAUST Repository Fayed, Hassan E. 2015-10-20 Flow of generalized Newtonian fluids in porous media can be modeled as a bundle of capillary tubes or a pore-scale network. In general, both approaches rely on the solution of Hagen–Poiseuille equation using power law to estimate the variations in the fluid viscosity due to the applied shear rate. Despite the effectiveness and simplicity, power law tends to provide unrealistic values for the effective viscosity especially in the limits of zero and infinite shear rates. Here, instead of using power law, Carreau model (bubbles, drops, and particles in non-Newtonian fluids. Taylor & Francis Group, New York, 2007) is used to determine the effective viscosity as a function of the shear strain rate. Carreau model can predict accurately the variation in the viscosity at all shear rates and provide more accurate solution for the flow physics in a single pore. Using the results for a single pore, normalized Fanning friction coefficient has been calculated and plotted as a function of the newly defined Reynolds number based on pressure gradient. For laminar flow, the variation in the friction coefficient with Reynolds number has been plotted and scaled. It is observed that generalized Newtonian fluid flows show Newtonian nature up to a certain Reynolds number. At high Reynolds number, deviation from the Newtonian behavior is observed. The main contribution of this paper is to present a closed-form solution for the flow in a single pore using Carreau model, which allows for fast evaluation of the relationship between flux and pressure gradient in an arbitrary pore diameter. In this way, we believe that our development will open the perspectives for using Carreau models in pore-network simulations at low computational costs to obtain more accurate prediction for generalized Newtonian fluid flows in porous media. 3. Investigation into the Impact and Buffering Characteristics of a Non-Newtonian Fluid Damper: Experiment and Simulation Directory of Open Access Journals (Sweden) Jingya Sun 2014-01-01 Full Text Available Dampers are widely applied to protect devices or human body from severe impact or harmful vibration circumstances. Considering that dampers with low velocity exponent have advantages in energy absorption, they have been widely used in antiseismic structures and shock buffering. Non-Newtonian fluid with strong shear-thinning effect is commonly adopted to achieve this goal. To obtain the damping mechanism and find convenient methods to design the nonlinear fluid damper, in this study, a hydraulic damper is filled with 500,000 cSt silicone oil to achieve a low velocity exponent. Drop hammer test is carried out to experimentally obtain its impact and buffering characteristics. Then a coupling model is built to analyze its damping mechanism, which consists of a model of impact system and a computational fluid dynamics (CFD model. Results from the coupling model can be consistent with the experiment results. Simulation method can help design non-Newtonian fluid dampers more effectively. 4. Morphological stability of an interface between two non-Newtonian fluids moving in a Hele-Shaw cell. Science.gov (United States) Martyushev, L M; Birzina, A I 2015-01-01 The problem of the morphological stability of an interface in the case of the displacement of one non-Newtonian fluid by another non-Newtonian fluid in a radial Hele-Shaw cell has been considered. Both fluids have been described by the two-parameter Ostwald-de Waele power-law model. The nonzero viscosity of the displacing fluid has been taken into account. A generalized Darcy's law for the system under consideration, as well as an equation for the determination of the critical size of morphological stability with respect to harmonic perturbations (linear analysis), has been derived. Morphological phase diagrams have been constructed, and the region of the parameters in which nonequilibrium reentrant morphological transitions are possible has been revealed. 5. Experimental model for non-Newtonian fluid viscosity estimation: Fit to mathematical expressions Directory of Open Access Journals (Sweden) Guillem Masoliver i Marcos 2017-01-01 Full Text Available The construction process of a viscometer, developed in collaboration with a final project student, is here presented. It is intended to be used by first year's students to know the viscosity as a fluid property, for both Newtonian and non-Newtonian flows. Viscosity determination is crucial for the fluids behaviour knowledge related to their reologic and physical properties. These have great implications in engineering aspects such as friction or lubrication. With the present experimental model device three different fluids are analyzed (water, kétchup and a mixture with cornstarch and water. Tangential stress is measured versus velocity in order to characterize all the fluids in different thermal conditions. A mathematical fit process is proposed to be done in order to adjust the results to expected analytical expressions, obtaining good results for these fittings, with R2 greater than 0.88 in any case. 6. The analysis of pneumatic atomization of Newtonian and non-Newtonian fluids for different medical nebulizers. Science.gov (United States) Ochowiak, Marek; Matuszak, Magdalena; Włodarczak, Sylwia 2017-08-01 The article contains results of the experimental studies on atomization process of inhaled drugs and aqueous solutions of glycerol with aqueous solutions of glycerol polyacrylamide (Rokrysol WF1) in pneumatic nebulizers. In experiments, the different concentration of aqueous solutions of glycerol polyacrylamide have been tested. In addition, the effect of nebulizer design on atomization process has been determined. The one of the main elements of medical pneumatic nebulizer is nebulizer cup. The experiment with this scope is new and is very important from the point of view of aerosol therapy. The results have been obtained by the use of the digital microphotography technique. In order to determine a physicochemical properties of tested liquids, a rheological measurements and measurements of the surface tension were carried out. The differences between characteristics of aerosol for the liquids have been observed. The analysis of the droplets size distributions shows that the different diameters of droplets for Newtonian and non-Newtonian fluids have been formed during atomization in pneumatic nebulizers equipped with different nebulizer cups. The effect of the mouthpiece location on the droplets diameters has been shown. Precise design of nebulizer and nebulizer cups, and also physicochemical properties of atomized liquids are of high importance in order to the effectiveness of drug delivery to patient's respiratory tracts. 7. Study on Flow Characteristic of Non-Newtonian fluid in Eccentric Annulus Directory of Open Access Journals (Sweden) Li Mingzhong 2013-08-01 Full Text Available This study studied the flow characteristic of non-newtonian in eccentric annulus of highly-deviated well. On the basis of dimensionless analysis of motion equations and continuity equation, Hele-Shaw model suitable for fluid flow in the annulus was derived. Combined with H-B rheological model, velocity and stream distribution model were founded and calculated by numerical method. Furthermore, two-dimensional flow characteristic in eccentric annulus was got and the influence of different factors (such as yield stress, pressure gradient or eccentricity on velocity distribution in condition of laminar flow was analyzed. Width of flow core in the annular is proportional to yield stress and inversely proportional to pressure gradient. In eccentric annulus, eccentricity influences the stream distribution remarkably: with the increment of eccentricity, the contour lines of stream function gradually centralize in the widest annular gap, however distribute the most loosely in the narrowest annular gap. Axial velocity is the largest in the widest gap. The larger eccentricity is, the larger contrast of axial velocity between in the widest gap and in the narrowest gap is. There is the largest azimuthal velocity in an annular gap of a certain azimuthal angle, however which equals to zero in the widest and narrowest annular gap separately. The larger eccentricity is, the more homogeneous azimuthal velocity is. The velocity contrast in the entire annulus can be smoothed by increasing pressure gradient, power law index or decreasing yield stress. 8. Gravity driven instabilities in miscible non-Newtonian fluid displacements in porous media Science.gov (United States) Freytes, V. M.; D'Onofrio, A.; Rosen, M.; Allain, C.; Hulin, J. P. 2001-02-01 Gravity driven instabilities in model porous packings of 1 mm diameter spheres are studied by comparing the broadening of the displacement front between fluids of slightly different densities in stable and unstable configurations. Water, water-glycerol and water-polymer solutions are used to vary independently viscosity and molecular diffusion and study the influence of shear-thinning properties. Both injected and displaced solutions are identical but for a different concentration of NaNO 3 salt used as an ionic tracer and to introduce the density contrast. Dispersivity in stable configuration increases with polymer concentration - as already reported for double porosity packings of porous grains. Gravity-induced instabilities are shown to develop below a same threshold Péclet number Pe for water and water-glycerol solutions of different viscosities and result in considerable increases of the dispersivity. Measured threshold Pe values decrease markedly on the contrary with polymer concentration. The quantitative analysis demonstrates that the development of the instabilities is controlled by viscosity through a characteristic gravity number G (ratio between hydrostatic and viscous pressure gradients). A single threshold value of G accounts for results obtained on Newtonian and non-Newtonian solutions. 9. Analysis of the flow of non-Newtonian viscoelastic fluids in fractal reservoir with the fractional derivative Institute of Scientific and Technical Information of China (English) TONG Dengke; WANG Ruihe 2004-01-01 In this paper, fractional order derivative, fractal dimension and spectral dimension are introduced into the seepage flow mechanics to establish the relaxation models of non-Newtonian viscoelastic fluids with the fractional derivative in fractal reservoirs. A new type integral transform is introduced, and the flow characteristics of non-Newtonian viscoelastic fluids with the fractional order derivative through a fractal reservoir are studied by using the integral transform, the discrete Laplace transform of sequential fractional derivatives and the generalized Mittag-Leffler function. Exact solutions are obtained for arbitrary fractional order derivative. The long-time and short-time asymptotic solutions for an infinite formation are also obtained. The pressure transient behavior of non-Newtonian viscoelastic fluids flow through an infinite fractal reservoir is studied by using the Stehfest's inversion method of the numerical Laplace transform. It is shown that the clearer the viscoelastic characteristics of the fluid, the more the fluid is sensitive to the order of the fractional derivative. The new type integral transform provides a new analytical tool for studying the seepage mechanics of fluid in fractal porous media. 10. Dynamics of stochastic non-Newtonian fluids driven by fractional Brownian motion with Hurst parameter $H \\in (1/4,1/2)$ CERN Document Server Li, Jin 2011-01-01 In this paper we consider the Stochastic isothermal, nonlinear, incompressible bipolar viscous fluids driven by a genuine cylindrical fractional Bronwnian motion with Hurst parameter $H \\in (1/4,1/2)$ under Dirichlet boundary condition on 2D square domain. First we prove the existence and regularity of the stochastic convolution corresponding to the stochastic non-Newtonian fluids. Then we obtain the existence and uniqueness results for the stochastic non-Newtonian fluids. Under certain condition, the random dynamical system generated by non-Newtonian fluids has a random attractor. 11. Gravity-Driven Flow of non-Newtonian Fluids in Heterogeneous Porous Media: a Theoretical and Experimental Analysis Science.gov (United States) Di Federico, V.; Longo, S.; Ciriello, V.; Chiapponi, L. 2015-12-01 A theoretical and experimental analysis of non-Newtonian gravity-driven flow in porous media with spatially variable properties is presented. The motivation for our study is the rheological complexity exhibited by several environmental contaminants (wastewater sludge, oil pollutants, waste produced by the minerals and coal industries) and remediation agents (suspensions employed to enhance the efficiency of in-situ remediation). Natural porous media are inherently heterogeneous, and this heterogeneity influences the extent and shape of the porous domain invaded by the contaminant or remediation agent. To grasp the combined effect of rheology and spatial heterogeneity, we consider: a) the release of a thin current of non-Newtonian power-law fluid into a 2-D, semi-infinite and saturated porous medium above a horizontal bed; b) perfectly stratified media, with permeability and porosity varying along the direction transverse (vertical) or parallel (horizontal) to the flow direction. This continuous variation of spatial properties is described by two additional parameters. In order to represent several possible spreading scenarios, we consider: i) instantaneous injection with constant mass; ii) continuous injection with time-variable mass; iii) instantaneous release of a mound of fluid, which can drain freely out of the formation at the origin (dipole flow). Under these assumptions, scalings for current length and thickness are derived in self similar form. An analysis of the conditions on model parameters required to avoid an unphysical or asymptotically invalid result is presented. Theoretical results are validated against multiple sets of experiments, conducted for different combinations of spreading scenarios and types of stratification. Two basic setups are employed for the experiments: I) direct flow simulation in an artificial porous medium constructed superimposing layers of glass beads of different diameter; II) a Hele-Shaw (HS) analogue made of two parallel 12. Entropy analysis of convective MHD flow of third grade non-Newtonian fluid over a stretching sheet Directory of Open Access Journals (Sweden) M.M. Rashidi 2017-03-01 Full Text Available The purpose of this article is to study and analyze the convective flow of a third grade non-Newtonian fluid due to a linearly stretching sheet subject to a magnetic field. The dimensionless entropy generation equation is obtained by solving the reduced momentum and energy equations. The momentum and energy equations are reduced to a system of ordinary differential equations by a similarity method. The optimal homotopy analysis method (OHAM is used to solve the resulting system of ordinary differential equations. The effects of the magnetic field, Biot number and Prandtl number on the velocity component and temperature are studied. The results show that the thermal boundary-layer thickness gets decreased with increasing the Prandtl number. In addition, Brownian motion plays an important role to improve thermal conductivity of the fluid. The main purpose of the paper is to study the effects of Reynolds number, dimensionless temperature difference, Brinkman number, Hartmann number and other physical parameters on the entropy generation. These results are analyzed and discussed. 13. Mathematical modeling of a non-Newtonian fluid flow in the main fracture inside permeable porous media Science.gov (United States) Ilyasov, A. M.; Bulgakova, G. T. 2016-08-01 This paper describes a mathematical model of the main fracture isolation in porous media by water-based mature gels. While modeling injection, water infiltration from the gel pack through fracture walls is taking into account, due to which the polymer concentration changes and the residual water resistance factor changes as a consequence. The salutation predicts velocity and pressure fields of the non-Newtonian incompressible fluid filtration for conditions of a non-deformable formation as well as a gel front trajectory in the fracture. The mathematical model of agent injection into the main fracture is based on the fundamental laws of continuum mechanics conservation describing the flow of non-Newtonian and Newtonian fluids separated by an interface plane in a flat channel with permeable walls. The mathematical model is based on a one-dimensional isothermal approximation, with dynamic parameters pressure and velocity, averaged over the fracture section. 14. Data on the mixing of non-Newtonian fluids by a Rushton turbine in a cylindrical tank. Science.gov (United States) Khapre, Akhilesh; Munshi, Basudeb 2016-09-01 The paper focuses on the data collected from the mixing of shear thinning non-Newtonian fluids in a cylindrical tank by a Rushton turbine. The data presented are obtained by using Computational Fluid Dynamics (CFD) simulation of fluid flow field in the entire tank volume. The CFD validation data for this study is reported in the research article 'Numerical investigation of hydrodynamic behavior of shear thinning fluids in stirred tank' (Khapre and Munshi, 2015) [1]. The tracer injection method is used for the prediction of mixing time and mixing efficiency of a Rushton turbine impeller. 15. Data on the mixing of non-Newtonian fluids by a Rushton turbine in a cylindrical tank Directory of Open Access Journals (Sweden) Akhilesh Khapre 2016-09-01 Full Text Available The paper focuses on the data collected from the mixing of shear thinning non-Newtonian fluids in a cylindrical tank by a Rushton turbine. The data presented are obtained by using Computational Fluid Dynamics (CFD simulation of fluid flow field in the entire tank volume. The CFD validation data for this study is reported in the research article ‘Numerical investigation of hydrodynamic behavior of shear thinning fluids in stirred tank’ (Khapre and Munshi, 2015 [1]. The tracer injection method is used for the prediction of mixing time and mixing efficiency of a Rushton turbine impeller. 16. An Improved Lattice Boltzmann Model for Non-Newtonian Flows with Applications to Solid-Fluid Interactions in External Flows Science.gov (United States) 2016-11-01 Fluid mechanics of non-Newtonian fluids, which arise in numerous settings, are characterized by non-linear constitutive models that pose certain unique challenges for computational methods. Here, we consider the lattice Boltzmann method (LBM), which offers some computational advantages due to its kinetic basis and its simpler stream-and-collide procedure enabling efficient simulations. However, further improvements are necessary to improve its numerical stability and accuracy for computations involving broader parameter ranges. Hence, in this study, we extend the cascaded LBM formulation by modifying its moment equilibria and relaxation parameters to handle a variety of non-Newtonian constitutive equations, including power-law and Bingham fluids, with improved stability. In addition, we include corrections to the moment equilibria to obtain an inertial frame invariant scheme without cubic-velocity defects. After preforming its validation study for various benchmark flows, we study the physics of non-Newtonian flow over pairs of circular and square cylinders in a tandem arrangement, especially the wake structure interactions and their effects on resulting forces in each cylinder, and elucidate the effect of the various characteristic parameters. 17. Characterising the rheology of non-Newtonian fluids using PFG-NMR and cumulant analysis. Science.gov (United States) Blythe, T W; Sederman, A J; Mitchell, J; Stitt, E H; York, A P E; Gladden, L F 2015-06-01 Conventional rheological characterisation using nuclear magnetic resonance (NMR) typically utilises spatially-resolved measurements of velocity. We propose a new approach to rheometry using pulsed field gradient (PFG) NMR which readily extends the application of MR rheometry to single-axis gradient hardware. The quantitative use of flow propagators in this application is challenging because of the introduction of artefacts during Fourier transform, which arise when realistic sampling strategies are limited by experimental and hardware constraints and when particular spatial and temporal resolution are required. The method outlined in this paper involves the cumulant analysis of the acquisition data directly, thereby preventing the introduction of artefacts and reducing data acquisition times. A model-dependent approach is developed to enable the pipe-flow characterisation of fluids demonstrating non-Newtonian power-law rheology, involving the use of an analytical expression describing the flow propagator in terms of the flow behaviour index. The sensitivity of this approach was investigated and found to be robust to the signal-to-noise ratio (SNR) and number of acquired data points, enabling an increase in temporal resolution defined by the SNR. Validation of the simulated results was provided by an experimental case study on shear-thinning aqueous xanthan gum solutions, whose rheology could be accurately characterised using a power-law model across the experimental shear rate range of 1-100 s(-1). The flow behaviour indices calculated using this approach were observed to be within 8% of those obtained using spatially-resolved velocity imaging and within 5% of conventional rheometry. Furthermore, it was shown that the number of points sampled could be reduced by a factor of 32, when compared to the acquisition of a volume-averaged flow propagator with 128 gradient increments, without negatively influencing the accuracy of the characterisation, reducing the 18. Squeeze film problems of long partial journal bearings for non-Newtonian couple stress fluids with pressure-dependent viscosity Energy Technology Data Exchange (ETDEWEB) Lin, Jaw-Ren; Hung, Chi-Ren; Lu, Rong-Fang [Nanya Institute of Technology, Jhongli, Taiwan (China). Dept. of Mechanical Engineering; Chu, Li-Ming [I-Shou Univ., Kaohsiung, Taiwan (China). Dept. of Mechanical and Automation Engineering 2011-08-15 According to the experimental work of C. Barus in Am. J. Sci. 45, 87 (1893), the dependency of liquid viscosity on pressure is exponential. Therefore, we extend the study of squeeze film problems of long partial journal bearings for Stokes non-Newtonian couple stress fluids by considering the pressure-dependent viscosity in the present paper. Through a small perturbation technique, we derive a first-order closed-form solution for the film pressure, the load capacity, and the response time of partial-bearing squeeze films. It is also found that the non-Newtonian couple-stress partial bearings with pressure-dependent viscosity provide better squeeze-film characteristics than those of the bearing with constant-viscosity situation. (orig.) 19. Squeeze Film Problems of Long Partial Journal Bearings for Non-Newtonian Couple Stress Fluids with Pressure-Dependent Viscosity Science.gov (United States) Lin, Jaw-Ren; Chu, Li-Ming; Hung, Chi-Ren; Lu, Rong-Fang 2011-09-01 According to the experimental work of C. Barus in Am. J. Sci. 45, 87 (1893) [1], the dependency of liquid viscosity on pressure is exponential. Therefore, we extend the study of squeeze film problems of long partial journal bearings for Stokes non-Newtonian couple stress fluids by considering the pressure-dependent viscosity in the present paper. Through a small perturbation technique, we derive a first-order closed-form solution for the film pressure, the load capacity, and the response time of partial-bearing squeeze films. It is also found that the non-Newtonian couple-stress partial bearings with pressure-dependent viscosity provide better squeeze-film characteristics than those of the bearing with constant-viscosity situation. 20. Numerical analysis of natural convection for non-Newtonian fluid conveying nanoparticles between two vertical parallel plates Science.gov (United States) Sahebi, S. A. R.; Pourziaei, H.; Feizi, A. R.; Taheri, M. H.; Rostamiyan, Y.; Ganji, D. D. 2015-12-01 In this paper, natural convection of non-Newtonian bio-nanofluids flow between two vertical flat plates is investigated numerically. Sodium Alginate (SA) and Sodium Carboxymethyl Cellulose (SCMC) are considered as the base non-Newtonian fluid, and nanoparticles such as Titania ( TiO2 and Alumina ( Al2O3 were added to them. The effective thermal conductivity and viscosity of nanofluids are calculated through Maxwell-Garnetts (MG) and Brinkman models, respectively. A fourth-order Runge-Kutta numerical method (NUM) and three Weighted Residual Methods (WRMs), Collocation (CM), Galerkin (GM) and Least-Square Method (LSM) and Finite-Element Method (FEM), are used to solve the present problem. The influence of some physical parameters such as nanofluid volume friction on non-dimensional velocity and temperature profiles are discussed. The results show that SCMC- TiO2 has higher velocity and temperature values than other nanofluid structures. 1. MHD flow and heat transfer from continuous surface in uniform free stream of non-Newtonian fluid Institute of Scientific and Technical Information of China (English) 2007-01-01 An analysis is carried out to study the steady flow and heat transfer characteristics from a continuous flat surface moving in a parallel free stream of an electrically conducting non-Newtonian viscoelastic fluid. The flow is subjected to a transverse uniform magnetic field. The constitutive equation of the fluid is modeled by that for a second grade fluid. Numerical results are obtained for the distribution of velocity and temperature profiles. The effects of various physical parameters like viscoelastic parameter, magnetic parameter and Prandtl number on various momentum and heat transfer characteristics are discussed in detail and shown graphically. 2. Electro-osmosis of non-Newtonian fluids in porous media using lattice Poisson-Boltzmann method. Science.gov (United States) Chen, Simeng; He, Xinting; Bertola, Volfango; Wang, Moran 2014-12-15 Electro-osmosis in porous media has many important applications in various areas such as oil and gas exploitation and biomedical detection. Very often, fluids relevant to these applications are non-Newtonian because of the shear-rate dependent viscosity. The purpose of this study was to investigate the behaviors and physical mechanism of electro-osmosis of non-Newtonian fluids in porous media. Model porous microstructures (granular, fibrous, and network) were created by a random generation-growth method. The nonlinear governing equations of electro-kinetic transport for a power-law fluid were solved by the lattice Poisson-Boltzmann method (LPBM). The model results indicate that: (i) the electro-osmosis of non-Newtonian fluids exhibits distinct nonlinear behaviors compared to that of Newtonian fluids; (ii) when the bulk ion concentration or zeta potential is high enough, shear-thinning fluids exhibit higher electro-osmotic permeability, while shear-thickening fluids lead to the higher electro-osmotic permeability for very low bulk ion concentration or zeta potential; (iii) the effect of the porous medium structure depends significantly on the constitutive parameters: for fluids with large constitutive coefficients strongly dependent on the power-law index, the network structure shows the highest electro-osmotic permeability while the granular structure exhibits the lowest permeability on the entire range of power law indices considered; when the dependence of the constitutive coefficient on the power law index is weaker, different behaviors can be observed especially in case of strong shear thinning. 3. Magnetic targeting in the impermeable microvessel with two-phase fluid model--non-Newtonian characteristics of blood. Science.gov (United States) Shaw, Sachin; Murthy, P V S N 2010-09-01 The present investigation deals with finding the trajectories of the drug dosed magnetic carrier particle in a microvessel with two-phase fluid model which is subjected to the external magnetic field. The radius of the microvessel is divided into the endothelial glycocalyx layer in which the blood is assumed to obey Newtonian character and a core and plug regions where the blood obeys the non-Newtonian Herschel-Bulkley character which is suitable for the microvessel of radius 50 microm. The carrier particles, bound with nanoparticles and drug molecules are injected into the vascular system upstream from malignant tissue, and captured at the tumor site using a local applied magnetic field. The applied magnetic field is produced by a cylindrical magnet positioned outside the body and near the tumor position. The expressions for the fluidic force for the carrier particle traversing in the two-phase fluid in the microvessel and the magnetic force due to the external magnetic field are obtained. Several factors that influence the magnetic targeting of the carrier particles in the microvasculature, such as the size of the carrier particle, the volume fraction of embedded magnetic nanoparticles, and the distance of separation of the magnet from the axis of the microvessel are considered in the present problem. An algorithm is given to solve the system of coupled equations for trajectories of the carrier particle in the invasive case. The trajectories of the carrier particle are found for both invasive and noninvasive targeting systems. A comparison is made between the trajectories in these cases. Also, the present results are compared with the data available for the impermeable microvessel with single-phase fluid flow. Also, a prediction of the capture of therapeutic magnetic nanoparticle in the impermeable microvasculature is made for different radii, distances and volume fractions in both the invasive and noninvasive cases. 4. Pulsatile magneto-hydrodynamic blood flows through porous blood vessels using a third grade non-Newtonian fluids model. Science.gov (United States) 2016-04-01 In this paper, the unsteady pulsatile magneto-hydrodynamic blood flows through porous arteries concerning the influence of externally imposed periodic body acceleration and a periodic pressure gradient are numerically simulated. Blood is taken into account as the third-grade non-Newtonian fluid. Besides the numerical solution, for small Womersley parameter (such as blood flow through arterioles and capillaries), the analytical perturbation method is used to solve the nonlinear governing equations. Consequently, analytical expressions for the velocity profile, wall shear stress, and blood flow rate are obtained. Excellent agreement between the analytical and numerical predictions is evident. Also, the effects of body acceleration, magnetic field, third-grade non-Newtonian parameter, pressure gradient, and porosity on the flow behaviors are examined. Some important conclusions are that, when the Womersley parameter is low, viscous forces tend to dominate the flow, velocity profiles are parabolic in shape, and the center-line velocity oscillates in phase with the driving pressure gradient. In addition, by increasing the pressure gradient, the mean value of the velocity profile increases and the amplitude of the velocity remains constant. Also, when non-Newtonian effect increases, the amplitude of the velocity profile. 5. Numerical description and experimental validation of a rheology model for non-Newtonian fluid flow in cancellous bone. Science.gov (United States) Widmer Soyka, René P; López, Alejandro; Persson, Cecilia; Cristofolini, Luca; Ferguson, Stephen J 2013-11-01 Fluids present or used in biology, medicine and (biomedical) engineering are often significantly non-Newtonian. Furthermore, they are chemically complex and can interact with the porous matrix through which they flow. The porous structures themselves display complex morphological inhomogeneities on a wide range of length scales. In vertebroplasty, a shear-thinning fluid, e.g. poly(methyl methacrylate) (PMMA), is injected into the cavities of vertebral trabecular bone for the stabilization of fractures and metastatic lesions. The main objective of this study was therefore to provide a protocol for numerically investigating the rheological properties of PMMA-based bone cements to predict its spreading behavior while flowing through vertebral trabecular bone. A numerical upscaling scheme based on a dimensionless formulation of the Navier-Stokes equation is proposed in order to relate the pore-scale rheological properties of the PMMA that were experimentally estimated using a plate rheometer, to the continuum-scale. On the pore length scale, a viscosity change on the order of one magnitude was observed whilst the shear-thinning properties caused a viscosity change on the order of only 10% on the continuum length scale and in a flow regime that is relevant for vertebroplasty. An experimental validation, performed on human cadaveric vertebrae (n=9), showed a significant improvement of the cement spreading prediction accuracy with a non-Newtonian formulation. A root mean square cement surface prediction error of 1.53mm (assuming a Newtonian fluid) and 1.37mm (assuming a shear-thinning fluid) was found. Our findings highlight the importance of incorporating the non-Newtonian fluids properties in computational models of porous media at the appropriate length scale. 6. Consistent prediction of streaming potential in non-Newtonian fluids: the effect of solvent rheology and confinement on ionic conductivity. Science.gov (United States) 2015-03-21 By considering an ion moving inside an imaginary sphere filled with a power-law fluid, we bring out the implications of the fluid rheology and the influence of the proximity of the other ions towards evaluating the conduction current in an ionic solution. We show that the variation of the conductivity as a function of the ionic concentration is both qualitatively and quantitatively similar to that predicted by the Kohlrausch law. We then utilize this consideration for estimating streaming potentials developed across narrow fluidic confinements as a consequence of the transport of ions in a convective medium constituting a power-law fluid. These estimates turn out to be in sharp contrast to the classical estimates of streaming potential for non-Newtonian fluids, in which the effect of rheology of the solvent is merely considered to affect the advection current, disregarding its contributions to the conduction current. Our results have potential implications of devising a new paradigm of consistent estimation of streaming potentials for non-Newtonian fluids, with combined considerations of the confinement effect and fluid rheology in the theoretical calculations. 7. Continuous separation of microparticles in a microfluidic channel via the elasto-inertial effect of non-Newtonian fluid. Science.gov (United States) Nam, Jeonghun; Lim, Hyunjung; Kim, Dookon; Jung, Hyunwook; Shin, Sehyun 2012-04-07 Pure separation and sorting of microparticles from complex fluids are essential for biochemical analyses and clinical diagnostics. However, conventional techniques require highly complex and expensive labeling processes for high purity separation. In this study, we present a simple and label-free method for separating microparticles with high purity using the elasto-inertial characteristic of a non-Newtonian fluid in microchannel flow. At the inlet, particle-containing sample flow was pushed toward the side walls by introducing sheath fluid from the center inlet. Particles of 1 μm and 5 μm in diameter, which were suspended in viscoelastic fluid, were successfully separated in the outlet channels: larger particles were notably focused on the centerline of the channel at the outlet, while smaller particles continued flowing along the side walls with minimal lateral migration towards the centerline. The same technique was further applied to separate platelets from diluted whole blood. Through cytometric analysis, we obtained a purity of collected platelets of close to 99.9%. Conclusively, our microparticle separation technique using elasto-inertial forces in non-Newtonian fluid is an effective method for separating and collecting microparticles on the basis of size differences with high purity. 8. Low-density lipoprotein accumulation within a carotid artery with multilayer elastic porous wall: fluid-structure interaction and non-Newtonian considerations. Science.gov (United States) Deyranlou, Amin; Niazmand, Hamid; Sadeghi, Mahmood-Reza 2015-09-18 Low-density lipoprotein (LDL), which is recognized as bad cholesterol, typically has been regarded as a main cause of atherosclerosis. LDL infiltration across arterial wall and subsequent formation of Ox-LDL could lead to atherogenesis. In the present study, combined effects of non-Newtonian fluid behavior and fluid-structure interaction (FSI) on LDL mass transfer inside an artery and through its multilayer arterial wall are examined numerically. Navier-Stokes equations for the blood flow inside the lumen and modified Darcy's model for the power-law fluid through the porous arterial wall are coupled with the equations of mass transfer to describe LDL distributions in various segments of the artery. In addition, the arterial wall is considered as a heterogeneous permeable elastic medium. Thus, elastodynamics equation is invoked to examine effects of different wall elasticity on LDL distribution in the artery. Findings suggest that non-Newtonian behavior of filtrated plasma within the wall enhances LDL accumulation meaningfully. Moreover, results demonstrate that at high blood pressure and due to the wall elasticity, endothelium pores expand, which cause significant variations on endothelium physiological properties in a way that lead to higher LDL accumulation. Additionally, results describe that under hypertension, by increasing angular strain, endothelial junctions especially at leaky sites expand more dramatic for the high elastic model, which in turn causes higher LDL accumulation across the intima layer and elevates atherogenesis risk. 9. On the existence of weak solution to the coupled fluid-structure interaction problem for non-Newtonian shear-dependent fluid OpenAIRE Hundertmark-Zaušková, A.; Lukáčová-Medviďová, M.; Nečasová, Š. (Šárka) 2016-01-01 We study the existence of weak solution for unsteady fluid-structure interaction problem for shear-thickening flow. The time dependent domain has at one part a flexible elastic wall. The evolution of fluid domain is governed by the generalized string equation with action of the fluid forces. The power-law viscosity model is applied to describe shear-dependent non-Newtonian fluids. 10. Characteristics of three-phase internal loop airlift bioreactors with complete gas recirculation for non-Newtonian fluids. Science.gov (United States) Wen, Jianping; Jia, Xiaoqiang; Cheng, Xianrui; Yang, Peng 2005-05-01 Hydrodynamic and gas-liquid mass transfer characteristics, such as liquid velocity, gas holdup, solid holdup and gas-liquid volumetric mass transfer coefficient, in the riser and downcomer of the gas-liquid-solid three-phase internal loop airlift bioreactors with complete gas recirculation for non-Newtonian fluids, were investigated. A mathematical model for the description of flow behavior and gas-liquid mass transfer of these bioreactors was developed. The predicted results of this model agreed well with the experimental data. 11. Heat Source/Sink in a Magneto-Hydrodynamic Non-Newtonian Fluid Flow in a Porous Medium: Dual Solutions. Science.gov (United States) Hayat, Tasawar; Awais, Muhammad; Imtiaz, Amna 2016-01-01 This communication deals with the properties of heat source/sink in a magneto-hydrodynamic flow of a non-Newtonian fluid immersed in a porous medium. Shrinking phenomenon along with the permeability of the wall is considered. Mathematical modelling is performed to convert the considered physical process into set of coupled nonlinear mathematical equations. Suitable transformations are invoked to convert the set of partial differential equations into nonlinear ordinary differential equations which are tackled numerically for the solution computations. It is noted that dual solutions for various physical parameters exist which are analyzed in detail. 12. SOLUTION OF THE RAYLEIGH PROBLEM FOR A POWER-LAW NON-NEWTONIAN CONDUCTING FLUID VIA GROUP METHOD Institute of Scientific and Technical Information of China (English) Mina B.Abd-el-Malek; Nagwa A.Badran; Hossam S.Hassan 2002-01-01 An investigation is made of the magnetic Rayleigh problem where a semi-infinite plate is given an impulsive motion and thereafter moves with constant velocity in a nonNewtonian power law fluid of infinite extent. The solution of this highly non-linear problem is obtained by means of the transformation group theoretic approach. The one-parameter group transformation reduces the number of independent variables by one and the governing partial differential equation with the boundary conditions reduce to an ordinary differential equation with the appropriate boundary conditions. Effect of the some parameters on the velocity u ( y, t) has been studied and the results are plotted. 13. Numerical investigation of non-Newtonian fluids in annular ducts with finite aspect ratio using lattice Boltzmann method. Science.gov (United States) Khali, S; Nebbali, R; Ameziani, D E; Bouhadef, K 2013-05-01 In this work the instability of the Taylor-Couette flow for Newtonian and non-Newtonian fluids (dilatant and pseudoplastic fluids) is investigated for cases of finite aspect ratios. The study is conducted numerically using the lattice Boltzmann method (LBM). In many industrial applications, the apparatuses and installations drift away from the idealized case of an annulus of infinite length, and thus the end caps effect can no longer be ignored. The inner cylinder is rotating while the outer one and the end walls are maintained at rest. The lattice two-dimensional nine-velocity (D2Q9) Boltzmann model developed from the Bhatnagar-Gross-Krook approximation is used to obtain the flow field for fluids obeying the power-law model. The combined effects of the Reynolds number, the radius ratio, and the power-law index n on the flow characteristics are analyzed for an annular space of finite aspect ratio. Two flow modes are obtained: a primary Couette flow (CF) mode and a secondary Taylor vortex flow (TVF) mode. The flow structures so obtained are different from one mode to another. The critical Reynolds number Re(c) for the passage from the primary to the secondary mode exhibits the lowest value for the pseudoplastic fluids and the highest value for the dilatant fluids. The findings are useful for studies of the swirling flow of non-Newtonians fluids in axisymmetric geometries using LBM. The flow changes from the CF to TVF and its structure switches from the two-cells to four-cells regime for both Newtonian and dilatant fluids. Contrariwise for pseudoplastic fluids, the flow exhibits 2-4-2 structure passing from two-cells to four cells and switches again to the two-cells configuration. Furthermore, the critical Reynolds number presents a monotonic increase with the power-law index n of the non-Newtonian fluid, and as the radius ratio grows, the transition flow regimes tend to appear for higher critical Reynolds numbers. 14. Numerical Simulation of Concentration Field on Liquid Side around Bubble during Rising and Coalescing Process in Non-Newtonian Fluid Institute of Scientific and Technical Information of China (English) 朱春英; 付涛涛; 高习群; 马友光 2011-01-01 On the basis of Navier-Stockes equation and convection-diffusion equation, combined with surface tension and penetration models, the equations of moment and mass transfer between bubble and the ambient non-Newtonian liquid were established. The formation of a single bubble from a submersed nozzle of 1.0 mm diameter and the mass transfer from an artificially fixed bubble into the ambient liquid were simulated by the volume-of-fluid （VOF） method. Good agreement between simulation results and experimental data confirmed the validity of the numerical method. Furthermore, the concentration distribution around rising bubbles in shear thinning non-Newtonian fluid was simulated. When the process of a single ellipsoidal bubble with the bubble deformation rate below 2.0 rises, the concentration distribution is a single-tail in the bubble＇s wake, but it is fractal when thebubble deformation rate is greater than 2.0. For the overtaking of two in-line rising bubbles, the concentration distribution area between two bubbles broadens gradually and then coalescence occurs. The bifurcation of concentration distribution appears in the rear of the resultant bubble. 15. Secondary flow in a curved artery model with Newtonian and non-Newtonian blood-analog fluids Science.gov (United States) Najjari, Mohammad Reza; Plesniak, Michael W. 2016-11-01 Steady and pulsatile flows of Newtonian and non-Newtonian fluids through a 180°-curved pipe were investigated using particle image velocimetry (PIV). The experiment was inspired by physiological pulsatile flow through large curved arteries, with a carotid artery flow rate imposed. Sodium iodide (NaI) and sodium thiocyanate (NaSCN) were added to the working fluids to match the refractive index (RI) of the test section to eliminate optical distortion. Rheological measurements revealed that adding NaI or NaSCN changes the viscoelastic properties of non-Newtonian solutions and reduces their shear-thinning property. Measured centerline velocity profiles in the upstream straight pipe agreed well with an analytical solution. In the pulsatile case, secondary flow structures, i.e. deformed-Dean, Dean, Wall and Lyne vortices, were observed in various cross sections along the curved pipe. Vortical structures at each cross section were detected using the d2 vortex identification method. Circulation analysis was performed on each vortex separately during the systolic deceleration phase, and showed that vortices split and rejoin. Secondary flow structures in steady flows were found to be morphologically similar to those in pulsatile flows for sufficiently high Dean number. supported by the George Washington University Center for Biomimetics and Bioinspired Engineering. 16. MHD mixed convection analysis in an open channel by obstructed Poiseuille flow of non-Newtonian power law fluid Science.gov (United States) Rabbi, Khan Md.; Rakib, Tawfiqur; Das, Sourav; Mojumder, Satyajit; Saha, Sourav 2016-07-01 This paper demonstrates magneto-hydrodynamic (MHD) mixed convection flow through a channel with a rectangular obstacle at the entrance region using non-Newtonian power law fluid. The obstacle is kept at uniformly high temperature whereas the inlet and top wall of the channel are maintained at a temperature lower than obstacle temperature. Poiseuille flow is implemented as the inlet velocity boundary condition. Grid independency test and code validation are performed to justify the computational accuracy before solving the present problem. Galerkin weighted residual method has been appointed to solve the continuity, momentum and energy equations. The problem has been solved for wide range of pertinent parameters like Richardson number (Ri = 0.1 - 10) at a constant Reynolds number (Re = 100), Hartmann number (Ha = 0 - 100), power index (n = 0.6 - 1.6). The flow and thermal field have been thoroughly discussed through streamline and isothermal lines respectively. The heat transfer performance of the given study has been illustrated by average Nusselt number plots. It is observed that increment of Hartmann number (Ha) tends to decrease the heat transfer rate up to a critical value (Ha = 20) and then let increase the heat transfer performance. Thus maximum heat transfer rate has been recorded for higher Hartmann number and Rayleigh number in case of pseudo-plastic (n = 0.6) non-Newtonian fluid flow. 17. Viscosity bio reducer Influence in a non-Newtonian fluid horizontal pipeline pressure gradient Directory of Open Access Journals (Sweden) Edgardo Jonathan Suarez-Dominguez 2014-03-01 Full Text Available Due to increased production of heavy and extra heavy crude in Mexico, it has led to the necessity touse chemicals to facilitate the transport in the pipe of our country. Experimental study was conductedto analyze the influence of a viscosity reducer of biological origin (BRV, on the rheological behaviorof heavy oil in the northern region of Mexico, finding that it exhibits a non-Newtonian viscoelasticbehavior, where a concentration increase of BRV leads to a consistency decrease and an increasedflow order, where dilatant behavior was observed in high temperatures. From these results it wasestimated the pressure losses by friction in a horizontal pipe for single phase and two phase flow. Wefound that in all cases the increase in the concentration of BRV reduces these losses. 18. Structural Optimization of Non-Newtonian Rectifiers DEFF Research Database (Denmark) Jensen, Kristian Ejlebjærg; Okkels, Fridolin . In this context it is natural to look for other sources of non-linearity and one possibility is to introduce a non-Newtonian working fluid. Non-Newtonian properties are due to stretching of large particles/molecules in the fluid and this is commonly seen for biological samples in “lab-on-a-chip” systems....... The strength of non-Newtonian effects does not depend on the device size. Furthermore a non-Newtonian working fluid removes symmetry properties such that geometry influence is reintroduced, and indeed non-Newtonian effects have been used in experimentally realized microfluidic rectitifiers[1]. The rectifiers...... optimization, which is a kind of design optimization where nothing is assumed about the topology of the design. We will apply a high-level implementation of topology optimization using the density method in a commercial finite element package[2]. However, the modeling of non-Newtonian fluids remains a major... 19. Peristaltic Motion of Non-Newtonian Fluid with Heat and Mass Transfer through a Porous Medium in Channel under Uniform Magnetic Field Directory of Open Access Journals (Sweden) Nabil T. M. Eldabe 2014-01-01 Full Text Available This paper is devoted to the study of the peristaltic motion of non-Newtonian fluid with heat and mass transfer through a porous medium in the channel under the effect of magnetic field. A modified Casson non-Newtonian constitutive model is employed for the transport fluid. A perturbation series’ method of solution of the stream function is discussed. The effects of various parameters of interest such as the magnetic parameter, Casson parameter, and permeability parameter on the velocity, pressure rise, temperature, and concentration are discussed and illustrated graphically through a set of figures. 20. Effects of impingement and friction of swirling air stream on prefilming airblast atomization of non-Newtonian fluids Institute of Scientific and Technical Information of China (English) HE Wen-zhi; LI Guang-ming; JIANG Zhao-hua; SUO Quan-ling 2007-01-01 Liquids to be broken up using a prefilming airblast atomizer are usually Newton liquids with relatively low viscosities. While in some industrial processes, such as spray drying, liquids to be atomized are high concentration suspensions or non-Newtonian fluids with high viscosities. In this paper, non-Newtonian fluids with viscosity up to 4. 4 Pa · s were effectively atomized using a specially designed prefilming airblast atomizer. The atomizer enabled liquid to extend to a thickness-adjustable film and forced the atomizing air stream to swirl with 30° or 45° through gas distributors with spiral slots. The liquid film was impinged by the swirling air stream resulting in the disintegration of the film into drops. Drop sizes were measured using a laser diffraction technique.An improved four-parameter mathematical model was established to relate the Sauter mean diameter of drops to the atomization conditions in terms of power dependencies on three dimensionless groups: Weber number,Ohnesorge number and air liquid mass ratio. The friction on the surface of the 1iquid film made by swirling air stream played an important role in the prefilming atomization at the conditions of low air velocity and low liquid viscosity. In this case, the liquid film was disintegrated into drops according to the classical wavy-sheet mechanism, thus thinner liquid films and high swirl levels of the atomizing air produced smaller drops. With the increase of the air velocity and the liquid viscosity, the effect of the friction on the prefilming atomization relatively weakened, whereas the impingement on the liquid film made by atomizing air stream in a direction normal to the liquid film and corresponding momentum transfer gradually strengthened and eventually dominated the disruption of liquid into drops, which induced that the initial thickness of the liquid film and the swirl of atomizing air stream exercised a minor influence on the drop sizes. 1. A hybrid molecular dynamics study on the non-Newtonian rheological behaviors of shear thickening fluid. Science.gov (United States) Chen, Kaihui; Wang, Yu; Xuan, Shouhu; Gong, Xinglong 2017-07-01 To investigate the microstructural evolution dependency on the apparent viscosity in shear-thickening fluids (STFs), a hybrid mesoscale model combined with stochastic rotation dynamics (SRD) and molecular dynamics (MD) is used. Muller-Plathe reverse perturbation method is adopted to analyze the viscosities of STFs in a two-dimensional model. The characteristic of microstructural evolution of the colloidal suspensions under different shear rate is studied. The effect of diameter of colloidal particles and the phase volume fraction on the shear thickening behavior is investigated. Under low shear rate, the two-atom structure is formed, because of the strong particle attractions in adjacent layers. At higher shear rate, the synergetic pair structure extends to layered structure along flow direction because of the increasing hydrodynamics action. As the shear rate rises continuously, the layered structure rotates and collides with other particles, then turned to be individual particles under extension or curve string structure under compression. Finally, at the highest shear rate, the strings curve more severely and get into two-dimensional cluster. The apparent viscosity of the system changes from shear-thinning behavior to the shear-thickening behavior. This work presents valuable information for further understanding the shear thickening mechanism. 2. Convection instability of non-Newtonian Walter's nanofluid along a vertical layer Directory of Open Access Journals (Sweden) Galal M. Moatimid 2017-04-01 Full Text Available The linear stability of viscoelastic nanofluid layer is investigated. The rheological behavior of the viscoelastic fluid is described through the Walter's model. The normal modes analysis is utilized to treat the equations of motion for stationary and oscillatory convection. The stability analysis resulted in a third-degree dispersion equation with complex coefficients. The Routh–Hurwitz theory is employed to investigate the dispersion relation. The stability criteria divide the plane into several parts of stable/unstable regions. This shows some analogy with the nonlinear stability theory. The relation between the elasticity and the longitudinal wave number is graphically analyzed. The numerical calculations show that viscoelastic flows are more stable than those of the Newtonian ones. 3. A constitutive framework for the non-Newtonian pressure tensor of a simple fluid under planar flows. Science.gov (United States) Hartkamp, Remco; Todd, B D; Luding, Stefan 2013-06-28 Non-equilibrium molecular dynamics simulations of an atomic fluid under shear flow, planar elongational flow, and a combination of shear and elongational flow are unified consistently with a tensorial model over a wide range of strain rates. A model is presented that predicts the pressure tensor for a non-Newtonian bulk fluid under a homogeneous planar flow field. The model provides a quantitative description of the strain-thinning viscosity, pressure dilatancy, deviatoric viscoelastic lagging, and out-of-flow-plane pressure anisotropy. The non-equilibrium pressure tensor is completely described through these four quantities and can be calculated as a function of the equilibrium material constants and the velocity gradient. This constitutive framework in terms of invariants of the pressure tensor departs from the conventional description that deals with an orientation-dependent description of shear stresses and normal stresses. The present model makes it possible to predict the full pressure tensor for a simple fluid under various types of flows without having to produce these flow types explicitly in a simulation or experiment. 4. MASS TRANSFER COEFFICIENTS FOR A NON-NEWTONIAN FLUID AND WATER WITH AND WITHOUT ANTI-FOAM AGENTS Energy Technology Data Exchange (ETDEWEB) Leishear, R. 2009-09-09 Mass transfer rates were measured in a large scale system, which consisted of an 8.4 meter tall by 0.76 meter diameter column containing one of three fluids: water with an anti-foam agent, water without an anti-foam agent, and AZ101 simulant, which simulated a non-Newtonian nuclear waste. The testing contributed to the evaluation of large scale mass transfer of hydrogen in nuclear waste tanks. Due to its radioactivity, the waste was chemically simulated, and due to flammability concerns oxygen was used in lieu of hydrogen. Different liquids were used to better understand the mass transfer processes, where each of the fluids was saturated with oxygen, and the oxygen was then removed from solution as air bubbled up, or sparged, through the solution from the bottom of the column. Air sparging was supplied by a single tube which was co-axial to the column, the decrease in oxygen concentration was recorded, and oxygen measurements were then used to determine the mass transfer coefficients to describe the rate of oxygen transfer from solution. Superficial, average, sparging velocities of 2, 5, and 10 mm/second were applied to each of the liquids at three different column fill levels, and mass transfer coefficient test results are presented here for combinations of superficial velocities and fluid levels. 5. Flow of a non-Newtonian fluid through channels with permeable wall Energy Technology Data Exchange (ETDEWEB) Martins-Costa, Maria Laura [Universidade Federal Fluminense, Niteroi, RJ (Brazil). Dept. de Engenharia Mecanica. Lab. de Matematica Teorica e Aplicada]. E-mail: [email protected]; Gama, Rogerio M. Saldanha da [Laboratorio Nacional de Computacao Cientifica (LNCC), Petropolis, RJ (Brazil)]. E-mail: [email protected]; Frey, Sergio [Rio Grande do Sul Univ., Porto Alegre, RS (Brazil). Dept. de Engenharia Mecanica. Grupo de Estudos Termicos e Energeticos 2000-07-01 In the present work the momentum transport in two adjacent flow regions is described by means of a continuum theory of mixtures, specially developed to model multiphase phenomena. A generalized Newtonian fluid flows through the permeable wall channel, originating a pure fluid region and a mixture region - where the fluid saturates the porous matrix. The fluid and the porous matrix are treated as continuous constituents of a binary mixture coexisting superposed, each of them occupying simultaneously the whole volume of the mixture. An Ostwald-de Waele behavior is assumed for both the fluid constituent (in the mixture region) and the fluid (in the so-called pure fluid region), while the porous matrix, represented by the solid constituent, is assumed rigid, homogeneous, isotropic and at rest. Compatibility conditions at the interface (pure fluid-mixture) for momentum transfer are proposed and discussed. Assuming no flow across the interface, the velocity should be zero on the solid parts of the boundary and should match the fluid diffusing velocity on the fluid parts of the boundary. Also the shear stress at the pure fluid region is to be balanced by a multiple of the partial shear stress at the mixture region. A minimum principle for the above-described problem, assuming fully developed flow in both regions, is presented, providing an easy and reliable way for carrying out numerical simulations. (author) 6. Entropy Generation Analysis of Power-Law Non-Newtonian Fluid Flow Caused by Micropatterned Moving Surface Directory of Open Access Journals (Sweden) M. H. Yazdi 2014-01-01 Full Text Available In the present study, the first and second law analyses of power-law non-Newtonian flow over embedded open parallel microchannels within micropatterned permeable continuous moving surface are examined at prescribed surface temperature. A similarity transformation is used to reduce the governing equations to a set of nonlinear ordinary differential equations. The dimensionless entropy generation number is formulated by an integral of the local rate of entropy generation along the width of the surface based on an equal number of microchannels and no-slip gaps interspersed between those microchannels. The velocity, the temperature, the velocity gradient, and the temperature gradient adjacent to the wall are substituted into this equation resulting from the momentum and energy equations obtained numerically by Dormand-Prince pair and shooting method. Finally, the entropy generation numbers, as well as the Bejan number, are evaluated. It is noted that the presence of the shear thinning (pseudoplastic fluids creates entropy along the surface, with an opposite effect resulting from shear thickening (dilatant fluids. 7. An analytical investigation on unsteady motion of vertically falling spherical particles in non-Newtonian fluid by Collocation Method Directory of Open Access Journals (Sweden) M. Rahimi-Gorji 2015-06-01 Full Text Available An analytical investigation is applied for unsteady motion of a rigid spherical particle in a quiescent shear-thinning power-law fluid. The results were compared with those obtained from Collocation Method (CM and the established Numerical Method (Fourth order Runge–Kutta scheme. It was shown that CM gave accurate results. Collocation Method (CM and Numerical Method are used to solve the present problem. We obtained that the CM which was used to solve such nonlinear differential equation with fractional power is simpler and more accurate than series method such as HPM which was used in some previous works by others but the new method named Akbari-Ganji’s Method (AGM is an accurate and simple method which is slower than CM for solving such problems. The terminal settling velocity—that is the velocity at which the net forces on a falling particle eliminate—for three different spherical particles (made of plastic, glass and steel and three flow behavior index n, in three sets of power-law non-Newtonian fluids was investigated, based on polynomial solution (CM. Analytical results obtained indicated that the time of reaching the terminal velocity in a falling procedure is significantly increased with growing of the particle size that validated with Numerical Method. Further, with approaching flow behavior to Newtonian behavior from shear-thinning properties of flow (n → 1, the transient time to achieving the terminal settling velocity is decreased. 8. Spreading dynamics and dynamic contact angle of non-Newtonian fluids. Science.gov (United States) Wang, X D; Lee, D J; Peng, X F; Lai, J Y 2007-07-17 The spreading dynamics of power-law fluids, both shear-thinning and shear-thickening fluids, that completely or partially wet solid substrate was investigated theoretically and experimentally. An evolution equation for liquid-film thickness was derived using a lubrication approximation, from which the dynamic contact angle versus the contact line moving velocity relationship was evaluated. In the capillary spreading regime, film thickness h is proportional to xi3/(n+2) (xi is the distance from the contact line), whereas in the gravitational regime, h is proportional to xi1/(n+2), relating to the rheological power exponent n. The derived model fit the experimental data well for a shear-thinning fluid (0.2% w/w xanthan solution) or a shear-thickening fluid (7.5% w/w 10 nm silica in polypropylene glycol) on a completely wetted substrate. The derived model was extended using Hoffmann's proposal for partially wetting fluids. Good agreement was also attained between model predictions and the shear-thinning fluid (1% w/w cmc solution) and shear-thickening fluid (10% w/w 15 nm silica) on partially wetted surfaces. 9. Diffusion Coefficients of L-arginine in Non-Newtonian Fluid%L-精氨酸在非牛顿流体中的扩散系数 Institute of Scientific and Technical Information of China (English) 朱春英; 马友光; 季喜燕 2008-01-01 L-Arginine is an important component of amino acid injection. Its diffusion in body fluid and blood is of key importance to understand drug diffusion and drug release. As a fundamental demand for study and being a considerably valuable reference for application, in this study, the diffusion coefficients of L-arginine in polyacryla-mide(PAM) aqueous solution used as non-Newtonian fluid similar to blood and body fluid were measured using a holographic interferometer. The effects of interaction among molecules and solution concentration on diffusion were analyzed and discussed, respectively. Based on the obstruction-scaling model, a novel modified model was presented for predicting diffusivity of solute in non-Newtonian fluid. Good agreement was achieved between the calculated value and the experimental data. 10. MHD non-Newtonian micropolar fluid flow and heat transfer in channel with stretching walls Institute of Scientific and Technical Information of China (English) M. ASHRAF; N. JAMEEL; K. ALI 2013-01-01 A study is presented for magnetohydrodynamics (MHD) flow and heat trans-fer characteristics of a viscous incompressible electrically conducting micropolar fluid in a channel with stretching walls. The micropolar model introduced by Eringen is used to describe the working fluid. The transformed self similar ordinary differential equations together with the associated boundary conditions are solved numerically by an algorithm based on quasi-linearization and multilevel discretization. The effects of some physical parameters on the flow and heat transfer are discussed and presented through tables and graphs. The present investigations may be beneficial in the flow and thermal control of polymeric processing. 11. ISPH modelling of landslide generated waves for rigid and deformable slides in Newtonian and non-Newtonian reservoir fluids Science.gov (United States) Yeylaghi, Shahab; Moa, Belaid; Buckham, Bradley; Oshkai, Peter; Vasquez, Jose; Crawford, Curran 2017-09-01 A comprehensive modeling of landslide generated waves using an in-house parallel Incompressible Smoothed Particle Hydrodynamics (ISPH) code is presented in this paper. The study of landslide generated waves is challenging due to the involvement of several complex physical phenomena, such as slide-water interaction, turbulence and complex free surface profiles. A numerical tool that can efficiently calculate both slide motion, impact with the surface and the resulting wave is needed for ongoing study of these phenomena. Mesh-less numerical methods, such as Smoothed Particle Hydrodynamics (SPH), handle the slide motion and the complex free surface profile with ease. In this paper, an in-house parallel explicit ISPH code is used to simulate both subaerial and submarine landslides in 2D and in more realistic 3D applications. Both rigid and deformable slides are used to generate the impulsive waves. A landslide case is simulated where a slide falls into a non-Newtonian reservoir fluid (water-bentonite mixture). A new technique is also proposed to calculate the motion of a rigid slide on an inclined ramp implicitly, without using the prescribed motion in SPH. For all the test cases, results generated from the proposed ISPH method are compared with available experimental data and show good agreement. 12. Flush mounted hot film anemometer measurement of wall shear stress distal to a tri-leaflet valve for Newtonian and non-Newtonian blood analog fluids. Science.gov (United States) Nandy, S; Tarbell, J M 1987-01-01 Wall shear stress has been measured by flush-mounted hot film anemometry distal to an Ionescu-Shiley tri-leaflet valve under pulsatile flow conditions. Both Newtonian (aqueous glycerol) and non-Newtonian (aqueous polyacrylamide) blood analog fluids were investigated. Significant differences in the axial distribution of wall shear stress between the two fluids are apparent in flows having nearly identical Reynolds numbers. The Newtonian fluid exhibits a (peak) wall shear rate which is maximized near the valve seat (30 mm) and then decays to a fully developed flow value (by 106 mm). In contrast, the shear rate of the non-Newtonian fluid at 30 mm is less than half that of the Newtonian fluid and at 106 mm is more than twice that of the Newtonian fluid. It is suggested that non-Newtonian rheology influences valve flow patterns either through alterations in valve opening associated with low shear separation zones behind valve leaflets, or because of variations in the rate of jet spreading. More detailed studies are required to clarify the mechanisms. The Newtonian wall shear stresses for this valve are low. The highest value observed anywhere in the aortic chamber was 2.85 N/m2 at a peak Reynolds number of 3694. 13. Study of blades inclination influence of gate impeller with a non-Newtonian fluid of Bingham Directory of Open Access Journals (Sweden) Rahmani Lakhdar 2016-01-01 Full Text Available A large number of chemical operations, biochemical or petrochemical industry is very depending on the rheological fluids nature. In this work, we study the case of highly viscous of viscoplastic fluids in a classical system of agitation: a cylindrical tank with plate bottom without obstacles agitated by gate impeller agitator. We are interested to the laminar, incompressible and isothermal flows. We devote to a numerical approach carried out using an industrial code CFD Fluent 6.3.26 based on the method of finites volumes discretization of Navier - Stokes equations formulated in variables (U.V.P. The threshold of flow related to the viscoplastic behavior is modeled by a theoretical law of Bingham. The results obtained are used to compare between the five configurations suggested of power consumption. We study the influence of inertia by the variation of Reynolds number. 14. PFG NMR and Bayesian analysis to characterise non-Newtonian fluids Science.gov (United States) Blythe, Thomas W.; Sederman, Andrew J.; Stitt, E. Hugh; York, Andrew P. E.; Gladden, Lynn F. 2017-01-01 Many industrial flow processes are sensitive to changes in the rheological behaviour of process fluids, and there therefore exists a need for methods that provide online, or inline, rheological characterisation necessary for process control and optimisation over timescales of minutes or less. Nuclear magnetic resonance (NMR) offers a non-invasive technique for this application, without limitation on optical opacity. We present a Bayesian analysis approach using pulsed field gradient (PFG) NMR to enable estimation of the rheological parameters of Herschel-Bulkley fluids in a pipe flow geometry, characterised by a flow behaviour index n , yield stress τ0 , and consistency factor k , by analysis of the signal in q -space. This approach eliminates the need for velocity image acquisition and expensive gradient hardware. We investigate the robustness of the proposed Bayesian NMR approach to noisy data and reduced sampling using simulated NMR data and show that even with a signal-to-noise ratio (SNR) of 100, only 16 points are required to be sampled to provide rheological parameters accurate to within 2% of the ground truth. Experimental validation is provided through an experimental case study on Carbopol 940 solutions (model Herschel-Bulkley fluids) using PFG NMR at a 1H resonance frequency of 85.2 MHz; for SNR > 1000, only 8 points are required to be sampled. This corresponds to a total acquisition time of non-Bayesian NMR methods demonstrates that the Bayesian NMR approach is in agreement with MR flow imaging to within the accuracy of the measurement. Furthermore, as we increase the concentration of Carbopol 940 we observe a change in rheological characteristics, probably due to shear history-dependent behaviour and the different geometries used. This behaviour highlights the need for online, or inline, rheological characterisation in industrial process applications. 15. Thermoelectric MHD non-Newtonian fluid with fractional derivative heat transfer Energy Technology Data Exchange (ETDEWEB) Ezzat, Magdy A., E-mail: [email protected] [Department of Mathematics, Faculty of Education, Alexandria University, Alexandria (Egypt) 2010-10-01 In this work, a new mathematical model of thermoelectric MHD theory has been constructed in the context of a new consideration of heat conduction with fractional order. This model is applied to Stokes' first problem for a viscoelastic fluid with heat sources. Laplace transforms and state-space techniques will be used to obtain the general solution for any set of boundary conditions. According to the numerical results and its graphs, conclusion about the new theory has been constructed. Some comparisons have been shown in figures to estimate the effects of the fractional order parameter on all the studied fields. 16. Accelerated Sedimentation Velocity Assessment for Nanowires Stabilized in a Non-Newtonian Fluid. Science.gov (United States) Chang, Chia-Wei; Liao, Ying-Chih 2016-12-27 In this work, the long-term stability of titanium oxide nanowire suspensions was accessed by an accelerated sedimentation with centrifugal forces. Titanium oxide (TiO2) nanoparticle (NP) and nanowire (NW) dispersions were prepared, and their sizes were carefully characterized. To replace the time-consuming visual observation, sedimentation velocities of the TiO2 NP and NW suspensions were measured using an analytical centrifuge. For an aqueous TiO2 NP suspension, the measured sedimentation velocities were linearly dependent on the relative centrifugal forces (RCF), as predicted by the classical Stokes law. A similar linear relationship was also found in the case of TiO2 NW aqueous suspensions. However, NWs preferred to settle parallel to the centrifugal direction under high RCF because of the lower flow resistance along the long axis. Thus, the extrapolated sedimentation velocity under regular gravity can be overestimated. Finally, a stable TiO2 NW suspension was formulated with a shear thinning fluid and showed great stability for weeks using visual observation. A theoretical analysis was deduced with rheological shear-thinning parameters to describe the nonlinear power-law dependence between the measured sedimentation velocities and RCF. The good agreement between the theoretical predictions and measurements suggested that the sedimentation velocity can be properly extrapolated to regular gravity. In summary, this accelerated assessment on a theoretical basis can yield quantitative information about long-term stability within a short time (a few hours) and can be further extended to other suspension systems. 17. Flow of Chemically Reactive non-Newtonian Fluids in Twin-Screw Extruders Science.gov (United States) Zhu, Weimin; Jaluria, Yogesh 1998-11-01 Many applications of twin-screw extruders are found in the processing of food, plastics, pharmaceutical materials and other highly viscous materials. In reactive extrusion, complex interactions in which the flow pattern, and the heat and mass transfer are affected by viscous dissipation, reaction energy, convection, residence time distribution and rheology of the materials may occur. The fluid flow, heat transfer and chemical reactions in a fully intermeshing, corotating and self wiping twin screw extruder were investigated numerically by using the finite volume method. The screw channel of a twin screw extruder are approximated as translation (parabolic) domain and intermeshing (elliptic) domain. The full governing equations were solved to determine the velocity components in the three coordinate directions. The energy equation is coupled with the equations of motion through viscosity. The Residence Time Distribution (RTD), was obtained by using a particle tracking method. The flow field, temperature field, pressure as well as RTD and chemical conversion were obtained by numerical simulation and the results yielded agreement with experimental measurements and expected physical characteristic of the process. 18. Local Liquid Side Mass Transfer Model in Gas-Liquid-Solid Three-Phase Flow Airlift Loop Reactor for Newtonian and Non-Newtonian Fluids Institute of Scientific and Technical Information of China (English) 闻建平; 贾晓强; 毛国柱 2004-01-01 A small scale isotropic mass transfer model was developed for the local liquid side mass transfer coefficients in gas-liquid-solid three-phase flow airlift loop reactor for Newtonian and non-Newtonian fluids. It is based on Higbie's penetration theory and Kolmogoroff's theory of isotropic turbulence with kl=3√2D∈11/3/π(η1-1/3-λf-1/3)where e1 is local rate of energy dissipation, Af is the local microscale, r/l is the local Kolmogoroff scale and D is the diffusion coefficient. The capability of the proposed model is discussed in the light of experimental data obtained from 12 L gas-liquid-solid three-phase flow airlift loop reactor using Newtonian and non-Newtonian fluids. Good agreement with the experimental data was obtained over a wide range of conditions suggesting a general applicability of the proposed model. 19. The effect of the inner cylinder rotation on the fluid dynamics of non-Newtonian fluids in concentric and eccentric annuli Directory of Open Access Journals (Sweden) J. L. Vieira Neto 2014-12-01 Full Text Available Helical flow in an annular space occurs during oil drilling operations. The correct prediction of flow of drilling fluid in an annular space between the wellbore wall and the drill pipe is essential to determine the variation in fluid pressure within the wellbore. This paper presents experimental and CFD simulation results of the pressure drop in the flow of non-Newtonian fluids through a concentric annular section and another section with fixed eccentricity (E = 0.75, using aqueous solutions of two distinct polymers (Xanthan Gum and Carboxymethylcellulose. The hydrodynamic behavior in this annular system was analyzed based on the experimental and CFD results, providing important information such as the formation of zones with preferential flows and stagnation regions. 20. 牛顿流体和非牛顿流体教学演示仪器的开发研究%Teaching Apparatus to Demonstrate the Differences Between Newtonian Fluid and Non-Newtonian Fluid Institute of Scientific and Technical Information of China (English) 葛雄; 张根广; 颜婷; 王华伟 2011-01-01 There are many differences between the Newtonian fluid and the non-Newtonian fluid, and there is a few or even not teaching apparatus to demonstrate the behavior differences between the Newtonian fluid and the non-Newtonian fluid. To strengthen understanding of the Newtonian fluid and the non-Newtonian fluid, a teaching apparatus to demonstrate the velocity distribution differences in laminar flow between the Newtonian fluid and the non-Newtonian fluid was designed and made based on the flow characteristics differences. The movement behaviors of the Newtonian fluid and the non-Newtonian fluid were displayed by the tracer liquid developed by us. The velocity distribution differences at laminar flow between the Newtonian fluid and the non-Newtonian fluid can be found easily and directly in the demonstration apparatus.%牛顿流体和非牛顿流体存在着较大的差异，有关牛顿流体和非牛顿流体特性差异方面的教学演示仪器还较少，有些还处于空白。为了加强学生对牛顿流体和非牛顿流体的认识，本文根据牛顿流体与非牛顿流体流动特性差异，设计制作了表现牛顿流体和非牛顿流体层流流速分布差异的演示仪器;利用自行研制的示踪液体演示了非牛顿流体和牛顿流体运动过程，直观的展示了牛顿流体与非牛顿流体层流流速分布的差异。 1. Dynamic scaling of unsteady shear-thinning non-Newtonian fluid flows in a large-scale model of a distal anastomosis Science.gov (United States) Gray, J. D.; Owen, I.; Escudier, M. P. 2007-10-01 Dimensional analysis has been applied to an unsteady pulsatile flow of a shear-thinning power-law non-Newtonian liquid. An experiment was then designed in which both Newtonian and non-Newtonian liquids were used to model blood flow through a large-scale (38.5 mm dia.), simplified, rigid arterial junction (a distal anastomosis of a femorodistal bypass). The flow field within the junction was obtained by Particle Imaging Velocimetry and near-wall velocities were used to calculate the wall shear stresses. Dimensionless wall shear stresses were obtained at different points in the cardiac cycle for two different but dynamically similar non-Newtonian fluids; the good agreement between the measured dimensionless wall shear stresses confirm the validity of the dimensional analysis. However, blood exhibits a constant viscosity at high-shear rates and to obtain complete dynamic similarity between large-scale experiments and life-scale flows, the high-shear viscosity also needs to be included in the analysis. How this might be done is discussed in the paper. 2. A Study on Mixed Convective, MHD Flow from a Vertical Plate Embedded in Non-Newtonian Fluid Saturated Non- Darcy Porous Medium with Melting Effect Directory of Open Access Journals (Sweden) 2016-01-01 Full Text Available We analyzed in this paper the problem of mixed convection along a vertical plate in a non-Newtonian fluid saturated non-Darcy porous medium in the presence of melting and thermal dispersion-radiation effects for aiding and opposing external flows. Similarity solution for the governing equations is obtained for the flow equations in steady state. The equations are numerically solved by using Runge-kutta fourth order method coupled with shooting technique. The effects of melting (M, thermal dispersion (D, radiation (R, magnetic field (MH, viscosity index (n and mixed convection (Ra/Pe on fluid velocity and temperature are examined for aiding and opposing external flows. 3. Group Theoretical Analysis of non-Newtonian Fluid Flow, Heat and Mass Transfer over a Stretching Surface in the Presence of Thermal Radiation Directory of Open Access Journals (Sweden) 2016-01-01 Full Text Available The present article examines the flow, heat and mass transfer of a non-Newtonian fluid known as Casson fluid over a stretching surface in the presence of thermal radiations effects. Lie Group analysis is used to reduce the governing partial differential equations into non-linear ordinary differential equations. These equations are then solved by an analytical technique known as Homotopy Analysis Method (HAM. A comprehensive study of the problem is being made for various parameters involving in the equations through tables and graphs. 4. In Vivo/Ex Vivo MRI-Based 3D Non-Newtonian FSI Models for Human Atherosclerotic Plaques Compared with Fluid/Wall-Only Models. Science.gov (United States) Yang, Chun; Tang, Dalin; Yuan, Chun; Hatsukami, Thomas S; Zheng, Jie; Woodard, Pamela K 2007-01-01 It has been recognized that fluid-structure interactions (FSI) play an important role in cardiovascular disease initiation and development. However, in vivo MRI multi-component FSI models for human carotid atherosclerotic plaques with bifurcation and quantitative comparisons of FSI models with fluid-only or structure-only models are currently lacking in the literature. A 3D non-Newtonian multi-component FSI model based on in vivo/ex vivo MRI images for human atherosclerotic plaques was introduced to investigate flow and plaque stress/strain behaviors which may be related to plaque progression and rupture. Both artery wall and plaque components were assumed to be hyperelastic, isotropic, incompressible and homogeneous. Blood flow was assumed to be laminar, non-Newtonian, viscous and incompressible. In vivo/ex vivo MRI images were acquired using histologically-validated multi-spectral MRI protocols. The 3D FSI models were solved and results were compared with those from a Newtonian FSI model and wall-only/fluid-only models. A 145% difference in maximum principal stresses (Stress-P(1)) between the FSI and wall-only models and 40% difference in flow maximum shear stress (MSS) between the FSI and fluid-only models were found at the throat of the plaque using a severe plaque sample (70% severity by diameter). Flow maximum shear stress (MSS) from the rigid wall model is much higher (20-40% in maximum MSS values, 100-150% in stagnation region) than those from FSI models. 5. Non-NewtonianFluid Flow and Heat Transfer over a Non- Linearly Stretching Surface Along With Porous Plate in Porous Medium Directory of Open Access Journals (Sweden) S.Jothimani 2014-08-01 Full Text Available This paper investigates the MHD flow and heat transfer of an electrically conducting non-newtonian power-law fluid over a non-linearly stretching surface along with porous plate in porous medium. The governing equations are reduced to non-linear ordinary differential equations by means of similarity transformations. These equations are then solved numerically with the help ofRunge – Kutta shooting method. The effect of various flow parameters in the form of dimensionless quantities on the flow field are discussed and presented graphically. 6. Capillary rise of a non-Newtonian power law liquid: impact of the fluid rheology and dynamic contact angle. Science.gov (United States) Digilov, Rafael M 2008-12-02 The impact of non-Newtonian behavior and the dynamic contact angle on the rise dynamics of a power law liquid in a vertical capillary is studied theoretically and experimentally for quasi-steady-state flow. An analytical solution for the time evolution of the meniscus height is obtained in terms of a Gaussian hypergeometric function, which in the case of a Newtonian liquid reduces to the Lucas-Washburn equation modified by the dynamic contact angle correction. The validity of the solution is checked against experimental data on the rise dynamics of a shear-thinning cmc solution in a glass microcapillary, and excellent agreement is found. 7. Evaluation of dispersive mixing, extension rate and bubble size distribution using numerical simulation of a non-Newtonian fluid in a twin-screw mixer Science.gov (United States) Rathod, Maureen L. Initially 3D FEM simulation of a simplified mixer was used to examine the effect of mixer configuration and operating conditions on dispersive mixing of a non-Newtonian fluid. Horizontal and vertical velocity magnitudes increased with increasing mixer speed, while maximum axial velocity and shear rate were greater with staggered paddles. In contrast, parallel paddles produced an area of efficient dispersive mixing between the center of the paddle and the barrel wall. This study was expanded to encompass the complete nine-paddle mixing section using power-law and Bird-Carreau fluid models. In the center of the mixer, simple shear flow was seen, corresponding with high [special character omitted]. Efficient dispersive mixing appeared near the barrel wall at all flow rates and near the barrel center with parallel paddles. Areas of backflow, improving fluid retention time, occurred with staggered paddles. The Bird-Carreau fluid showed greater influence of paddle motion under the same operating conditions due to the inelastic nature of the fluid. Shear-thinning behavior also resulted in greater maximum shear rate as shearing became easier with decreasing fluid viscosity. Shear rate distributions are frequently calculated, but extension rate calculations have not been made in a complex geometry since Debbaut and Crochet (1988) defined extension rate as the ratio of the third to the second invariant of the strain rate tensor. Extension rate was assumed to be negligible in most studies, but here extension rate is shown to be significant. It is possible to calculate maximum stable bubble diameter from capillary number if shear and extension rates in a flow field are known. Extension rate distributions were calculated for Newtonian and non-Newtonian fluids. High extension and shear rates were found in the intermeshing region. Extension is the major influence on critical capillary number and maximum stable bubble diameter, but when extension rate values are low shear rate has 8. The role of the rheological properties of non-newtonian fluids in controlling dispersive mixing in a batch electrophoretic cell with Joule heating Directory of Open Access Journals (Sweden) M.A. Bosse 2001-03-01 Full Text Available The problem of the effect of Joule heating generation on the hydrodynamic profile and the solute transport found in electrophoretic devices is addressed in this article. The research is focused on the following two problems: The first one is centered around the effect of Joule heating on the hydrodynamic velocity profile and it is referred to as "the carrier fluid problem." The other one is related to the effect of Joule heating on the solute transport inside electrophoretic cells and it is referred to as "the solute problem". The hydrodynamic aspects were studied first to yield the velocity profiles required for analysis of the solute transport problem. The velocity profile obtained in this study is analytical and the results are valid for non-Newtonian fluids carriers. To this end, the power-law model was used to study the effect of the rheology of the material in conjunction with the effect of Joule heating generation inside batch electrophoretic devices. This aspect of the research was then effectively used to study the effect of Joule heating generation on the motion of solutes (such as macromolecules under the influence of non-Newtonian carriers. This aspect of the study was performed using an area-averaging approach that yielded analytical results for the effective diffusivity of the device. 9. Local overall volumetric gas-liquid mass transfer coefficients in gas-liquid-solid reversed flow jet loop bioreactor with a non-Newtonian fluid. Science.gov (United States) Jianping; Ping; Lin; Yunlin 2000-07-01 The local overall volumetric gas-liquid mass transfer coefficients at the specified point in a gas-liquid-solid three-phase reversed flow jet loop bioreactor (JLB) with a non-Newtonian fluid was experimentally investigated by a transient gassing-in method. The effects of liquid jet flow rate, gas jet flow rate, particle density, particle diameter, solids loading, nozzle diameter and CMC concentration on the local overall volumetric gas-liquid mass transfer coefficient (K(L)a) profiles were discussed. It was observed that local overall K(L)a profiles in the three-phase reversed flow JLB with non-Newtonian fluid increased with the increase of gas jet flow rate, liquid jet flow rate, particle density and particle diameter, but decreased with the increase of the nozzle diameter and CMC concentration. The presence of solids at a low concentration increased the local overall K(L)a profiles, and the optimum of solids loading for a maximum profile of the local overall K(L)a was found to be 0.18x10(-3)m(3) corresponding to a solids volume fraction, varepsilon(S)=2.8%. 10. Study on flow field characteristics of Non-Newtonian fluid in hydrocyclone%非牛顿流体在分离旋流器内流场特性研究 Institute of Scientific and Technical Information of China (English) 艾志久; 汪利霞; 刘晓明 2011-01-01 运用计算流体动力学(CFD)方法分别对旋流器内非牛顿流体与牛顿流体的流场分别进行数值模拟.分析采用RSM(SSG)雷诺应力模型,得到非牛顿流体与牛顿流体的速度场、压力场以及表观粘度分布规律.研究结果揭示了非牛顿流体在分离旋流器内的流场特性:由于受非牛顿流体表现粘度随剪切速率变化的影响,在同一位置处,非牛顿流体的静压力、轴向速度以及径向速度都大于牛顿流体,而切向速度小于牛顿流体；同时,旋流器内非牛顿流体的零轴速包络面(LZVV)比牛顿流体更加靠近器壁,这导致旋流器中非牛顿流体在同等条件下比牛顿流体的分离效率低；以上这些特性为进一步充分认识用于分离非牛顿流体的旋流器分离机理提供依据.%The method of Computational Fluid Dynamics was used to simulate the fluid field of the non-Newtonian fluid and the Newtonian fluid in the hydrocyclone. The Analysis was operated by reynolds stress model and obtained the velocity field,the pressure field as well as the apparent viscosity distribution of non-Newtonian fluid and the Newtonian fluid. The results revealed the Flow field Characteristics of the non-Newtonian fluids in the hydrocyclone: In the same position .with the effect of apparent viscosity changes of non-Newtonian fluid,the static pressure,the axial velocity as well as radial velocity of non-Newtonian fluid were bigger than Newtonian fluid's.and the radial velocity was less; The locus of zero vertical velocity of hydrocyclone of non-Newtonian was closer to the wall;these characteristics led to lower separation efficiency of non-Newtonian fluid than Newtonian fluid's under the same conditions. This study provided the evidence for new understand of the separation mechanism of non-Newtonian fluid. 11. MHD mixed convection flow of power law non-Newtonian fluids over an isothermal vertical wavy plate Science.gov (United States) 2015-09-01 Mixed convection flow of electrically conducting power law fluids along a vertical wavy surface in the presence of a transverse magnetic field is studied numerically. Prandtl coordinate transformation together with the spline alternating direction implicit method is employed to solve the boundary layer equations. The influences of both flow structure and dominant convection mode on the overall parameters of flow and heat transfer are well discussed. Also, the role of magnetic field in controlling the boundary layers is investigated. The variation of Nusselt number and skin friction coefficient are studied as functions of wavy geometry, magnetic field, buoyancy force and material parameters. Results reveal the interrelation of the contributing factors. 12. The quantification of hemodynamic parameters downstream of a Gianturco Zenith stent wire using newtonian and non-newtonian analog fluids in a pulsatile flow environment. Science.gov (United States) Walker, Andrew M; Johnston, Clifton R; Rival, David E 2012-11-01 Although deployed in the vasculature to expand vessel diameter and improve blood flow, protruding stent struts can create complex flow environments associated with flow separation and oscillating shear gradients. Given the association between magnitude and direction of wall shear stress (WSS) and endothelial phenotype expression, accurate representation of stent-induced flow patterns is critical if we are to predict sites susceptible to intimal hyperplasia. Despite the number of stents approved for clinical use, quantification on the alteration of hemodynamic flow parameters associated with the Gianturco Z-stent is limited in the literature. In using experimental and computational models to quantify strut-induced flow, the majority of past work has assumed blood or representative analogs to behave as Newtonian fluids. However, recent studies have challenged the validity of this assumption. We present here the experimental quantification of flow through a Gianturco Z-stent wire in representative Newtonian and non-Newtonian blood analog environments using particle image velocimetry (PIV). Fluid analogs were circulated through a closed flow loop at physiologically appropriate flow rates whereupon PIV snapshots were acquired downstream of the wire housed in an acrylic tube with a diameter characteristic of the carotid artery. Hemodynamic parameters including WSS, oscillatory shear index (OSI), and Reynolds shear stresses (RSS) were measured. Our findings show that the introduction of the stent wire altered downstream hemodynamic parameters through a reduction in WSS and increases in OSI and RSS from nonstented flow. The Newtonian analog solution of glycerol and water underestimated WSS while increasing the spatial coverage of flow reversal and oscillatory shear compared to a non-Newtonian fluid of glycerol, water, and xanthan gum. Peak RSS were increased with the Newtonian fluid, although peak values were similar upon a doubling of flow rate. The introduction of the 13. Dean vortex membrane microfiltration non-Newtonian viscosity effects NARCIS (Netherlands) Schutyser, M.A.I.; Belfort, G. 2002-01-01 Many industrial feeds behave as non-Newtonian fluids, and little understanding exists as to their influence on cross-flow microfiltration (CMF) performance. The viscosity effects of a model non-Newtonian shear-thickening fluid were investigated in CMF with and without suspended silica particles in t 14. Dean vortex membrane microfiltration non-Newtonian viscosity effects NARCIS (Netherlands) Schutyser, M.A.I.; Belfort, G. 2002-01-01 Many industrial feeds behave as non-Newtonian fluids, and little understanding exists as to their influence on cross-flow microfiltration (CMF) performance. The viscosity effects of a model non-Newtonian shear-thickening fluid were investigated in CMF with and without suspended silica particles in 15. Analysis of non-Newtonian effects on Low-Density Lipoprotein accumulation in an artery. Science.gov (United States) Iasiello, Marcello; Vafai, Kambiz; Andreozzi, Assunta; Bianco, Nicola 2016-06-14 In this work, non-Newtonian effects on Low-Density Lipoprotein (LDL) transport across an artery are analyzed with a multi-layer model. Four rheological models (Carreau, Carreau-Yasuda, power-law and Newtonian) are used for the blood flow through the lumen. For the non-Newtonian cases, the arterial wall is modeled with a generalized momentum equation. Convection-diffusion equation is used for the LDL transport through the lumen, while Staverman-Kedem-Katchalsky, combined with porous media equations, are used for the LDL transport through the wall. Results are presented in terms of filtration velocity, Wall Shear Stresses (WSS) and concentration profiles. It is shown that non-Newtonian effects on mass transport are negligible for a healthy intramural pressure value. Non-Newtonian effects increase slightly with intramural pressure, but Newtonian assumption can still be considered reliable. Effects of arterial size are also analyzed, showing that Newtonian assumption can be considered valid for both medium and large arteries, in predicting LDL deposition. Finally, non-Newtonian effects are also analyzed for an aorta-common iliac bifurcation, showing that Newtonian assumption is valid for mass transport at low Reynolds numbers. At a high Reynolds number, it has been shown that a non-Newtonian fluid model can have more impact due to the presence of flow recirculation. 16. Flow characteristics of Newtonian and non-Newtonian fluids in a vessel stirred by a 60° pitched blade impeller Directory of Open Access Journals (Sweden) Jamshid M. Nouri 2008-03-01 Full Text Available Mean and rms velocity characteristics of two Newtonian flows at Reynolds numbers of 12,800 (glycerin solution and 48,000 (water and of a non-Newtonian flow (0.2% CMC solution, at a power number similar to the Newtonian glycerin flow in a mixing vessel stirred by a 60° pitched blade impeller have been measured by laser Doppler velocimetry (LDV. The velocity measurements, resolved over 360° and 1.08° of impeller rotation, showed that the mean flow of the two power number matched glycerin and CMC flows were similar to within 3% of the impeller tip velocity and the turbulence intensities generally lower in the CMC flow by up to 5% of the tip velocity. The calculated mean flow quantities showed similar discharge coefficient and pumping efficiency in all three flows and similar strain rate between the two power number matched glycerin and CMC flows; the strain rate of the higher Reynolds number Newtonian flow was found to be slightly higher. The energy balance around the impeller indicated that the CMC flow dissipated up to 9% more of the total input power and converted 7% less into the turbulence compared to the glycerin flow with the same power input which could lead to less effective mixing processes where the micro-mixing is important. 17. Evolution of vortical structures in a curved artery model with non-Newtonian blood-analog fluid under pulsatile inflow conditions Science.gov (United States) Najjari, Mohammad Reza; Plesniak, Michael W. 2016-06-01 Steady flow and physiological pulsatile flow in a rigid 180° curved tube are investigated using particle image velocimetry. A non-Newtonian blood-analog fluid is used, and in-plane primary and secondary velocity fields are measured. A vortex detection scheme ( d 2-method) is applied to distinguish vortical structures. In the pulsatile flow case, four different vortex types are observed in secondary flow: deformed-Dean, Dean, Wall and Lyne vortices. Investigation of secondary flow in multiple cross sections suggests the existence of vortex tubes. These structures split and merge over time during the deceleration phase and in space as flow progresses along the 180° curved tube. The primary velocity data for steady flow conditions reveal additional vortices rotating in a direction opposite to Dean vortices—similar to structures observed in pulsatile flow—if the Dean number is sufficiently high. 18. MHD Effects on Non-Newtonian Power-Law Fluid Past a Continuously Moving Porous Flat Plate with Heat Flux and Viscous Dissipation Science.gov (United States) Kishan, N.; Shashidar Reddy, B. 2013-06-01 The problem of a magneto-hydro dynamic flow and heat transfer to a non-Newtonian power-law fluid flow past a continuously moving flat porous plate in the presence of sucion/injection with heat flux by taking into consideration the viscous dissipation is analysed. The non-linear partial differential equations governing the flow and heat transfer are transformed into non-linear ordinary differential equations using appropriate transformations and then solved numerically by an implicit finite difference scheme. The solution is found to be dependent on various governing parameters including the magnetic field parameter M, power-law index n, suction/injection parameter ƒw, Prandtl number Pr and Eckert number Ec. A systematical study is carried out to illustrate the effects of these major parameters on the velocity profiles, temperature profile, skin friction coefficient and rate of heat transfer and the local Nusslet number. 19. Effect of Viscous Dissipation on MHD Free Convection Flow Heat and Mass Transfer of Non-Newtonian Fluids along a Continuously Moving Stretching Sheet Directory of Open Access Journals (Sweden) K.C. Saha 2015-04-01 Full Text Available The effects of MHD free convection heat and mass transfer of power-law Non-Newtonian fluids along a stretching sheet with viscous dissipation has been analyzed. This has been done under the simultaneous action of suction, thermal radiation and uniform transverse magnetic field. The stretching sheet is assumed to continuously moving with a power-law velocity and maintaining a uniform surface heat-flux. The governing non-linear partial differential equations are transformed into non-linear ordinary differential equations, using appropriate similarity transformations and the resulting problem is solved numerically using Nachtsheim-Swigert shooting iteration technique along with sixth order Runge-Kutta integration scheme. A parametric study of the parameters arising in the problem such as the Eckert number due to viscous dissipation, radiation number, buoyancy parameter, Schmidt number, Prandtl number etc are studied and the obtained results are shown graphically and the physical aspects of the problem are discussed. 20. Particle migration using local variation of the viscosity (LVOV) model in flow of a non-Newtonian fluid for ceramic tape casting DEFF Research Database (Denmark) Jabbaribehnam, Mirmasoud; Spangenberg, Jon; Hattel, Jesper Henri 2016-01-01 In this paper, the migration of secondary particles in a non-Newtonian ceramic slurry inthe tape casting process is investigated with the purpose of understanding the particle distribution patterns along the casting direction. The Ostwald-de Waele power law model for the non-Newtonian flow... 1. Influence of yield stress on free convective boundary-layer flow of a non-Newtonian nanofluid past a vertical plate in a porous medium Energy Technology Data Exchange (ETDEWEB) Hady, F. M.; Ibrahim, F. S. [Assiut University, Assiut (Egypt); Abdel-Gaied, S. M.; Eid, M. R. [Assiut University, The New Valley (Egypt) 2011-08-15 The effect of yield stress on the free convective heat transfer of dilute liquid suspensions of nanofluids flowing on a vertical plate saturated in porous medium under laminar conditions is investigated considering the nanofluid obeys the mathematical model of power-law. The model used for non-Newtonian nanofluid incorporates the effects of Brownian motion and thermophoresis. The governing boundary- layer equations are cast into dimensionless system which is solved numerically using a deferred correction technique and Newton iteration. This solution depends on yield stress parameter {Omega}, a power-law index n, Lewis number Le, a buoyancy-ratio number Nr, a Brownian motion number Nb, and a thermophoresis number Nt. Analyses of the results found that the reduced Nusselt and Sherwood numbers are decreasing functions of the higher yield stress parameter for each dimensionless numbers, n and Le, except the reduced Sherwood number is an increasing function of higher Nb for different values of yield stress parameter. 2. Unsteady Boundary-Layer Flow over Jerked Plate Moving in a Free Stream of Viscoelastic Fluid Directory of Open Access Journals (Sweden) Sufian Munawar 2014-01-01 Full Text Available This study aims to investigate the unsteady boundary-layer flow of a viscoelastic non-Newtonian fluid over a flat surface. The plate is suddenly jerked to move with uniform velocity in a uniform stream of non-Newtonian fluid. Purely analytic solution to governing nonlinear equation is obtained. The solution is highly accurate and valid for all values of the dimensionless time 0≤τ<∞. Flow properties of the viscoelastic fluid are discussed through graphs. 3. Stability Analysis of Non-Newtonian Rimming Flow CERN Document Server Fomin, Sergei; Haine, Peter 2015-01-01 The rimming flow of a viscoelastic thin film inside a rotating horizontal cylinder is studied theoretically. Attention is given to the onset of non-Newtonian free-surface instability in creeping flow. This non-inertial instability has been observed in experiments, but current theoretical models of Newtonian fluids can neither describe its origin nor explain its onset. This study examines two models of non Newtonian fluids to see if the experimentally observed instability can be predicted analytically. The non-Newtonian viscosity and elastic properties of the fluid are described by the Generalized Newtonian Fluid (GNF) and Second Order Viscoelastic Fluid (SOVF) constitutive models, respectively. With linear stability analysis, it is found that, analogously to the Newtonian fluid, rimming flow of viscous non-Newtonian fluids (modeled by GNF) is neutrally stable. However, the viscoelastic properties of the fluid (modeled by SOVF) are found to contribute to the flow destabilization. The instability is shown to in... 4. Non newtonian annular alloy solidification in mould Energy Technology Data Exchange (ETDEWEB) Moraga, Nelson O.; Garrido, Carlos P. [Universidad de La Serena, Departamento de Ingenieria Mecanica, La Serena (Chile); Castillo, Ernesto F. [Universidad de Santiago de Chile, Departamento de Ingenieria Mecanica, Santiago (Chile) 2012-08-15 The annular solidification of an aluminium-silicon alloy in a graphite mould with a geometry consisting of horizontal concentric cylinders is studied numerically. The analysis incorporates the behavior of non-Newtonian, pseudoplastic (n=0.2), Newtonian (n=1), and dilatant (n=1.5) fluids. The fluid mechanics and heat transfer coupled with a transient model of convection diffusion are solved using the finite volume method and the SIMPLE algorithm. Solidification is described in terms of a liquid fraction of a phase change that varies linearly with temperature. The final results make it possible to infer that the fluid dynamics and heat transfer of solidification in an annular geometry are affected by the non-Newtonian nature of the fluid, speeding up the process when the fluid is pseudoplastic. (orig.) 5. An experimental study of convective heat transfer, friction, and rheology for non-Newtonian fluids: Polymer solutions, suspensions of fibers, and suspensions of particulates Science.gov (United States) Matthys, E. F. The convective heat transfer, friction, and rheological properties of various types of nonNewtonian fluid in circular tube flows were investigated. If an apparent Reynolds number is used and if the temperature and degradation effects are properly taken into account, the reduced turbulent friction and heat transfer results, respectively, are then shown to be well correlated by the same expressions for different fluids, regardless of the nature of the fluids and whether they are shear-thinning or shear-thickening. This representation can also separate the reductions in turbulent heat transfer and friction that are induced by viscoelasticity from those induced by pseudoplasticity. Polyacrylamide solutions inducing asymptotic and intermediate drag reduction regimes were investigated over a broad range of Reynolds numbers. A kerosene-based antimisting polymer solution was also studied. Suspensions of bentonite of various concentrations were investigated in laminar and turbulent regimes, and the results for fully developed and entrance flows were well correlated by Newtonian relationships when an adequate wall viscosity concept was used. 6. NUMERICAL INVESTIGATION OF NON-NEWTONIAN DRILLING FLUIDS DURING THE OCCURRENCE OF A GAS KICK IN A PETROLEUM RESERVOIR Directory of Open Access Journals (Sweden) F. F. Oliveira Full Text Available Abstract In this work, a simplified kick simulator is developed using the ANSYS® CFX software in order to better understand the phenomena called kick. This simulator is based on the modeling of a petroleum well where a gas kick occurs. Dynamic behavior of some variables like pressure, viscosity, density and volume fraction of the fluid is analyzed in the final stretch of the modeled well. In the simulations nine different drilling fluids are used of two rheological categories, Ostwald de Waele, also known as Power-Law, and Bingham fluids, and the results are compared among them. In these comparisons what fluid allows faster or slower invasion of gas is analyzed, as well as how the gas spreads into the drilling fluid. The pressure behavior during the kick process is also compared t. It is observed that, for both fluids, the pressure behavior is similar to a conventional leak in a pipe. 7. Exact solutions for the flow of non-Newtonian fluid with fractional derivative in an annular pipe Science.gov (United States) Tong, Dengke; Wang, Ruihe; Yang, Heshan 2005-08-01 This paper deals with some unsteady unidirectional transient flows of Oldroyd-B fluid in an annular pipe. The fractional calculus approach in the constitutive relationship model Oldroyd-B fluid is introduced and a generalized Jeffreys model with the fractional calculus has been built. Exact solutions of some unsteady flows of Oldroyd-B fluid in an annular pipe are obtained by using Hankel transform and Laplace transform for fractional calculus. The following four problems have been studied: (1) Poiseuille flow due to a constant pressure gradient; (2) axial Couette flow in an annulus; (3) axial Couette flow in an annulus due to a longitudinal constant shear; (4) Poiseuille flow due to a constant pressure gradient and a longitudinal constant shear. The well-known solutions for Navier-Stokes fluid, as well as those corresponding to a Maxwell fluid and a second grade one, appear as limited cases of our solutions. 8. Exact solutions for the flow of non-Newtonian fluid with fractional derivative in an annular pipe Institute of Scientific and Technical Information of China (English) TONG Dengke; WANG Ruihe; YANG Heshan 2005-01-01 This paper deals with some unsteady unidirectional transient flows of Oldroyd-B fluid in an annular pipe. The fractional calculus approach in the constitutive relationship model Oldroyd-B fluid is introduced and a generalized Jeffreys model with the fractional calculus has been built. Exact solutions of some unsteady flows of Oldroyd-B fluid in an annular pipe are obtained by using Hankel transform and Laplace transform for fractional calculus. The following four problems have been studied: (1) Poiseuille flow due to a constant pressure gradient; (2) axial Couette flow in an annulus; (3) axial Couette flow in an annulus due to a longitudinal constant shear; (4) Poiseuille flow due to a constant pressure gradient and a longitudinal constant shear. The well-known solutions for Navier-Stokes fluid, as well as those corresponding to a Maxwell fluid and a second grade one, appear as limited cases of our solutions. 9. The flow of a non-Newtonian fluid induced due to the oscillations of a porous plate Directory of Open Access Journals (Sweden) S. Asghar 2004-01-01 Full Text Available An analytic solution of the flow of a third-grade fluid on a porous plate is constructed. The porous plate is executing oscillations in its own plane with superimposed injection or suction. An increasing or decreasing velocity amplitude of the oscillating porous plate is also examined. It is also shown that in case of third-grade fluid, a combination of suction/injection and decreasing/increasing velocity amplitude is possible as well. Several limiting situations with their implications are given and discussed. 10. Simple Navier’s slip boundary condition for the non-Newtonian Lattice Boltzmann fluid dynamics solver DEFF Research Database (Denmark) Svec, Oldrich; Skoček, Jan 2013-01-01 The ability of the Lattice Boltzmann method, as the fluid dynamics solver, to properly simulate macroscopic Navier’s slip boundary condition is investigated. An approximate equation relating the Lattice Boltzmann variable slip boundary condition with the macroscopic Navier’s slip boundary condition... 11. Using a tracer technique to identify the extent of non-ideal flows in the continuous mixing of non-Newtonian fluids Directory of Open Access Journals (Sweden) Mehrvar M. 2013-05-01 Full Text Available The identification of non-ideal flows in a continuous-flow mixing of non-Newtonian fluids is a challenging task for various chemical industries: plastic manufacturing, water and wastewater treatment, and pulp and paper manufacturing. Non-ideal flows such as channelling, recirculation, and dead zones significantly affect the performance of continuous-flow mixing systems. Therefore, the main objective of this paper was to develop an identification protocol to measure non-ideal flows in the continuous-flow mixing system. The extent of non-ideal flows was quantified using a dynamic model that incorporated channelling, recirculation, and dead volume in the mixing vessel. To estimate the dynamic model parameters, the system was excited using a frequency-modulated random binary input by injecting the saline solution (as a tracer into the fresh feed stream prior to being pumped into the mixing vessel. The injection of the tracer was controlled by a computer-controlled on-off solenoid valve. Using the trace technique, the extent of channelling and the effective mixed volume were successfully determined and used as mixing quality criteria. Such identification procedures can be applied at various areas of chemical engineering in order to improve the mixing quality. 12. Numerical investigation of MHD free convection flow of a non-Newtonian fluid past an impulsively started vertical plate in the presence of thermal diffusion and radiation absorption Directory of Open Access Journals (Sweden) M. Umamaheswar 2016-09-01 Full Text Available A numerical investigation is carried out on an unsteady MHD free convection flow of a well-known non-Newtonian visco elastic second order Rivlin-Erickson fluid past an impulsively started semi-infinite vertical plate in the presence of homogeneous chemical reaction, thermal radiation, thermal diffusion, radiation absorption and heat absorption with constant mass flux. The presence of viscous dissipation is also considered at the plate under the influence of uniform transverse magnetic field. The flow is governed by a coupled nonlinear system of partial differential equations which are solved numerically by using finite difference method. The effects of various physical parameters on the flow quantities viz. velocity, temperature, concentration, Skin friction, Nusselt number and Sherwood number are studied numerically. The results are discussed with the help of graphs. We observed that the velocity decreases with an increase in magnetic field parameter, Schmidt number, and Prandtl number while it increases with an increase in Grashof number, modified Grashof number, visco-elastic parameter and Soret number. Temperature increases with an increase in radiation absorption parameter, Eckert number and visco-elastic parameter while it decreases with increasing values of radiation parameter, Prandtl number and heat absorption parameter. Concentration increases with increase in Soret number while it decreases with an increase in Schmidt number and chemical reaction parameter. 13. Effects of variable viscosity and thermal conductivity on unsteady MHD flow of non-Newtonian fluid over a stretching porous sheet Directory of Open Access Journals (Sweden) Rahman Abdel-Gamal M. 2013-01-01 Full Text Available The unsteady flow and heat transfer in an incompressible laminar, electrically conducting and non-Newtonian fluid over a non-isothermal stretching sheet with the variation in the viscosity and thermal conductivity in a porous medium by the influence of an external transverse magnetic field have been obtained and studied numerically. By using similarity analysis the governing differential equations are transformed into a set of non-linear coupled ordinary differential equations which are solved numerically. Numerical results were presented for velocity and temperature profiles for different parameters of the problem as power law parameter, unsteadiness parameter, radiation parameter, magnetic field parameter, porous medium parameter, temperature buoyancy parameter, Prandtl parameter, modified Eckert parameter, Joule heating parameter , heat source/sink parameter and others. A comparison with previously published work has been carried out and the results are found to be in good agreement. Also the effects of the pertinent parameters on the skin friction and the rate of heat transfer are obtained and discussed numerically and illustrated graphically. 14. Using a tracer technique to identify the extent of non-ideal flows in the continuous mixing of non-Newtonian fluids Science.gov (United States) Patel, D.; Ein-Mozaffari, F.; Mehrvar, M. 2013-05-01 The identification of non-ideal flows in a continuous-flow mixing of non-Newtonian fluids is a challenging task for various chemical industries: plastic manufacturing, water and wastewater treatment, and pulp and paper manufacturing. Non-ideal flows such as channelling, recirculation, and dead zones significantly affect the performance of continuous-flow mixing systems. Therefore, the main objective of this paper was to develop an identification protocol to measure non-ideal flows in the continuous-flow mixing system. The extent of non-ideal flows was quantified using a dynamic model that incorporated channelling, recirculation, and dead volume in the mixing vessel. To estimate the dynamic model parameters, the system was excited using a frequency-modulated random binary input by injecting the saline solution (as a tracer) into the fresh feed stream prior to being pumped into the mixing vessel. The injection of the tracer was controlled by a computer-controlled on-off solenoid valve. Using the trace technique, the extent of channelling and the effective mixed volume were successfully determined and used as mixing quality criteria. Such identification procedures can be applied at various areas of chemical engineering in order to improve the mixing quality. 15. Existence and uniqueness theorem for flow and heat transfer of a non-Newtonian fluid over a stretching sheet Institute of Scientific and Technical Information of China (English) SAHOO Bikash; SHARMA H.G. 2007-01-01 Analysis is carried out to study the existence, uniqueness and behavior of exact solutions of the fourth order nonlinear coupled ordinary differential equations arising in the flow and heat transfer of a viscoelastic, electrically conducting fluid past a continuously stretching sheet. The ranges of the parametric values are obtained for which the system has a unique pair of solutions,a double pair of solutions and infinitely many solutions. 16. Influence of the combined effect of magnetic field and rotation on the onset of a non-Newtonian viscoelastic nanofluid layer: Linear and nonlinear analyses Science.gov (United States) Khurana, Meenakshi; Rana, Puneet; Srivastava, Sangeet 2016-12-01 In the present paper, we present both linear and nonlinear analyses to investigate thermal instability on a rotating non-Newtonian viscoelastic nanofluid layer under the influence of a magnetic field. In the linear stability analysis, the stationary and oscillatory modes of convection are obtained for various controlling parameters using the normal mode technique. Both Nusselt and Sherwood numbers are calculated after employing the minimal truncated Fourier series to steady and unsteady state. The main findings conclude that rotation and strain retardation parameter increase the value of the critical Rayleigh number in the neutral stability curve which delays the onset of convection in the nanofluid layer while the stress relaxation parameter enhances the convection. The magnetic field stabilizes the system for low values of the Taylor number (rotation) but an inverse trend is observed for high Taylor number. Both Nusselt and Sherwood numbers initially oscillate with time until the steady state prevails and they decrease with both Chandrasekhar and Taylor numbers. The magnitude of the streamlines and the contours of both isotherms and iso-nanohalines concentrate near the boundaries for large values of Ra, indicating an increase in convection. 17. Shock Wave Solutions for Some Nonlinear Flow Models Arising in the Study of a Non-Newtonian Third Grade Fluid Directory of Open Access Journals (Sweden) Taha Aziz 2013-01-01 Full Text Available This study is based upon constructing a new class of closed-form shock wave solutions for some nonlinear problems arising in the study of a third grade fluid model. The Lie symmetry reduction technique has been employed to reduce the governing nonlinear partial differential equations into nonlinear ordinary differential equations. The reduced equations are then solved analytically, and the shock wave solutions are constructed. The conditions on the physical parameters of the flow problems also fall out naturally in the process of the derivation of the solutions. 18. Time-dependent electrokinetic flows of non-Newtonian fluids in microchannel-array for energy conversion Science.gov (United States) Chun, Myung-Suk; Chun, Byoungjin; Lee, Ji-Young; Complex Fluids Team 2016-11-01 We investigate the externally time-dependent pulsatile electrokinetic viscous flows by extending the previous simulations concerning the electrokinetic microfluidics for different geometries. The external body force originated from between the nonlinear Poisson-Boltzmann field and the flow-induced electric field is employed in the Cauchy momentum equation, and then the Nernst-Planck equation in connection with the net current conservation is coupled. Our explicit model allows one to quantify the effects of the oscillating frequency and conductance of the Stern layer, considering the shear thinning effect and the strong electric double layer interaction. This presentation reports the new results regarding the implication of optimum frequency pressure pulsations toward realizing mechanical to electrical energy transfer with high conversion efficiencies. These combined factors for different channel dimension are examined in depth to obtain possible enhancements of streaming current, with taking advantage of pulsating pressure field. From experimental verifications by using electrokinetic power chip, it is concluded that our theoretical framework can serve as a useful basis for micro/nanofluidics design and potential applications to the enhanced energy conversion. NRF of Korea (No.2015R1A2A1A15052979) and KIST (No.2E26490). 19. Structural Optimization of non-Newtonian Microfluidics DEFF Research Database (Denmark) Jensen, Kristian Ejlebjærg 2013-01-01 Many of the biological fluids analyzed in Lab-on-a-Chip systems contain elastic components, which gives the fluids elastic character. Such fluids are said to be non-Newtonian or, more precisely, viscoelastic. They can give rise to exotic effects on the macroscale, which are never seen for fluids...... with components relying on viscoelastic effects, but the non-intuitive nature of these fluids complicates the design process. This thesis combines the method of topology optimization with differential constitutive equations, which govern the flow of viscoelastic fluids. The optimization method iteratively...... experimentally, and compared the results with the established hyperbolic designs. We found superior performance in the parameter regime of the optimization as well as similar optimal performance [P3]. The cross-slot geometry is known to exhibit bistability for viscoelastic fluids. We studied this geometry... 20. Non-Newtonian Momentum Transfer past an Isothermal Stretching Sheet with Applied Suction Science.gov (United States) Veena, P. H.; Suresh, B.; Pravin, V. K.; Goud, A. M. 2017-08-01 The paper discusses the flow of an incompressible non-Newtonian fluid due to stretching of a plane elastic surface in a saturated porous medium in the approximation of boundary layer theory. An exact analytical solution of non-linear MHD momentum equation governing the self-similar flow is given. The skin friction co-efficient decreases with an increase in the visco-elastic parameter k1 and increase in the values of both the magnetic parameter and permeability parameter. 1. 外形任意的多孔介质轴对称物体中充满非Newton幂律流体时的自然对流%Natural Convection of Non-Newtonian Power-Law Fluid Over Axisymmetric and Two-Dimensional Bodies of Arbitrary Shape in a Fluid-Saturated Porous Medium Institute of Scientific and Technical Information of China (English) S·M·阿布德尔-盖德; M·R·伊德 2011-01-01 在一个轴对称、外形任意的多孔介质二维体中,充满了有屈服应力的非Newton幂律流体时,数值分析其自由对流及其传热/传质问题,利用相似变换,将边界层控制方程及其边界条件变换为无量纲形式,然后用有限差分法求解该方程组.所研究的参数为流变常数、浮力比和Lewis数.给出并讨论了典型的速度、温度及浓度曲线,发现屈服应力参数值和非Newton流体的幂律指数对结果有着显著的影响.%Numerical analysis of free convection coupled heat and mass transfer was presented for non-Newtonian power-law fluids with yield stress flowing over two-dimensional or axisymmetric body of arbitrary shape in a fluid-saturated porous medium.The governing boundary layer equations and boundary conditions were cast into a dimensionless form by similarity transformation and the resulting system of equations was solved by a finite difference method.The parameters studied were the rheologicai constants, the buoyancy ratio, and the Lewis number.Representative velocity as well as temperature and concentration profiles were presented and discussed.It was found that the result depend strongly on the values of the yield stress parameter, and the power-law index of non-Newtonian fluid. 2. Numerical analysis of non-Newtonian rheology effect on hydrocyclone flow field Directory of Open Access Journals (Sweden) Lin Yang 2015-03-01 Full Text Available In view of the limitations of the existing Newton fluid effects on the vortex flow mechanism study, numerical analysis of non Newton fluid effects was presented. Using Reynolds stress turbulence model (RSM and mixed multiphase flow model (Mixture of FLUENT (fluid calculation software and combined with the constitutive equation of apparent viscosity of non-Newtonian fluid, the typical non-Newtonian fluid (drilling fluid, polymer flooding sewage and crude oil as medium and Newton flow field (water as medium were compared by quantitative analysis. Based on the research results of water, the effects of non-Newtonian rheology on the key parameters including the combined vortex motion index n and tangential velocity were analyzed. The study shows that: non-Newtonian rheology has a great effect on tangential velocity and n value, and tangential velocity decreases with non-Newtonian increasing. The three kinds of n values (constant segment are: 0.564(water, 0.769(polymer flooding sewage, 0.708(drilling fluid and their variation amplitudes are larger than Newtonian fluid. The same time, non-Newtonian rheology will lead to the phenomenon of turbulent drag reduction in the vortex flow field. Compared with the existing formula calculation results shown, the calculation result of non-Newtonian rheology is most consistent with the simulation result, and the original theory has large deviations. The study provides reference for theory research of non-Newtonian cyclone separation flow field. 3. Physiological non-Newtonian blood flow through single stenosed artery Science.gov (United States) 2016-07-01 A numerical simulation to investigate the Non-Newtonian modelling effects on physiological flows in a three dimensional idealized artery with a single stenosis of 85% severity. The wall vessel is considered to be rigid. Oscillatory physiological and parabolic velocity profile has been imposed for inlet boundary condition. Where the physiological waveform is performed using a Fourier series with sixteen harmonics. The investigation has a Reynolds number range of 96 to 800. Low Reynolds number k - ω model is used as governing equation. The investigation has been carried out to characterize two Non-Newtonian constitutive equations of blood, namely, (i) Carreau and (ii) Cross models. The Newtonian model has also been investigated to study the physics of fluid. The results of Newtonian model are compared with the Non-Newtonian models. The numerical results are presented in terms of pressure, wall shear stress distributions and the streamlines contours. At early systole pressure differences between Newtonian and Non-Newtonian models are observed at pre-stenotic, throat and immediately after throat regions. In the case of wall shear stress, some differences between Newtonian and Non-Newtonian models are observed when the flows are minimum such as at early systole or diastole. 4. Crossover phenomena in non-Newtonian viscous fingers at a finite viscosity ratio Science.gov (United States) Nagatani, Takashi 1990-04-01 A viscous fingering of non-Newtonian fluids at a finite viscosity ratio is considered in order to study the effect of non-Newtonian fluid on crossover phenomena. The crossover from the fractal pattern to the dense structure is investigated by using a two-parameter position-space renormalization-group method. The global flow diagrams in two-parameter space are obtained. It is found that there are two nontrivial fixed points: the fractal point and the Eden point. When the viscosity ratio is finite, the pattern must eventually cross over to the dense structure. The dependences of the crossover phenomena on the parameter k, which describes the different non-Newtonian fluids, are shown. It is found that the non-Newtonian fluids have important effects on the fractal point and the crossover line but the crossover exponent is independent of the non-Newtonian property. 5. Implicit Partitioned Cardiovascular Fluid-Structure Interaction of the Heart Cycle Using Non-newtonian Fluid Properties and Orthotropic Material Behavior. Science.gov (United States) Muehlhausen, M-P; Janoske, U; Oertel, H 2015-03-01 Although image-based methods like MRI are well-developed, numerical simulation can help to understand human heart function. This function results from a complex interplay of biochemistry, structural mechanics, and blood flow. The complexity of the entire system often causes one of the three parts to be neglected, which limits the truth to reality of the reduced model. This paper focuses on the interaction of myocardial stress distribution and ventricular blood flow during diastole and systole in comparison to a simulation of the same patient-specific geometry with a given wall movement (Spiegel, Strömungsmechanischer Beitrag zur Planung von Herzoperationen, 2009). The orthotropic constitutive law proposed by Holzapfel et al. (Philos. Trans. R. Soc. Lond. Ser. A, 367:3445-3475, 2009) was implemented in a finite element package to model the passive behavior of the myocardium. Then, this law was modified for contraction. Via the ALE method, the structural model was coupled to a flow model which incorporates blood rheology and the circulatory system (Oertel, Prandtl-Essentials of Fluid Mechanics, 3rd edn, Springer Science + Business Media, 2010; Oertel et al., Modelling the Human Cardiac Fluid Mechanics, 3rd edn, Universitätsverlag Karlsruhe, 2009). Comparison reveals a good quantitative and qualitative agreement with respect to fluid flow. The motion of the myocardium is consistent with physiological observations. The calculated stresses and the distribution are within the physiological range and appear to be reasonable. The coupled model presented contains many features essential to cardiac function. It is possible to calculate wall stresses as well as the characteristic ventricular fluid flow. Based on the simulations we derive two characteristics to assess the health state quantitatively including solid and fluid mechanical aspects. 6. Boundary layer flow and heat transfer to Carreau fluid over a nonlinear stretching sheet OpenAIRE Masood Khan; Hashim 2015-01-01 This article studies the Carreau viscosity model (which is a generalized Newtonian model) and then use it to obtain a formulation for the boundary layer equations of the Carreau fluid. The boundary layer flow and heat transfer to a Carreau model over a nonlinear stretching surface is discussed. The Carreau model, adequate for many non-Newtonian fluids, is used to characterize the behavior of the fluids having shear thinning properties and fluids with shear thickening properties for numerical ... 7. Inelastic non-Newtonian flow over heterogeneously slippery surfaces NARCIS (Netherlands) Haase, A. Sander; Wood, Jeffery A.; Sprakel, Lisette M.J.; Lammertink, Rob G.H. 2017-01-01 In this study, we investigated inelastic non-Newtonian fluid flow over heterogeneously slippery surfaces. First, we simulated the flow of aqueous xanthan gum solutions over a bubble mattress, which is a superhydrophobic surface consisting of transversely positioned no-slip walls and no-shear gas bub 8. Semiclassical law for the apparent viscosity of non-Newtonian fluids: An analogy between thixotropy of fluids and sintering of solids Science.gov (United States) Mezzasalma, Stefano A. 2000-08-01 A theory is presented to describe the apparent viscosity of thixotropic fluids as a function of the rate of shear. It represents the extension of a semiclassical approach that was previously formulated to deal with matter densification phenomena in solids starting from the state equation of the medium. In this context, the Debye expression for the Helmholtz free energy has been provided with a density of vibrational modes that accounts for atomic and microstructural changes occurring at the frequency scale of momentum transport (see diffusion). Working out the steady-state condition with respect to time gives an equation relating reduced apparent viscosity (η˜) and shear rate (γ˜) through the temperature value (θ) that is energetically equivalent to the medium vibrations implied. Viscosity also turns out to depend on the Debye temperature θD (see φ˜θ/θD) and an equivalent Gruneisen parameter (μ), defined with respect to viscosity variations. Increasing φ in pseudoplastic and dilatant media, respectively, increases and decreases η˜, which always increases with increasing μ. The analogy between dilatancy/sintering and pseudoplasticity/desintering is suggested, and a correspondence between matter and momentum transports is traced on the basis of the phononic spectrum properties. Application to experimental measurements are presented and discussed for aqueous monodispersions of polystyrene (PS) latex particles, aqueous glycerol solutions of partially hydrolyzed polyacrylamide (PHPAA) at different sodium chloride (NaCl) concentrations, polymethylmethacrylate (PMMA) suspensions in dioctylphthalate (DOP), and for a molecularly thin liquid film of octamethylciclotetrasiloxane (OMCTS). Best fit coefficients for φ and μ have been constrained to the Debye temperature and the effective low-shear viscosity (η0) according to their dependences upon the suspended volume fraction (φ), θD=θD(φ), and η0=η0(φ), and the agreement with experimental data is 9. Weakly nonlinear analysis of Rayleigh-Bénard convection in a non-Newtonian fluid between plates of finite conductivity: Influence of shear-thinning effects. Science.gov (United States) Bouteraa, Mondher; Nouar, Chérif 2015-12-01 Finite-amplitude thermal convection in a shear-thinning fluid layer between two horizontal plates of finite thermal conductivity is considered. Weakly nonlinear analysis is adopted as a first approach to investigate nonlinear effects. The rheological behavior of the fluid is described by the Carreau model. As a first step, the critical conditions for the onset of convection are computed as a function of the ratio ξ of the thermal conductivity of the plates to the thermal conductivity of the fluid. In agreement with the literature, the critical Rayleigh number Ra(c) and the critical wave number k(c) decrease from 1708 to 720 and from 3.11 to 0, when ξ decreases from infinity to zero. In the second step, the critical value α(c) of the shear-thinning degree above which the bifurcation becomes subcritical is determined. It is shown that α(c) increases with decreasing ξ. The stability of rolls and squares is then investigated as a function of ξ and the rheological parameters. The limit value ξ(c), below which squares are stable, decreases with increasing shear-thinning effects. This is related to the fact that shear-thinning effects increase the nonlinear interactions between sets of rolls that constitute the square patterns [M. Bouteraa et al., J. Fluid Mech. 767, 696 (2015)]. For a significant deviation from the critical conditions, nonlinear convection terms and nonlinear viscous terms become stronger, leading to a further diminution of ξ(c). The dependency of the heat transfer on ξ and the rheological parameters is reported. It is consistent with the maximum heat transfer principle. Finally, the flow structure and the viscosity field are represented for weakly and highly conducting plates. 10. Boundary Layer Equations and Lie Group Analysis of a Sisko Fluid Directory of Open Access Journals (Sweden) Gözde Sarı 2012-01-01 Full Text Available Boundary layer equations are derived for the Sisko fluid. Using Lie group theory, a symmetry analysis of the equations is performed. A partial differential system is transferred to an ordinary differential system via symmetries. Resulting equations are numerically solved. Effects of non-Newtonian parameters on the solutions are discussed. 11. 低气速条件下CO2在牛顿及非牛顿流体中气含率%Gas holdup of CO2 in Newtonian and non-Newtonian fluid at low gas velocity Institute of Scientific and Technical Information of China (English) 李少白; 马友光; 付涛涛; 朱春英 2012-01-01 实验测定了低气速下CO2气泡群在牛顿流体、剪切变稀流体及黏弹性流体中的气含率.讨论了流体的流变性、质量分数及表观气速对气含率的影响.结果表明:在3种不同性质的流体中,气含率均随表观气速的增大而增大.同时发现流体性质对气含率具有不同的影响:对于牛顿流体,表观气速较低时,质量分数对气含率影响可忽略；对于非牛顿流体,气含率随着流动指数n的减小而减小,即剪切变稀效应对气含率有负作用,而黏弹性对气含率的影响可忽略.气含率是气液传质过程设计中最重要的参数,因此研究结果为进一步研究CO2气泡群在非牛顿流体中的传质奠定了一定基础.%The gas holdups of CO2 bubble swarm at low superficial gas velocity in Newtonian fluid, shear thinned fluid and viscoelastic fl uid were measured. The influences of rheological property, mass fraction and superficial gas velocity on the gas holdup were investigated. The results show that the gas holdups in three fluids all increase with the increase of superficial gas velocity, and different fluid has different effects on the gas holdup. The effect of mass fraction on the gas holdup in Newtonian fluid at low superficial gas velocity is negligible. For non-Newtonian fluid, the gas holdup decreases with the decrease of flow index n, that is, shear thinning effect has negative impact on the gas holdup, and the influence of viscoelasticity on the gas holdup is negligible. The gas holdup is the most important parameter for the gas-liquid mass transfer process design, and it provides a solid foundation for the research on mass transfer of CO2 bubble swarm in non-Newtonian fluid. 12. Numerical study of flow and heat transfer of non-Newtonian Tangent Hyperbolic fluid from a sphere with Biot number effects Directory of Open Access Journals (Sweden) S. Abdul Gaffar 2015-12-01 Full Text Available In this article, we investigate the nonlinear steady boundary layer flow and heat transfer of an incompressible Tangent Hyperbolic fluid from a sphere. The transformed conservation equations are solved numerically subject to physically appropriate boundary conditions using implicit finite-difference Keller Box technique. The numerical code is validated with previous studies. The influence of a number of emerging non-dimensional parameters, namely Weissenberg number (We, power law index (n, Prandtl number (Pr, Biot number (γ and dimensionless tangential coordinate (ξ on velocity and temperature evolution in the boundary layer regime is examined in detail. Furthermore, the effects of these parameters on heat transfer rate and skin friction are also investigated. Validation with earlier Newtonian studies is presented and excellent correlation is achieved. It is found that the velocity, Skin friction and the Nusselt number (heat transfer rate are decreased with increasing Weissenberg number (We, whereas the temperature is increased. Increasing power law index (n increases the velocity and the Nusselt number (heat transfer rate but decreases the temperature and the Skin friction. An increase in the Biot number (γ is observed to increase velocity, temperature, local skin friction and Nusselt number. The study is relevant to chemical materials processing applications. 13. Approximate analytical solutions and approximate value of skin friction coefficient for boundary layer of power law fluids Institute of Scientific and Technical Information of China (English) SU Xiao-hong; ZHENG Lian-cun; JIANG Feng 2008-01-01 This paper presents a theoretical analysis for laminar boundary layer flow in a power law non-Newtonian fluids.The Adomian analytical decomposition technique is presented and an approximate analytical solution is obtained.The approximate analytical solution can be expressed in terms of a rapid convergent power series with easily computable terms.Reliability and efficiency of the approximate solution are verified by comparing with numerical solutions in the literature.Moreover,the approximate solution can be successfully applied to provide values for the skin friction coefficient of the laminar boundary layer flow in power law non-Newtonian fluids. 14. Non-Newtonian particulate flow simulation: A direct-forcing immersed boundary-lattice Boltzmann approach Science.gov (United States) Amiri Delouei, A.; Nazari, M.; Kayhani, M. H.; Kang, S. K.; Succi, S. 2016-04-01 In the current study, a direct-forcing immersed boundary-non-Newtonian lattice Boltzmann method (IB-NLBM) is developed to investigate the sedimentation and interaction of particles in shear-thinning and shear-thickening fluids. In the proposed IB-NLBM, the non-linear mechanics of non-Newtonian particulate flows is detected by combination of the most desirable features of immersed boundary and lattice Boltzmann methods. The noticeable roles of non-Newtonian behavior on particle motion, settling velocity and generalized Reynolds number are investigated by simulating benchmark problem of one-particle sedimentation under the same generalized Archimedes number. The effects of extra force due to added accelerated mass are analyzed on the particle motion which have a significant impact on shear-thinning fluids. For the first time, the phenomena of interaction among the particles, such as Drafting, Kissing, and Tumbling in non-Newtonian fluids are investigated by simulation of two-particle sedimentation and twelve-particle sedimentation. The results show that increasing the shear-thickening behavior of fluid leads to a significant increase in the kissing time. Moreover, the transverse position of particles for shear-thinning fluids during the tumbling interval is different from Newtonian and the shear-thickening fluids. The present non-Newtonian particulate study can be applied in several industrial and scientific applications, like the non-Newtonian sedimentation behavior of particles in food industrial and biological fluids. 15. Aspects of non-Newtonian flow and displacement in porous media Energy Technology Data Exchange (ETDEWEB) Shah, C.; Yortsos, Y.C. 1993-02-01 The rheology of many heavy oils has been shown to be non-Newtonian, Bingham plastics being one manifestation of heavy oil flow. In EOR applications, non-Newtonian fluids such as low concentration polymer solutions, emulsions, gels etc. are simultaneously injected to increase the viscosity of driving agents that displace oil. Such rheologically complex fluids are used to improve sweep efficiencies, divert displacing fluids and block swept zones. The present study has been undertaken to understand the flow of non-Newtonian fluids through porous media. The work considered involves the numerical (pore network) modeling of both single and multiphase flow of power-law and Bingham plastic fluids in network-like porous media. We consider aspects of both single- and multi-phase flow and displacement. Section 2 describes elementary aspects of non-Newtonian flow and some simple models for porous media. Viscoelastic effects in the flow of non-Newtonian fluids are also discussed. The section includes a brief literature review on non-Newtonian flow in porous media. Section 3 describes single-phase flow. 16. Non-Newtonian ink transfer in gravure-offset printing Energy Technology Data Exchange (ETDEWEB) Ghadiri, Fatemeh; Ahmed, Dewan Hasan [Department of Mechanical Engineering, KAIST, 291 Daehak-ro, Yuseong-gu, Daejeon 305-701 (Korea, Republic of); Sung, Hyung Jin, E-mail: [email protected] [Department of Mechanical Engineering, KAIST, 291 Daehak-ro, Yuseong-gu, Daejeon 305-701 (Korea, Republic of); Shirani, Ebrahim [Department of Mechanical Engineering, Isfahan University of Technology, 841568311, Isfahan (Iran, Islamic Republic of) 2011-02-15 The inks used in gravure-offset printing are non-Newtonian fluids with higher viscosities and lower surface tensions than Newtonian fluids. This paper examines the transfer of a non-Newtonian ink between a flat plate and a groove when the plate is moved upward with a constant velocity while the groove is held fixed. Numerical simulations were carried out with the Carreau model to explore the behavior of this non-Newtonian ink in gravure-offset printing. The volume of fluid (VOF) method was implemented to capture the interface during the ink transfer process. The effects of varying the contact angle of the ink on the flat plate and groove walls and geometrical parameters such as the groove angle and the groove depth on the breakup time of the liquid filament that forms between the plate and the groove and the ink transfer ratio were determined. Our results indicate that increasing the groove contact angle and decreasing the flat plate contact angle enhance the ink transfer ratio and the breakup time. However, increasing the groove depth and the groove angle decreases the transfer ratio and the breakup time. By optimizing these parameters, it is possible to achieve an ink transfer from the groove to the flat plate of approximately 92%. Moreover, the initial width and the vertical velocity of the neck of the ink filament have significant influences on the ink transfer ratio and the breakup time. 17. Influence of Non-Newtonian rheology on magma degassing CERN Document Server Divoux, Thibaut; Ripepe, Maurizio; Géminard, Jean-Christophe 2011-01-01 Many volcanoes exhibit temporal changes in their degassing process, from rapid gas puffing to lava fountaining and long-lasting quiescent passive degassing periods. This range of behaviors has been explained in terms of changes in gas flux and/or magma input rate. We report here a simple laboratory experiment which shows that the non- Newtonian rheology of magma can be responsible, alone, for such intriguing behavior, even in a stationary gas flux regime. We inject a constant gas flow-rate Q at the bottom of a non-Newtonian fluid column, and demonstrate the existence of a critical flow rate Q* above which the system spontaneously alternates between a bubbling and a channeling regime, where a gas channel crosses the entire fluid column. The threshold Q* depends on the fluid rheological properties which are controlled, in particular, by the gas volume fraction (or void fraction) {\\phi}. When {\\phi} increases, Q* decreases and the degassing regime changes. Non-Newtonian properties of magma might therefore play a... 18. Technical Report on NETL's Non Newtonian Multiphase Slurry Workshop: A path forward to understanding non-Newtonian multiphase slurry flows Energy Technology Data Exchange (ETDEWEB) Edited by Guenther, Chris; Garg, Rahul 2013-08-19 The Department of Energy’s (DOE) National Energy Technology Laboratory (NETL) sponsored a workshop on non-Newtonian multiphase slurry at NETL’s Morgantown campus August 19 and 20, 2013. The objective of this special two-day meeting of 20-30 invited experts from industry, National Labs and academia was to identify and address technical issues associated with handling non-Newtonian multiphase slurries across various facilities managed by DOE. Particular emphasis during this workshop was placed on applications managed by the Office of Environmental Management (EM). The workshop was preceded by two webinars wherein personnel from ORP and NETL provided background information on the Hanford WTP project and discussed the critical design challenges facing this project. In non-Newtonian fluids, viscosity is not constant and exhibits a complex dependence on applied shear stress or deformation. Many applications under EM’s tank farm mission involve non-Newtonian slurries that are multiphase in nature; tank farm storage and handling, slurry transport, and mixing all involve multiphase flow dynamics, which require an improved understanding of the mechanisms responsible for rheological changes in non-Newtonian multiphase slurries (NNMS). To discuss the issues in predicting the behavior of NNMS, the workshop focused on two topic areas: (1) State-of-the-art in non-Newtonian Multiphase Slurry Flow, and (2) Scaling up with Confidence and Ensuring Safe and Reliable Long-Term Operation. 19. The role of the rheological properties of non-newtonian fluids in controlling dispersive mixing in a batch electrophoretic cell with Joule heating OpenAIRE M.A. Bosse; Arce, P; S.A. Troncoso; A. Vasquez 2001-01-01 The problem of the effect of Joule heating generation on the hydrodynamic profile and the solute transport found in electrophoretic devices is addressed in this article. The research is focused on the following two problems: The first one is centered around the effect of Joule heating on the hydrodynamic velocity profile and it is referred to as "the carrier fluid problem." The other one is related to the effect of Joule heating on the solute transport inside electrophoretic cells and it is r... 20. Effects of Navier slip on unsteady flow of a reactive variable viscosity non- Newtonian fluid through a porous saturated medium with asymmetric convecti- ve boundary conditions Institute of Scientific and Technical Information of China (English) RUNDORA Lazarus; MAKINDE Oluwole Daniel 2015-01-01 A study on the effects of Navier slip, in conjunction with other flow parameters, on unsteady flow of reactive variable viscosity third-grade fluid through a porous saturated medium with asymmetric convective boundary conditions is presented. The channel walls are assumed to be subjected to asymmetric convective heat exchange with the ambient, and exothermic chemical reactions take place within the flow system. The heat exchange with the ambient obeys Newton’s law of cooling. The coupled equations, arising from the law of conservation of momentum and the first law of thermodynamics, then the derived system are non- dimensionalised and solved using a semi-implicit finite difference scheme. The lower wall slip parameter is observed to increase the fluid velocity profiles, whereas the upper wall slip parameter retards them because of backflow at the upper channel wall. Heat pro- duction in the fluid is seen to increase with the slip parameters. The wall shear stress increases with the slip parameters while the wall heat transfer rate is largely unaltered by the lower wall slip parameter but marginally increased by the upper wall slip parameter. 1. Time Decay Rates of the Isotropic Non-Newtonian Flows in Rn Institute of Scientific and Technical Information of China (English) Bo-Qing Dong 2007-01-01 This paper is concerned with time decay rates for weak solutions to a class system of isotropic incompressible non-Newtonian fluid motion in Rn. With the use of the spectral decomposition methods of Stokes operator, the optimal decay estimates of weak solutions in L2 norm are derived under the different conditions on the initial velocity. Moreover, the error estimates of the difference between non-Newtonian flow and Navier-Stokes flow are also investigated. 2. Stability of a flow down an incline with respect to two-dimensional and three-dimensional disturbances for Newtonian and non-Newtonian fluids. Science.gov (United States) Allouche, M H; Millet, S; Botton, V; Henry, D; Ben Hadid, H; Rousset, F 2015-12-01 Squire's theorem, which states that the two-dimensional instabilities are more dangerous than the three-dimensional instabilities, is revisited here for a flow down an incline, making use of numerical stability analysis and Squire relationships when available. For flows down inclined planes, one of these Squire relationships involves the slopes of the inclines. This means that the Reynolds number associated with a two-dimensional wave can be shown to be smaller than that for an oblique wave, but this oblique wave being obtained for a larger slope. Physically speaking, this prevents the possibility to directly compare the thresholds at a given slope. The goal of the paper is then to reach a conclusion about the predominance or not of two-dimensional instabilities at a given slope, which is of practical interest for industrial or environmental applications. For a Newtonian fluid, it is shown that, for a given slope, oblique wave instabilities are never the dominant instabilities. Both the Squire relationships and the particular variations of the two-dimensional wave critical curve with regard to the inclination angle are involved in the proof of this result. For a generalized Newtonian fluid, a similar result can only be obtained for a reduced stability problem where some term connected to the perturbation of viscosity is neglected. For the general stability problem, however, no Squire relationships can be derived and the numerical stability results show that the thresholds for oblique waves can be smaller than the thresholds for two-dimensional waves at a given slope, particularly for large obliquity angles and strong shear-thinning behaviors. The conclusion is then completely different in that case: the dominant instability for a generalized Newtonian fluid flowing down an inclined plane with a given slope can be three dimensional. 3. Effects of Slip Condition, Variable Viscosity and Inclined Magnetic Field on the Peristaltic Motion of a Non-Newtonian Fluid in an Inclined Asymmetric Channel Directory of Open Access Journals (Sweden) A. Afsar Khan 2016-01-01 Full Text Available The peristaltic motion of a third order fluid due to asymmetric waves propagating on the sidewalls of a inclined asymmetric channel is discussed. The key features of the problem includes longwavelength and low-Reynolds number assumptions. A mathematical analysis has been carried out to investigate the effect of slip condition, variable viscosity and magnetohydrodynamics (MHD. Followed by the nondimensionalization of the nonlinear governing equations along with the nonlinear boundary conditions, a perturbation analysis is made. For the validity of the approximate solution, a numerical solution is obtained using the iterative collocation technique. 4. Mathematical analysis of non-Newtonian blood flow in stenosis narrow arteries. Science.gov (United States) Sriyab, Somchai 2014-01-01 The flow of blood in narrow arteries with bell-shaped mild stenosis is investigated that treats blood as non-Newtonian fluid by using the K-L model. When skin friction and resistance of blood flow are normalized with respect to non-Newtonian blood in normal artery, the results present the effect of stenosis length. When skin friction and resistance of blood flow are normalized with respect to Newtonian blood in stenosis artery, the results present the effect of non-Newtonian blood. The effect of stenosis length and effect of non-Newtonian fluid on skin friction are consistent with the Casson model in which the skin friction increases with the increase of either stenosis length or the yield stress but the skin friction decreases with the increase of plasma viscosity coefficient. The effect of stenosis length and effect of non-Newtonian fluid on resistance of blood flow are contradictory. The resistance of blood flow (when normalized by non-Newtonian blood in normal artery) increases when either the plasma viscosity coefficient or the yield stress increases, but it decreases with the increase of stenosis length. The resistance of blood flow (when normalized by Newtonian blood in stenosis artery) decreases when either the plasma viscosity coefficient or the yield stress increases, but it decreases with the increase of stenosis length. 5. Air Sparging for Mixing Non-Newtonian Slurries Energy Technology Data Exchange (ETDEWEB) Bamberger, Judith A.; Enderlin, Carl W.; Tzemos, Spyridon 2010-01-01 The mechanics of air sparger systems have been primarily investigated for aqueous-based Newtonian fluids. Tilton et al. (1982) [1] describes the fluid mechanics of air sparging systems in non-Newtonian fluids as having two primary flow regions. A center region surrounding the sparger, referred to as the region of bubbles (ROB), contains upward flow due to the buoyant driving force of the rising bubbles. In an annular region, outside the ROB, referred to as the zone of influence (ZOI), the fluid flow is reversed and is opposed to the direction of bubble rise. Outside the ZOI the fluid is unaffected by the air sparger system. The flow regime in the ROB is often turbulent, and the flow regime in the ZOI is laminar; the flow regime outside the ZOI is quiescent. Tests conducted with shear thinning non-Newtonian fluid in a 34-in. diameter tank showed that the ROB forms an approximately inverted cone that is the envelop of the bubble trajectories. The depth to which the air bubbles reach below the sparger nozzle is a linear function of the air-flow rate. The recirculation time through the ZOI was found to vary proportionally with the inverse square of the sparging air-flow rate. Visual observations of the ROB were made in both water and Carbopol®. The bubbles released from the sparge tube in Carbopol® were larger than those in water 6. AC electric field controlled non-Newtonian filament thinning and droplet formation on the microscale. Science.gov (United States) Huang, Y; Wang, Y L; Wong, T N 2017-08-22 Monodispersity and fast generation are innate advantages of microfluidic droplets. Other than the normally adopted simple Newtonian fluids such as a water/oil emulsion system, fluids with complex rheology, namely, non-Newtonian fluids, which are being widely adopted in industries and bioengineering, have gained increasing research interest on the microscale. However, challenges occur in controlling the dynamic behavior due to their complex properties. In this sense, the AC electric field with merits of fast response and easiness in fulfilling "Lab on a chip" has attracted our attention. We design and fabricate flow-focusing microchannels with non-contact types of electrodes for the investigation. We firstly compare the formation of a non-Newtonian droplet with that of a Newtonian one under an AC electric field and discover that viscoelasticity contributes to the discrepancies significantly. Then we explore the effect of AC electric fields on the filament thinning and droplet formation dynamics of one non-Newtonian fluid which has a similar rheological behavior to bio samples, such as DNA or blood samples. We investigate the dynamics of the thinning process of the non-Newtonian filament under the influence of an AC electric field and implement a systematic exploration of the non-Newtonian droplet generation influenced by parameters such as the flow conditions (flow rate Q, capillary number Ca), fluid property (Weissenberg number Wi), applied voltage (U) and frequency (f) of the AC electric field. We present the dependencies of the flow condition and electric field on the non-Newtonian droplet formation dynamics, and conclude with an operating diagram, taking into consideration all the above-mentioned parameters. Results show that the electric field plays a critical role in controlling the thinning process of the filament and the size of the generated droplet. Furthermore, for the first time, we quantitatively measure the flow field of the non-Newtonian droplet 7. Numerical simulation of the non-Newtonian blood flow through a mechanical aortic valve. Non-Newtonian blood flow in the aortic root Science.gov (United States) De Vita, F.; de Tullio, M. D.; Verzicco, R. 2016-04-01 This work focuses on the comparison between Newtonian and non-Newtonian blood flows through a bileaflet mechanical heart valve in the aortic root. The blood, in fact, is a concentrated suspension of cells, mainly red blood cells, in a Newtonian matrix, the plasma, and consequently its overall behavior is that of a non-Newtonian fluid owing to the action of the cells' membrane on the fluid part. The common practice, however, assumes the blood in large vessels as a Newtonian fluid since the shear rate is generally high and the effective viscosity becomes independent of the former. In this paper, we show that this is not always the case even in the aorta, the largest artery of the systemic circulation, owing to the pulsatile and transitional nature of the flow. Unexpectedly, for most of the pulsating cycle and in a large part of the fluid volume, the shear rate is smaller than the threshold level for the blood to display a constant effective viscosity and its shear thinning character might affect the system dynamics. A direct inspection of the various flow features has shown that the valve dynamics, the transvalvular pressure drop and the large-scale features of the flow are very similar for the Newtonian and non-Newtonian fluid models. On the other hand, the mechanical damage of the red blood cells (hemolysis), induced by the altered stress values in the flow, is larger for the non-Newtonian fluid model than for the Newtonian one. 8. Non-Newtonian mechanics of oscillation centers Science.gov (United States) Dodin, I. Y.; Fisch, N. J. 2008-10-01 Classical particles oscillating in high-frequency or static fields effectively exhibit a modified rest mass meff which determines the oscillation center motion. Unlike the true mass, meff depends on the field parameters and can be a nonanalytic function of the particle average velocity and the oscillation energy; hence non-Newtonian "metaplasmas" that permit a new type of plasma maser, signal rectification, frequency doubling, and one-way walls. 9. COMPUTER SIMULATION OF NON-NEWTONIAN FLOW AND MASS TRANSPORT THROUGH CORONARY ARTERIAL STENOSIS Institute of Scientific and Technical Information of China (English) 李新宇; 温功碧; 李丁 2001-01-01 A numerical analysis of Newtonian and non-Newtonian flow in an axi-symmetric tube with a local constriction simulating a stenosed artery under steady and pulsatile flow conditions was carried out. Based on these results, the concentration fields of LDL ( low density lipoprotein ) and Albumin were discussed. According to the results, in great details the macromolecule transport influences of wall shear stress, non-Newtonian fluid character and the scale of the molecule etc are given. The results of Newtonian fluid flow and non Newtonian fluid flow , steady flow and pulsatile flow are compared. These investigations can provide much valuable information about the correlation between the flow properties, the macromolecule transport and the development of atherosclerosis. 10. Effect of a Non-Newtonian Load on Signature S2 for Quartz Crystal Microbalance Measurements Directory of Open Access Journals (Sweden) Jae-Hyeok Choi 2014-01-01 Full Text Available The quartz crystal microbalance (QCM is increasingly used for monitoring the interfacial interaction between surfaces and macromolecules such as biomaterials, polymers, and metals. Recent QCM applications deal with several types of liquids with various viscous macromolecule compounds, which behave differently from Newtonian liquids. To properly monitor such interactions, it is crucial to understand the influence of the non-Newtonian fluid on the QCM measurement response. As a quantitative indicator of non-Newtonian behavior, we used the quartz resonator signature, S2, of the QCM measurement response, which has a consistent value for Newtonian fluids. We then modified De Kee’s non-Newtonian three-parameter model to apply it to our prediction of S2 values for non-Newtonian liquids. As a model, we chose polyethylene glycol (PEG400 with the titration of its volume concentration in deionized water. As the volume concentration of PEG400 increased, the S2 value decreased, confirming that the modified De Kee’s three-parameter model can predict the change in S2 value. Collectively, the findings presented herein enable the application of the quartz resonator signature, S2, to verify QCM measurement analysis in relation to a wide range of experimental subjects that may exhibit non-Newtonian behavior, including polymers and biomaterials. 11. Dynamics of Non-Newtonian Liquid Droplet Collision Science.gov (United States) Chen, Xiaodong; Yang, Vigor 2012-11-01 Collision of Newtonian liquid droplets has been extensively investigated both experimentally and numerically for decades. Limited information, however, is available about non-Newtonian droplet collision dynamics. In the present work, high-fidelity numerical simulations were performed to study the situation associated with shear-thinning non-Newtonian liquids. The formulation is based on a complete set of conservation equations for the liquid and the surrounding gas phases. An improved volume-of-fluid (VOF) method, combined with an innovative topology-oriented adaptive mesh refinement (TOAMR) technique, was developed and implemented to track the interfacial dynamics. The complex evolution of the droplet surface over a broad range of length scales was treated accurately and efficiently. In particular, the thin gas film between two approaching droplets and subsequent breakup of liquid threads were well-resolved. Various types of droplet collision were obtained, including coalescence, bouncing, and reflexive and stretching separations. A regime diagram was developed and compared with that for Newtonian liquids. Fundamental mechanisms and key parameters that dictate droplet behaviors were identified. In addition, collision-induced atomization was addressed. This work was sponsored by the U.S. Army Research Office under the Multi-University Research Initiative under contract No. W911NF-08-1-0124. The support and encouragement provided by Dr. Ralph Anthenien are gratefully acknowledged. 12. Unsteady Non-Newtonian Solver on Unstructured Grid for the Simulation of Blood Flow Directory of Open Access Journals (Sweden) Guojie Li 2013-01-01 Full Text Available Blood is in fact a suspension of different cells with yield stress, shear thinning, and viscoelastic properties, which can be represented by different non-Newtonian models. Taking Casson fluid as an example, an unsteady solver on unstructured grid for non-Newtonian fluid is developed to simulate transient blood flow in complex flow region. In this paper, a steady solver for Newtonian fluid is firstly developed with the discretization of convective flux, diffusion flux, and source term on unstructured grid. For the non-Newtonian characteristics of blood, the Casson fluid is approximated by the Papanastasiou's model and treated as Newtonian fluid with variable viscosity. Then considering the transient property of blood flow, an unsteady non-Newtonian solver based on unstructured grid is developed by introducing the temporal term by first-order upwind difference scheme. Using the proposed solver, the blood flows in carotid bifurcation of hypertensive patients and healthy people are simulated. The result shows that the possibility of the genesis and development of atherosclerosis is increased, because of the increase in incoming flow shock and backflow areas of the hypertensive patients, whose WSS was 20~87.1% lower in outer vascular wall near the bifurcation than that of the normal persons and 3.7~5.5% lower in inner vascular wall downstream the bifurcation. 13. Beyond the Virtual Intracranial Stenting Challenge 2007: non-Newtonian and flow pulsatility effects. Science.gov (United States) Cavazzuti, Marco; Atherton, Mark; Collins, Michael; Barozzi, Giovanni 2010-09-17 The Virtual Intracranial Stenting Challenge 2007 (VISC'07) is becoming a standard test case in computational minimally invasive cerebrovascular intervention. Following views expressed in the literature and consistent with the recommendations of a report, the effects of non-Newtonian viscosity and pulsatile flow are reported. Three models of stented cerebral aneurysms, originating from VISC'07 are meshed and the flow characteristics simulated using commercial computational fluid dynamics (CFD) software. We conclude that non-Newtonian and pulsatile effects are important to include in order to discriminate more effectively between stent designs. 14. Study on Forced Convective Heat Transfer of Non-Newtonian Nanofluids Institute of Scientific and Technical Information of China (English) Yurong He; Yubin Men; Xing Liu; Huilin Lu; Haisheng Chen; Yulong Ding 2009-01-01 This paper is concerned with the forced convective heat transfer of dilute liquid suspensions of nanoparticles (nanofluids) flowing through a straight pipe under laminar conditions. Stable nanofluids are formulated by using the high shear mixing and ultrasonication methods. They are then characterised for their size, surface charge, thermal and rheological properties and tested for their convective heat transfer behaviour. Mathematical model-ling is performed to simulate the convective heat transfer of nanofluids using a single phase flow model and con-sidering nanofluids as both Newtonian and non-Newtonian fluid. Both experiments and mathematical modelling show that nanofluids can substantially enhance the convective heat transfer. Analyses of the results suggest that the non-Newtonian character of nanofluids influences the overall enhancement, especially for nanofluids with an obvious non-Newtonian character. 15. STUDIES OF THE REDUCTION OF PIPE FRICTION WITH THE NON-NEWTONIAN ADDITIVE CMC, Science.gov (United States) water can remarkably reduce the frictional resistance to flow. The material sodium carboxymethylcellulose was added to fresh water and subjected to...pipe friction tests under a wide range of shear rates, additive concentration, and temperature conditions. The frictional data are characterized by application of the power law expression for non-Newtonian fluids. (Author) 16. A Colorful Mixing Experiment in a Stirred Tank Using Non-Newtonian Blue Maize Flour Suspensions Science.gov (United States) Trujilo-de Santiago, Grissel; Rojas-de Gante, Cecillia; García-Lara, Silverio; Ballesca´-Estrada, Adriana; Alvarez, Marion Moise´s 2014-01-01 A simple experiment designed to study mixing of a material of complex rheology in a stirred tank is described. Non-Newtonian suspensions of blue maize flour that naturally contain anthocyanins have been chosen as a model fluid. These anthocyanins act as a native, wide spectrum pH indicator exhibiting greenish colors in alkaline environments, blue… 17. Lie group analysis of flow and heat transfer of non-Newtonian nanofluid over a stretching surface with convective boundary condition Science.gov (United States) Afify, Ahmed A.; El-Aziz, Mohamed Abd 2017-02-01 The steady two-dimensional flow and heat transfer of a non-Newtonian power-law nanofluid over a stretching surface under convective boundary conditions and temperature-dependent fluid viscosity has been numerically investigated. The power-law rheology is adopted to describe non-Newtonian characteristics of the flow. Four different types of nanoparticles, namely copper (Cu), silver (Ag), alumina (Al 2 O 3) and titanium oxide (TiO 2) are considered by using sodium alginate (SA) as the base non-Newtonian fluid. Lie symmetry group transformations are used to convert the boundary layer equations into non-linear ordinary differential equations. The transformed equations are solved numerically by using a shooting method with fourth-order Runge-Kutta integration scheme. The results show that the effect of viscosity on the heat transfer rate is remarkable only for relatively strong convective heating. Moreover, the skin friction coefficient and the rate of heat transfer increase with an increase in Biot number. 18. Lie group analysis of flow and heat transfer of non-Newtonian nanofluid over a stretching surface with convective boundary condition AHMED A AFIFY; MOHAMED ABD EL-AZIZ 2017-02-01 The steady two-dimensional flow and heat transfer of a non-Newtonian power-law nanofluid over a stretching surface under convective boundary conditions and temperature-dependent fluid viscosity has been numerically investigated. The power-law rheology is adopted to describe non-Newtonian characteristics of the flow. Four different types of nanoparticles, namely copper (Cu), silver (Ag), alumina (Al$_2$O$_3$) and titanium oxide (TiO$_2$) are considered by using sodium alginate (SA) as the base non-Newtonian fluid. Lie symmetry group transformations are used to convert the boundary layer equations into non-linear ordinary differential equations. The transformed equations are solved numerically by using a shooting method with fourth-order Runge–Kutta integration scheme. The results show that the effect of viscosity on the heat transfer rate is remarkable only for relatively strong convective heating. Moreover, the skin friction coefficient and the rate of heat transfer increasewith an increase in Biot number. 19. Effects of non Newtonian spiral blood flow through arterial stenosis Science.gov (United States) Hasan, Md. Mahmudul; Maruf, Mahbub Alam; Ali, Mohammad 2016-07-01 The spiral component of blood flow has both beneficial and detrimental effects in human circulatory system. A numerical investigation is carried out to analyze the effect of spiral blood flow through an axisymmetric three dimensional artery having 75% stenosis at the center. Blood is assumed as a Non-Newtonian fluid. Standard k-ω model is used for the simulation with the Reynolds number of 1000. A parabolic velocity profile with spiral flow is used as inlet boundary condition. The peak values of all velocity components are found just after stenosis. But total pressure gradually decreases at downstream. Spiral flow of blood has significant effects on tangential component of velocity. However, the effect is mild for radial and axial velocity components. The peak value of wall shear stress is at the stenosis zone and decreases rapidly in downstream. The effect of spiral flow is significant for turbulent kinetic energy. Detailed investigation and relevant pathological issues are delineated throughout the paper. 20. Inline Ultrasonic Rheometry of a Non-Newtonian Waste Simulant Energy Technology Data Exchange (ETDEWEB) Pfund, David M.; Pappas, Richard A. 2004-03-31 This is a discussion of non-invasive determination of the viscosity of a non-Newtonian fluid in laminar pipe flow over the range of shear rates present in the pipe. The procedure requires knowledge of the flow profile in and the pressure drop along the long straight run of pipe. The profile is determined by using a pulsed ultrasonic Doppler velocimeter. This approach is ideal for making non-invasive, real-time measurements for monitoring and control. Rheograms of a shear thinning, thixotropic gel which is often used as a Hanford waste simulant are presented. The operating parameters and limitations of the ultrasound based instrument will be discussed. The component parts of the instrument have been packaged into a unit for field use. The presentation also discusses the features and engineering optimizations done to enhance field usability of the instrument. 1. A novel investigation of a micropolar fluid characterized by nonlinear constitutive diffusion model in boundary layer flow and heat transfer Science.gov (United States) Sui, Jize; Zhao, Peng; Cheng, Zhengdong; Zheng, Liancun; Zhang, Xinxin 2017-02-01 The rheological and heat-conduction constitutive models of micropolar fluids (MFs), which are important non-Newtonian fluids, have been, until now, characterized by simple linear expressions, and as a consequence, the non-Newtonian performance of such fluids could not be effectively captured. Here, we establish the novel nonlinear constitutive models of a micropolar fluid and apply them to boundary layer flow and heat transfer problems. The nonlinear power law function of angular velocity is represented in the new models by employing generalized "n-diffusion theory," which has successfully described the characteristics of non-Newtonian fluids, such as shear-thinning and shear-thickening fluids. These novel models may offer a new approach to the theoretical understanding of shear-thinning behavior and anomalous heat transfer caused by the collective micro-rotation effects in a MF with shear flow according to recent experiments. The nonlinear similarity equations with a power law form are derived and the approximate analytical solutions are obtained by the homotopy analysis method, which is in good agreement with the numerical solutions. The results indicate that non-Newtonian behaviors involving a MF depend substantially on the power exponent n and the modified material parameter K 0 introduced by us. Furthermore, the relations of the engineering interest parameters, including local boundary layer thickness, local skin friction, and Nusselt number are found to be fitted by a quadratic polynomial to n with high precision, which enables the extraction of the rapid predictions from a complex nonlinear boundary-layer transport system. 2. Determination of the flow and heat transfer characteristics in non-Newtonian media agitated using the electrochemical technique Energy Technology Data Exchange (ETDEWEB) Broniarz-Press, Lubomira; Rozanska, Sylwia [Department of Chemical Engineering and Equipment, Faculty of Chemical Technology, Poznan University of Technology, pl. M. Sklodowskiej-Curie 2, PL 60-965 Poznan (Poland) 2008-02-15 In the study the results of the friction factor in boundary layer and the distribution of heat transfer coefficient in non-Newtonian liquid agitated by different impellers, have been presented. It has been established that for studies in Na-CMC and guar gum aqueous solutions by the electrochemical method the following solution of 0.005 (kmol m{sup -3}) K{sub 3}[Fe(CN){sub 6}], 0.005 (kmol m{sup -3}) K{sub 4}[Fe(CN){sub 6}] and 0.3 (kmol m{sup -3}) K{sub 2}SO{sub 4} can be recommended. The common relationship (for a given type of an impeller) between local values of friction coefficient and heat transfer coefficient and Reynolds number proposed by Metzner and Otto [A.B. Metzner, R.E. Otto, Agitation of non-Newtonian fluids, AIChe J. 3 (1957) 3-10] for all power-law fluids, have been obtained. (author) 3. Local liquid side mass transfer model in gas-liquid-solid three-phase flow airlift loop reactor for Newtonian and non-Newtonian fluids%气-液-固三相气升式环流反应器中牛顿型及非牛顿型流体局部液相传质模型 Institute of Scientific and Technical Information of China (English) 闻建平; 贾晓强; 毛国柱 2004-01-01 A small scale isotropic mass transfer model was developed for the local liquid side mass transfer coefficients in gas-liquid-solid three-phase flow airlift loop reactor for Newtonian and non-Newtonian fluids.It is based on Higbie's penetration theory and Kolmogoroff's theory of isotropic turbulence with k1 = 3√2Dε1/3 1/π(η-1/3 1 -λ-1/3 f),where ε1 is local rate of energy dissipation,λf is the local microscale,η1 is the local Kolmogoroff scale and D is the diffusion coefficient.The capability of the proposed model is discussed in the light of experimental data obtained from 12 L gas-liquid-solid three-phase flow airlift loop reactor using Newtonian and non-Newtonian fluids.Good agreement with the experimental data was obtained over a wide range of conditions suggesting a general applicability of the proposed model. 4. NUMERICAL ANALYSIS OF THE NON-NEWTONIAN BLOOD FLOW IN THE NON-PLANAR ARTERY WITH BIFURCATION Institute of Scientific and Technical Information of China (English) CHEN Jie; LU Xi-yun; ZHUANG Li-xian; WANG Wen 2004-01-01 A numerical analysis of non-Newtonian fluid flow in non-planar artery with bifurcation was performed by using a finite element method to solve the three-dimensional Navier-Stokes equations coupled with the non-Newtonian constitutive models, including Carreau,Cross and Bingham models. The objective of this study is to investigate the effects of the non-Newtonian properties of blood as well as curvature and out-of-plane geometry in the non-planar daughter vessels on the velocity distribution and wall shear stress. The results of this study support the view that the non-planarity of blood vessels and the non-Newtonian properties of blood are of important in hemodynamics and play a significant role in vascular biology and pathophysiology. 5. Studying mixing in Non-Newtonian blue maize flour suspensions using color analysis. Science.gov (United States) Trujillo-de Santiago, Grissel; Rojas-de Gante, Cecilia; García-Lara, Silverio; Ballescá-Estrada, Adriana; Alvarez, Mario Moisés 2014-01-01 Non-Newtonian fluids occur in many relevant flow and mixing scenarios at the lab and industrial scale. The addition of acid or basic solutions to a non-Newtonian fluid is not an infrequent operation, particularly in Biotechnology applications where the pH of Non-Newtonian culture broths is usually regulated using this strategy. We conducted mixing experiments in agitated vessels using Non-Newtonian blue maize flour suspensions. Acid or basic pulses were injected to reveal mixing patterns and flow structures and to follow their time evolution. No foreign pH indicator was used as blue maize flours naturally contain anthocyanins that act as a native, wide spectrum, pH indicator. We describe a novel method to quantitate mixedness and mixing evolution through Dynamic Color Analysis (DCA) in this system. Color readings corresponding to different times and locations within the mixing vessel were taken with a digital camera (or a colorimeter) and translated to the CIELab scale of colors. We use distances in the Lab space, a 3D color space, between a particular mixing state and the final mixing point to characterize segregation/mixing in the system. Blue maize suspensions represent an adequate and flexible model to study mixing (and fluid mechanics in general) in Non-Newtonian suspensions using acid/base tracer injections. Simple strategies based on the evaluation of color distances in the CIELab space (or other scales such as HSB) can be adapted to characterize mixedness and mixing evolution in experiments using blue maize suspensions. 6. Studying mixing in Non-Newtonian blue maize flour suspensions using color analysis. Directory of Open Access Journals (Sweden) Grissel Trujillo-de Santiago Full Text Available BACKGROUND: Non-Newtonian fluids occur in many relevant flow and mixing scenarios at the lab and industrial scale. The addition of acid or basic solutions to a non-Newtonian fluid is not an infrequent operation, particularly in Biotechnology applications where the pH of Non-Newtonian culture broths is usually regulated using this strategy. METHODOLOGY AND FINDINGS: We conducted mixing experiments in agitated vessels using Non-Newtonian blue maize flour suspensions. Acid or basic pulses were injected to reveal mixing patterns and flow structures and to follow their time evolution. No foreign pH indicator was used as blue maize flours naturally contain anthocyanins that act as a native, wide spectrum, pH indicator. We describe a novel method to quantitate mixedness and mixing evolution through Dynamic Color Analysis (DCA in this system. Color readings corresponding to different times and locations within the mixing vessel were taken with a digital camera (or a colorimeter and translated to the CIELab scale of colors. We use distances in the Lab space, a 3D color space, between a particular mixing state and the final mixing point to characterize segregation/mixing in the system. CONCLUSION AND RELEVANCE: Blue maize suspensions represent an adequate and flexible model to study mixing (and fluid mechanics in general in Non-Newtonian suspensions using acid/base tracer injections. Simple strategies based on the evaluation of color distances in the CIELab space (or other scales such as HSB can be adapted to characterize mixedness and mixing evolution in experiments using blue maize suspensions. 7. Mounding of a non-Newtonian jet impinging on a solid substrate. Energy Technology Data Exchange (ETDEWEB) Schunk, Peter Randall; Grillet, Anne Mary; Roberts, Scott A.; Baer, Thomas A. (Procter & Gamble, Cincinnati, OH); Rao, Rekha Ranjana 2010-06-01 When a fluid jet impinges on a solid substrate, a variety of behaviors may occur around the impact region. One example is mounding, where the fluid enters the impact region faster than it can flow away, forming a mound of fluid above the main surface. For some operating conditions, this mound can destabilize and buckle, entraining air in the mound. Other behaviors include submerging flow, where the jet impinges into an otherwise steady pool of liquid, entraining a thin air layer as it enters the pool. This impact region is one of very high shear rates and as such, complex fluids behave very differently than do Newtonian fluids. In this work, we attempt to characterize this range of behavior for Newtonian and non-Newtonian fluids using dimensionless parameters. We model the fluid as a modified Bingham-Carreau-Yasuda fluid, which exhibits the full range of pseudoplastic flow properties throughout the impact region. Additionally, we study viscoelastic effects through the use of the Giesekus model. Both 2-D and 3-D numerical simulations are performed using a variety of finite element method techniques for tracking the jet interface, including Arbitrary Lagrangian Eulerian (ALE), diffuse level sets, and a conformal decomposition finite element method (CDFEM). The presence of shear-thinning characteristics drastically reduces unstable mounding behavior, yet can lead to air entrainment through the submerging flow regime. We construct an operating map to understand for what flow parameters mounding and submerging flows will occur, and how the fluid rheology affects these behaviors. This study has many implications in high-speed industrial bottle filling applications. 8. Dynamic viscosity measurement in non-Newtonian graphite nanofluids. Science.gov (United States) Duan, Fei; Wong, Ting Foong; Crivoi, Alexandru 2012-07-02 : The effective dynamic viscosity was measured in the graphite water-based nanofluids. The shear thinning non-Newtonian behavior is observed in the measurement. On the basis of the best fitting of the experimental data, the viscosity at zero shear rate or at infinite shear rate is determined for each of the fluids. It is found that increases of the particle volume concentration and the holding time period of the nanofluids result in an enhancement of the effective dynamic viscosity. The maximum enhancement of the effective dynamic viscosity at infinite rate of shear is more than 24 times in the nanofluids held for 3 days with the volume concentration of 4% in comparison with the base fluid. A transmission electron microscope is applied to reveal the morphology of aggregated nanoparticles qualitatively. The large and irregular aggregation of the particles is found in the 3-day fluids in the drying samples. The Raman spectra are extended to characterize the D and G peaks of the graphite structure in the nanofluids. The increasing intensity of the D peak indicates the nanoparticle aggregation growing with the higher concentration and the longer holding time of the nanofluids. The experimental results suggest that the increase on effective dynamic viscosity of nanofluids is related to the graphite nanoparticle aggregation in the fluids. 9. Increasing heat transfer of non-Newtonian nanofluid in rectangular microchannel with triangular ribs Science.gov (United States) Shamsi, Mohammad Reza; Akbari, Omid Ali; Marzban, Ali; Toghraie, Davood; Mashayekhi, Ramin 2017-09-01 In this study, computational fluid dynamics and the laminar flow of the non-Newtonian fluid have been numerically studied. The cooling fluid includes water and 0.5 wt% Carboxy methyl cellulose (CMC) making the non-Newtonian fluid. In order to make the best of non-Newtonian nanofluid in this simulation, solid nanoparticles of Aluminum Oxide have been added to the non-Newtonian fluid in volume fractions of 0-2% with diameters of 25, 45 and 100 nm. The supposed microchannel is rectangular and two-dimensional in Cartesian coordination. The power law has been used to speculate the dynamic viscosity of the cooling nanofluid. The field of numerical solution is simulated in the Reynolds number range of 5 ribs with angle of attacks of 30°, 45° and 60° is studied on flow parameters and heat transfer due to the fluid flow. The results show that an increase in the volume fraction of nanoparticles as well as the use for nanoparticles with smaller diameters lead to greater heat transfer. Among all the studied forms, the triangular rib from with an angle of attack 30° has the biggest Nusselt number and the smallest pressure drop along the microchannel. Also, an increase in the angle of attack and as a result of a sudden contact between the fluid and the ribs and also a reduction in the coflowing length (length of the rib) cause a cut in heat transfer by the fluid in farther parts from the solid wall (tip of the rib). 10. Pulsatile Non-Newtonian Laminar Blood Flows through Arterial Double Stenoses Directory of Open Access Journals (Sweden) Mir Golam Rabby 2014-01-01 Full Text Available The paper presents a numerical investigation of non-Newtonian modeling effects on unsteady periodic flows in a two-dimensional (2D pipe with two idealized stenoses of 75% and 50% degrees, respectively. The governing Navier-Stokes equations have been modified using the Cartesian curvilinear coordinates to handle complex geometries. The investigation has been carried out to characterize four different non-Newtonian constitutive equations of blood, namely, the (i Carreau, (ii Cross, (iii Modified Casson, and (iv Quemada models. The Newtonian model has also been analyzed to study the physics of fluid and the results are compared with the non-Newtonian viscosity models. The numerical results are represented in terms of streamwise velocity, pressure distribution, and wall shear stress (WSS as well as the vorticity, streamlines, and vector plots indicating recirculation zones at the poststenotic region. The results of this study demonstrate a lower risk of thrombogenesis at the downstream of stenoses and inadequate blood supply to different organs of human body in the Newtonian model compared to the non-Newtonian ones. 11. Non-Newtonian Viscosity Modeling of Crude Oils—Comparison Among Models Science.gov (United States) Ramírez-González, Patsy V.; Aguayo, Juan Pablo; Quiñones-Cisneros, Sergio E.; Deiters, Ulrich K. 2009-04-01 The presence of precipitated wax or even just low temperatures may induce non-Newtonian rheological behavior in crude oils. Such behavior can be found at operating conditions, for instance, in reservoirs at deep-water conditions. Therefore, reliable rheological models for crude oils applicable over the wide range of conditions the fluid may encounter are essential for a large number of oil technology applications. Such models must also be composition dependent, as many applications require predicting the rheological behavior of the fluid under strong compositional changes, e.g., recovery applications such as vapor extraction (VAPEX) processes or blending of fluids for improved rheological characteristics for piping, among many other applications. In this study, a comparative analysis between some published models applicable to the description of the non-Newtonian behavior of crude oils is carried out. Emphasis is placed on the stability of the model predictions within the wide range of conditions that may be encountered. 12. Inelastic non-Newtonian flow over heterogeneously slippery surfaces Science.gov (United States) Haase, A. Sander; Wood, Jeffery A.; Sprakel, Lisette M. J.; Lammertink, Rob G. H. 2017-02-01 In this study, we investigated inelastic non-Newtonian fluid flow over heterogeneously slippery surfaces. First, we simulated the flow of aqueous xanthan gum solutions over a bubble mattress, which is a superhydrophobic surface consisting of transversely positioned no-slip walls and no-shear gas bubbles. The results reveal that for shear-thinning fluids wall slip can be increased significantly, provided that the system is operated in the shear-thinning regime. For a 0.2 wt% xanthan gum solution with a power-law index of n =0.4 , the numerical results indicate that wall slip can be enhanced 3.2 times when compared to a Newtonian liquid. This enhancement factor was also predicted from a theoretical analysis, which gave an expression for the maximum slip length that can be attained over flat, heterogeneously slippery surfaces. Although this equation was derived for a no-slip/no-shear unit length that is much larger than the typical size of the system, we found that it can also be used to predict the enhancement in the regime where the slip length is proportional to the size of the no-shear region or the bubble width. The results could be coupled to the hydrodynamic development or entrance length of the system, as maximum wall slip is only reached when the fluid flow can fully adapt to the no-slip and no-shear conditions at the wall. 13. Non-Newtonian Aspects of Artificial Intelligence Science.gov (United States) Zak, Michail 2016-05-01 The challenge of this work is to connect physics with the concept of intelligence. By intelligence we understand a capability to move from disorder to order without external resources, i.e., in violation of the second law of thermodynamics. The objective is to find such a mathematical object described by ODE that possesses such a capability. The proposed approach is based upon modification of the Madelung version of the Schrodinger equation by replacing the force following from quantum potential with non-conservative forces that link to the concept of information. A mathematical formalism suggests that a hypothetical intelligent particle, besides the capability to move against the second law of thermodynamics, acquires such properties like self-image, self-awareness, self-supervision, etc. that are typical for Livings. However since this particle being a quantum-classical hybrid acquires non-Newtonian and non-quantum properties, it does not belong to the physics matter as we know it: the modern physics should be complemented with the concept of the information force that represents a bridge to intelligent particle. As a follow-up of the proposed concept, the following question is addressed: can artificial intelligence (AI) system composed only of physical components compete with a human? The answer is proven to be negative if the AI system is based only on simulations, and positive if digital devices are included. It has been demonstrated that there exists such a quantum neural net that performs simulations combined with digital punctuations. The universality of this quantum-classical hybrid is in capability to violate the second law of thermodynamics by moving from disorder to order without external resources. This advanced capability is illustrated by examples. In conclusion, a mathematical machinery of the perception that is the fundamental part of a cognition process as well as intelligence is introduced and discussed. 14. Non-Newtonian fluid flow and heat transfer in a porous medium. 2nd Report. Prediction of porous inertia based on a three-dimensional numerical model; Takoshitsutainai no hi Newton ryutai no netsuryudo. 2. Sanjigen suchi model ni yoru takoshitsutai kansei koka no yosoku Energy Technology Data Exchange (ETDEWEB) Inoue, M.; Nakayama, A. [Shizuoka University, Shizuoka (Japan). Faculty of Engineering 1996-09-25 Three-dimensional numerical calculations have been performed to simulate the viscous and porous inertia effects on the pressure drop in a non-Newtonian fluid flow through a porous medium. Cubes placed in an infinite space have been proposed as a three-dimensional model of microscopic porous structure. A full set of three-dimensional momentum equations is solved along with the continuity equation at a pore scale, so as to simulate a flow through an infinite number of obstacles arranged in a regular pattern. The microscopic numerical results, thus obtained, are processed to extract the macroscopic relationship between the pressure gradient-mass flow rate. Comparing the results based on the two- and three-dimensional models, it has been found that only the three-dimensional model can capture the porous inertia effects on the pressure drop correctly. 13 refs., 6 figs. 15. Modeling and prediction of non-Newtonian viscosity of crude oils Energy Technology Data Exchange (ETDEWEB) Ramirez-Gonzalez, P.V. [Univ. Nacional Autonoma de Mexico (Mexico). Dept. de Ingenieria Quimica; Quinones-Cisneros, S.E.; Manero, O. [Univ. Nacional Autonoma de Mexico (Mexico). Dept. de Reologia, Inst. de Investigaciones en Materiales; Creek, J. [Chevron Energy Technology Co., Houston, TX (United States); Deiters, U.K. [Cologne Univ., Cologne (Germany). Inst. of Physical Chemistry 2008-07-01 Non-Newtonian rheological behaviour in crude oils can be induced by the presence of precipitated wax in reservoir fluids or by low ambient temperatures in heavy oils. This type of behaviour exists at low temperature operating conditions in deep-water production, or in the case of vapor extraction (VAPEX) processes of heavy oils involving strong compositional related changes to the already non-Newtonian viscosity of the oil. Reliable rheological models are needed for crude oils over the wide range of conditions that the fluid may encounter. The models should be of a compositional nature because the rheological behaviour of the fluid must be predicted in many applications, including VAPEX processes or fluid blending for piping. This study compared some published models that describe the non-Newtonian behaviour of crude oils. The emphasis was on the stability of the models predictions within the wide range of conditions that may be encountered. The study also evaluated the prediction potential of the analyzed models. 16. On predicting the onset of transient convection in porous media saturated with Non-Newtonian liquid Science.gov (United States) Tan, K. K.; Pua, S. Y.; Yang, A. 2017-06-01 The onset of transient convection in non-Newtonian liquid immersing porous media was simulated using a Computational Fluid Dynamics (CFD) package for the thermal boundary condition of Fixed Surface Temperature (FST). Most of the simulated values of stability criteria were found to be in good agreement with the predicted and theoretical values of transient critical Rayleigh number for non-Newtonian liquid defined by Tan and Thorpe (1992) for power-law fluids. The critical transient Rayleigh numbers for convection in porous media were found to be in good agreement with theoretical values by using apparent viscosity µapp at zero shear. The critical time and critical depth for transient heat conduction were then determined accurately that 17. Simulation of Non-Newtonian Blood Flow by Lattice Boltzman Method Institute of Scientific and Technical Information of China (English) JI Yu-Pin; KANG Xiu-Ying; LIU Da-He 2010-01-01 @@ Blood flow under various conditions of vessel is simulated as a non-Newtonian fluid by the two-dimensional Lattice Boltzmann method,in which the Casson model is used to express the relationship between viscosity and shear rate of the blood.The flow field distributions at certain sites near the narrowing and bifurcation of the vessel explain the hemodynamic mechanism of the preclilection of the atherosclerotic lesions for these sites which are consistent with that found by medical studies. 18. Simulation of non-Newtonian ink transfer between two separating plates for gravure-offset printing Energy Technology Data Exchange (ETDEWEB) Ahmed, Dewan Hasan [Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, 373-1, Guseong-dong, Yuseong-gu, Daejeon 305-701 (Korea, Republic of); Sung, Hyung Jin, E-mail: [email protected] [Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, 373-1, Guseong-dong, Yuseong-gu, Daejeon 305-701 (Korea, Republic of); Kim, Dong-Soo [Nano-Mechanical Systems Research Division, Korea Institute of Machinery and Materials, 171, Jang-dong, Yuseong-gu, Daejeon 305-343 (Korea, Republic of) 2011-02-15 The inks used in gravure-offset printing are non-Newtonian fluids with higher viscosities and lower surface tensions compared to Newtonian fluids. This paper examines the transfer of a non-Newtonian ink between two parallel plates when the top plate is moved upward with a constant velocity while the bottom plate is held fixed. Numerical simulations were carried out using the Carreau model to explore the behavior of a non-Newtonian ink in gravure-offset printing. The volume of fluid (VOF) model was adopted to demonstrate the stretching and break-up behaviors of the ink. The results indicate that the ink transfer ratio is greatly influenced by the contact angle, especially the contact angle at the upper plate ({alpha}). For lower values of {alpha}, oscillatory or unstable behavior of the position of minimum thickness of the ink between the two parallel plates during the stretching period is observed. This oscillation gradually diminishes as the contact angle at the upper plate is increased. Moreover, the number of satellite droplets increases as the velocity of the upper plate is increased. The surface tension of the conductive ink shows a positive impact on the ink transfer ratio to the upper plate. Indeed, the velocity of the upper plate has a significant influence on the ink transfer in gravure-offset printing when the Capillary number (Ca) is greater than 1 and the surface tension dominates over the ink transfer process when Ca is less than 1. 19. Pore-Scale Modeling of Non-Newtonian Flow in Porous Media CERN Document Server Sochi, Taha 2010-01-01 The thesis investigates the flow of non-Newtonian fluids in porous media using pore-scale network modeling. Non-Newtonian fluids show very complex time and strain dependent behavior and may have initial yield stress. Their common feature is that they do not obey the simple Newtonian relation of proportionality between stress and rate of deformation. They are generally classified into three main categories: time-independent, time-dependent and viscoelastic. Two three-dimensional networks representing a sand pack and Berea sandstone were used. An iterative numerical technique is used to solve the pressure field and obtain the flow rate and apparent viscosity. The time-independent category is investigated using two fluid models: Ellis and Herschel-Bulkley. The analysis confirmed the reliability of the non-Newtonian network model used in this study. Good results are obtained, especially for the Ellis model, when comparing the network model results to experimental data sets found in the literature. The yield-stres... 20. 饱和纳米流体多孔介质中竖直嵌入板上边界层的非Newton流%Boundary-Layer Non-Newtonian Flow Over a Vertical Plate in a Porous MediumSaturated With a Nanofluid Institute of Scientific and Technical Information of China (English) F·M·哈迪; F·S·艾伯拉赫门; S·M·阿卜杜勒·盖德; M·R·艾德; 吴承平; 张禄坤 2011-01-01 在层流条件下,对饱和多孔介质中的竖直板,研究幂指数型非Newton流的自由对流热交换.非Newton纳米流体服从幂指数型的数学模型,模型综合考虑了Brown运动和热泳的影响.通过相似变换,将问题的偏微分控制方程组,转化为常微分方程组,得到了常微分方程组的数值解.数值解依赖于幂指数n,Lewis数Le,浮力比Nr,Brown运动参数Nb,以及热泳参数Nt.在n和Le的不同取值下,研究并讨论了对相关流体性质参数的影响和简化的Nusselt数.%The free convective heat transfer to the power-law non-Newtonian from a vertical plate in a porous medium saturated with nanofluid under laminar conditions was investigated. It was considered that the non-Newtonian nanofluid obeys the mathematical model of power-law. The model used for the nanofluid incorporates the effects of Brownian motion and thermo-phoresis. The partial differential system governing the problem was transformed into an ordinary system via a usual similarity transformation. The numerical solutions of the resulting ordinary system were obtained. These solutions depend on the power-law index n, Lewis number Le, buoyancy-ratio number Nr, Brownian motion number Nb and thermophoresis number Nt. For various values of n and Le, the effect of the influence parameters on the fluid behavior as well as the reduced Nusselt number was presented and discussed. 1. Experiments on densely-loaded non-Newtonian slurries in laminar and turbulent pipe flows Science.gov (United States) Park, J. T.; Mannheimer, R. J.; Grimley, T. A.; Morrow, T. B. 1988-05-01 An experimental description of the flow structure of non-Newtonian slurries in the laminar, transitional, and full turbulent pipe flow regimes is the primary objective of this research. Measurements include rheological characterization of the fluid and local fluid velocity measurements with a Laser Doppler Velocimeter (LDV). Optical access to the flow is gained through a test section and model slurry which are both transparent. The model slurry is formulated from silica gel particles and hydrocarbon liquid mixture whose indices of refraction are matched so that light is not scattered from the particles. Experiments are being conducted in a large-scale pipe slurry. Flow measurements including turbulence quantities such as Reynolds stress were measured with a two-component two-color LDV. The present research indicates that non-Newtonian slurries are possible with concentrations of a few percent by weight of small particles whose sizes are two microns or less. A non-Newtonian slurry from small particles could maintain large particles (one millimeter size) at high concentrations in suspension almost indefinitely. Such a slurry would prevent particle fallout and its associated problems. 2. A corrected smo othed particle hydro dynamics approach to solve the non-isothermal non-Newtonian viscous fluid flow problems%非等温非牛顿黏性流体流动问题的修正光滑粒子动力学方法模拟 Institute of Scientific and Technical Information of China (English) 蒋涛; 任金莲; 徐磊; 陆林广 2014-01-01 为准确、有效地模拟非等温非牛顿黏性流体的流动问题，本文基于一种不含核导数计算的核梯度修正格式和不可压缩条件给出了一种改进光滑粒子动力学(SPH)离散格式，它较传统SPH离散格式具有较高精度和较好稳定性。同时，为准确地描述温度场的演化过程，建立了非牛顿黏性的SPH温度离散模型。通过对等温Poiseuille流、喷射流和非等温Couette流、4：1收缩流进行模拟，并与其他数值结果作对比，分别验证了改进SPH方法模拟非牛顿黏性流动问题的可靠性和提出的SPH温度离散模型求解非等温流动问题的有效性和准确性。随后，运用改进SPH方法结合SPH温度离散模型对环形腔和C形腔内非等温非牛顿黏性流体的充模过程进行了试探性模拟研究，分析了数值模拟的收敛性，讨论了不同位置处热流参数对温度和流动的影响。%In this paper, a corrected smoothed particle hydrodynamics (SPH) method is proposed to solve the problems of non-isothermal non-Newtonian viscous fluid. The proposed particle method is based on the corrected kernel derivative scheme under no kernel derivative and incompressible conditions, which possesses higher accuracy and better stability than the traditional SPH method. Meanwhile, a temperature-discretization scheme is deduced by the concept of SPH method for the purpose of precisely describing the evolutionary process of the temperature field. Reliability of the corrected SPH method for simulating the non-Newtonian viscous fluid flow is demonstrated by simulating the isothermal Poiseuille flow and the jet fluid of filling process; and the validity and accuracy of the proposed SPH discrete scheme in a temperature model for solving the non-isothermal fluid flow are tested by solving the non-isothermal Couette flow and 4:1 contraction flow. Subsequently, the proposed corrected SPH method combined with the SPH temperature 3. Comment on 'Application of the homotopy method for analytical solution of non-Newtonian channel flows' Energy Technology Data Exchange (ETDEWEB) Lipscombe, T C [Johns Hopkins University, 2715 North Charles Street, Baltimore, MD 21218 (United States)], E-mail: [email protected] 2010-03-15 We solve exactly the Poiseuille and Couette flows of a non-Newtonian fluid discussed by Roohi et al (2009 Phys. Scr. 79 065009) and thereby show that the approximate analytical solutions provided by the homotopy method must be used with caution. 4. Are Non-Newtonian Effects Important in Hemodynamic Simulations of Patients With Autogenous Fistula? Science.gov (United States) Javid Mahmoudzadeh Akherat, S M; Cassel, Kevin; Boghosian, Michael; Dhar, Promila; Hammes, Mary 2017-04-01 Given the current emphasis on accurate computational fluid dynamics (CFD) modeling of cardiovascular flows, which incorporates realistic blood vessel geometries and cardiac waveforms, it is necessary to revisit the conventional wisdom regarding the influences of non-Newtonian effects. In this study, patient-specific reconstructed 3D geometries, whole blood viscosity data, and venous pulses postdialysis access surgery are used as the basis for the hemodynamic simulations of renal failure patients with native fistula access. Rheological analysis of the viscometry data initially suggested that the correct choice of constitutive relations to capture the non-Newtonian behavior of blood is important because the end-stage renal disease (ESRD) patient cohort under observation experience drastic variations in hematocrit (Hct) levels and whole blood viscosity throughout the hemodialysis treatment. For this purpose, various constitutive relations have been tested and implemented in CFD practice, namely Quemada and Casson. Because of the specific interest in neointimal hyperplasia and the onset of stenosis in this study, particular attention is placed on differences in nonhomeostatic wall shear stress (WSS) as that drives the venous adaptation process that leads to venous geometric evolution over time in ESRD patients. Surprisingly, the CFD results exhibit no major differences in the flow field and general flow characteristics of a non-Newtonian simulation and a corresponding identical Newtonian counterpart. It is found that the vein's geometric features and the dialysis-induced flow rate have far greater influence on the WSS distribution within the numerical domain. 5. CFD analysis of Newtonian and non-Newtonian droplets impinging on heated hydrophilic and hydrophobic surfaces Science.gov (United States) Khojasteh, Danial; Mousavi, Seyed Mahmood; Kamali, Reza 2016-11-01 In the present study, the behaviors of Newtonian and shear-thinning non-Newtonian droplets impinging on heated hydrophilic and hydrophobic surfaces have been investigated numerically using Ansys-Fluent. In this context, the volume-of-fluid technique is applied to track the free-surface of the liquid, and variable time-step is also utilized to control the Courant number. Furthermore, we have considered the dependence of viscosity, density and surface tension on temperature during the simulation. The results are compared to available experimental data at the same conditions, such as boundary conditions. The results demonstrate that there is a good agreement between the obtained results and the experimental trends, concerning normalized diameter profiles at various Weber numbers. Therefore, the focus of the present study is an assessment of the effects of variations in Weber number, contact angle and surface temperature for Newtonian and non-Newtonian liquids on dynamics behavior of droplet in collision with hydrophobic and hydrophilic surfaces. The results represent that the behaviors of Newtonian and non-Newtonian droplets are totally different, indicating the droplet sensitivity to the working parameters. 6. Investigating the impact of non-Newtonian blood models within a heart pump. Science.gov (United States) Al-Azawy, Mohammed G; Turan, A; Revell, A 2017-01-01 A detailed computational fluid dynamics (CFD) study of transient, turbulent blood flow through a positive displacement left ventricular assist device is performed. Two common models for non-Newtonian blood flow are compared to the Newtonian model to investigate their impact on predicted levels of shear rate and wall shear stress. Given that both parameters are directly relevant to the evaluation of risk from thrombus and haemolysis, there is a need to assess the sensitivity to modelling non-Newtonian flow effects within a pulsatile turbulent flow, in order to identify levels of uncertainly in CFD. To capture the effects of turbulence, the elliptic blending Reynolds stress model is used in the present study, on account of superior performance of second moment closure schemes previously identified by the present authors. The CFD configuration includes two cyclically rotating valves and a moving pusher plate to periodically vary the chamber volume. An overset mesh algorithm is used for each instance of mesh motion, and a zero gap technique was employed to ensure full valve closure. The left ventricular assist device was operated at a pumping rate of 86 BPM (beats per minute) and a systolic duration of 40% of the pumping cycle, in line with existing experimental data to which comparisons are made. The sensitivity of the variable viscosity models is investigated in terms of mean flow field, levels of turbulence and global shear rate, and a non-dimensional index is used to directly evaluate the impact of non-Newtonian effects. The clinical relevance of the results is reported along with a discussion of modelling uncertainties, observing that the turbulent kinetic energy is generally predicted to be higher in non-Newtonian flow than that observed in Newtonian flow. Copyright © 2016 John Wiley & Sons, Ltd. 7. Experimental Investigation and Optimization of Solid Suspension in Non-Newtonian Liquids at High Solid Concentration Directory of Open Access Journals (Sweden) Roozbeh Mollaabbasi 2016-01-01 Full Text Available This research deals with experimental work on solid suspension and dispersion in stirred tank reactors that operate with complex fluids. Only suspended speed (Njs throughout the vessel was characterized using Gamma-Ray Densitometry. The outcomes of this study help to understand solid suspension mechanisms involving changes the rheology of the fluid and provide engineering data for designing stirred tanks. All experiments were based on classic radial and axial flow impellers, i.e., Rushton Turbine (RT and Pitched Blade Turbine in down pumping mode (PBT-D. Three different liquids (water, water+CMC, and water+PAA were employed in several concentrations. The CMC solution introduced as a pseudo plastic fluid and PAA solution was applied as a Herschel Bulkley fluid. The rheological properties of these fluids were characterized separately. According to the findings, the critical impeller speeds for solid suspension for non-Newtonian fluids were more eminent than those for water. Experiments were performed to characterize the effects of solid loading, impeller clearance and viscosity on Njs. Also the PSO method is employed to find suitable parameters of Zwietering's correlation for prediction of Njs in Non Newtonian fluids. 8. A Lagrangian finite element method for the simulation of flow of non-newtonian liquids DEFF Research Database (Denmark) Hassager, Ole; Bisgaard, C 1983-01-01 A Lagrangian method for the simulation of flow of non-Newtonian liquids is implemented. The fluid mechanical equations are formulated in the form of a variational principle, and a discretization is performed by finite elements. The method is applied to the slow of a contravariant convected Maxwell...... liquid around a sphere moving axially in a cylinder. The simulations show that the friction factor for a sphere in a narrow cylinder is a rapidly decreasing function of the Deborah number, while the friction factor for a sphere in a very wide cylinder is not significantly affected by fluid elasticity... 9. Pressure Drop of Non-Newtonian Liquid Flow Through Elbows Institute of Scientific and Technical Information of China (English) 2000-01-01 Experimental data on the pressure drop across different types of elbow for non-Newtonian pseudoplastic liquid flow in laminar condition have been presented. A generalized correlation has been developed for predicting the frictional pressure drop across the elbows in the horizontal plane. 10. On preconditioning incompressible non-Newtonian flow problems NARCIS (Netherlands) He, X.; Neytcheva, M.; Vuik, C. 2013-01-01 This paper deals with fast and reliable numerical solution methods for the incompressible non-Newtonian Navier-Stokes equations. To handle the nonlinearity of the governing equations, the Picard and Newton methods are used to linearize these coupled partial differential equations. For space discreti 11. Sinking of spherical slablets through a non-Newtonian mantle Science.gov (United States) Crameri, Fabio; Stegman, Dave; Petersen, Robert; Tackley, Paul 2014-05-01 The dominant driving force for plate tectonics is slab pull, in which sinking slabs pull the trailing plate. Forward plate velocities are typically similar in magnitude (7 cm/yr) as estimates for sinking velocities of slabs through the upper mantle. However, these estimates are based on data for slabs that are coherent into the transition zone as well as models that considered the upper mantle to be entirely Newtonian. Dislocation creep in the upper mantle can strongly influence mantle flow, and is likely activated for flow around vertically sinking slabs in the uppermost mantle. Thus, it is possible that in some scenarios, a non-Newtonian mantle will have an influence on plate motions but it is unclear to what degree. To address this question, we investigate how the non-Newtonian rheology modifies the sinking velocities of slablets (spherical, negatively buoyant and highly viscous blobs). The model set-up is similar to a Stokes sphere sinking, but is in 2-D cartesian with temperature-and stress-dependent rheology. For these numerical models, we use the Stag-YY code (e.g., Tackley 2008) and apply a pseudo-free surface using the 'sticky-air' approach (Matsumoto and Tomoda 1983; Schmeling et al, 2008, Crameri et al., 2012). The sinking blob is both highly viscous and compositionally dense, but is the same temperature as the background fluid which eliminates thermal diffusion and associated variations in thermal buoyancy. The model domain is 2x1 or 4x1 and allows enough distance to the sidewalls so that sinking velocities are not influenced by the boundary conditions. We compare our results with those previously obtained for salt diapirs rising through a power-law rheology mantle/crust (Weinberg, 1993; Weinberg and Podladchikov, 1994), which provided both numerical and analytic results. Previous results indicate a speed-up of an order of magnitude is possible. Finally, we then extend the models and analysis to mantle convection systems that include for single 12. Viscous Dissipation Effects on the Motion of Casson Fluid over an Upper Horizontal Thermally Stratified Melting Surface of a Paraboloid of Revolution: Boundary Layer Analysis Directory of Open Access Journals (Sweden) T. M. Ajayi 2017-01-01 Full Text Available The problem of a non-Newtonian fluid flow past an upper surface of an object that is neither a perfect horizontal/vertical nor inclined/cone in which dissipation of energy is associated with temperature-dependent plastic dynamic viscosity is considered. An attempt has been made to focus on the case of two-dimensional Casson fluid flow over a horizontal melting surface embedded in a thermally stratified medium. Since the viscosity of the non-Newtonian fluid tends to take energy from the motion (kinetic energy and transform it into internal energy, the viscous dissipation term is accommodated in the energy equation. Due to the existence of internal space-dependent heat source; plastic dynamic viscosity and thermal conductivity of the non-Newtonian fluid are assumed to vary linearly with temperature. Based on the boundary layer assumptions, suitable similarity variables are applied to nondimensionalized, parameterized and reduce the governing partial differential equations into a coupled ordinary differential equations. These equations along with the boundary conditions are solved numerically using the shooting method together with the Runge-Kutta technique. The effects of pertinent parameters are established. A significant increases in Rex1/2Cfx is guaranteed with St when magnitude of β is large. Rex1/2Cfx decreases with Ec and m. 13. Non-Newtonian viscosity in magnetized plasma CERN Document Server Johnson, Robert W 2007-01-01 The particle and momentum balance equations can be solved on concentric circular flux surfaces to determine the effective viscous drag present in a magnetized tokamak plasma in the low aspect ratio limit. An analysis is developed utilizing the first-order Fourier expansion of the poloidal variation of quantities on the flux surface akin to that by Stacey and Sigmar [Phys. Fluids, 28, 9 (1985)]. Expressions to determine the poloidal variations of density, poloidal velocity, toroidal velocity, radial electric field, poloidal electric field, and other radial profiles are presented in a multi-species setting. Using as input experimental data for the flux surface averaged profiles of density, temperature, toroidal current, toroidal momentum injection, and the poloidal and toroidal rotations of at least one species of ion, one may solve the equations numerically for the remaining profiles. The resultant effective viscosities are compared to those predicted by Stacey and Sigmar and Shaing, et al., [Nuclear Fusion, 2... 14. 有限长线接触非牛顿热弹流润滑分析%A Thermal EHL Model for Finite Line Contact with Non-Newtonian Fluids Institute of Scientific and Technical Information of China (English) 刘明勇; 朱才朝; 刘怀举 2014-01-01 A thermal elastohydrodynamic lubrication (TEHL)finite line contact model was proposed for a cylindrical roller with Ree-Eyring fluid and Power-Law fluid.The results show that with the increase of contact line length,end effect is decreased.The Eyring shear stress for Ree-Eyring fluid has a great influence on the temperature rise and shear stress, while has a little influence on the pressure and film thickness.For Power-Law fluid,the film thickness is decreased dramat-ically with the increase of Power-Law exponent.Compared with the isothermal results,the thermal results are quite different with the increase of load and speed,and there is a more significant influence on friction coefficient for thermal effects.%建立有限长圆柱滚子的非牛顿流体热弹流润滑模型，选取Ree-Eyring流体和Power-Law流体进行有限长线接触弹流润滑分析。研究表明：随着接触线长度增大，端部效应减弱；Ree-Eyring流体特征剪切力对润滑温升和剪切力影响较大，而对润滑压力与膜厚影响甚微；Power-Law流体随指数增大，润滑膜厚明显降低；随载荷、转速升高，热解与等温润滑结果差异增大，热效应对摩擦因数的影响尤其显著。 15. Simulation of non-Newtonian oil-water core annular flow through return bends Science.gov (United States) Jiang, Fan; Wang, Ke; Skote, Martin; Wong, Teck Neng; Duan, Fei 2017-07-01 The volume of fluid (VOF) model is used together with the continuum surface force (CSF) model to numerically simulate the non-Newtonian oil-water core annular flow across return bends. A comprehensive study is conducted to generate the profiles of pressure, velocity, volume fraction and wall shear stress for different oil properties, flow directions, and bend geometries. It is revealed that the oil core may adhere to the bend wall under certain operating conditions. Through the analysis of the total pressure gradient and fouling angle, suitable bend geometric parameters are identified for avoiding the risk of fouling. 16. Non-Newtonian behavior and molecular structure of Cooee bitumen under shear flow DEFF Research Database (Denmark) Lemarchand, Claire; Bailey, Nicholas; Daivis, Peter 2015-01-01 The rheology and molecular structure of a model bitumen (Cooee bitumen) under shear are investigated in the non-Newtonian regime using non-equilibrium molecular dynamics simulations. The shear viscosity, normal stress differences, and pressure of the bitumen mixture are computed at different shear...... rates and different temperatures. The model bitumen is shown to be a shear-thinning fluid at all temperatures. In addition, the Cooee model is able to reproduce experimental results showing the formation of nanoaggregates composed of stacks of flat aromatic molecules in bitumen. These nanoaggregates... 17. Weak solutions for a non-Newtonian diffuse interface model with different densities Science.gov (United States) Abels, Helmut; Breit, Dominic 2016-11-01 We consider weak solutions for a diffuse interface model of two non-Newtonian viscous, incompressible fluids of power-law type in the case of different densities in a bounded, sufficiently smooth domain. This leads to a coupled system of a nonhomogenouos generalized Navier-Stokes system and a Cahn-Hilliard equation. For the Cahn-Hilliard part a smooth free energy density and a constant, positive mobility is assumed. Using the {{L}∞} -truncation method we prove existence of weak solutions for a power-law exponent p>\\frac{2d+2}{d+2} , d = 2, 3. 18. Non-Newtonian Liquid Flow through Small Diameter Piping Components: CFD Analysis Science.gov (United States) Bandyopadhyay, Tarun Kanti; Das, Sudip Kumar 2016-10-01 Computational Fluid Dynamics (CFD) analysis have been carried out to evaluate the frictional pressure drop across the horizontal pipeline and different piping components, like elbows, orifices, gate and globe valves for non-Newtonian liquid through 0.0127 m pipe line. The mesh generation is done using GAMBIT 6.3 and FLUENT 6.3 is used for CFD analysis. The CFD results are verified with our earlier published experimental data. The CFD results show the very good agreement with the experimental values. 19. Structure of a binary mixture under shear: non-Newtonian effects from computer simulation Energy Technology Data Exchange (ETDEWEB) Hanley, H.J.M.; Evans, D.J.; Hess, S. 1983-02-01 A binary equimolar dense fluid mixture is subjected to a shear. The orientational distribution of particles of type i around particles of type j (i, j = 1, 2) and the distortion of the radial distribution function is discussed for planar Couette flow. Results are presented in terms of a mixture of soft spheres, for which one species differs substantially in size and mass from the other, simulated on the computer using the technique of shear nonequilbrium molecular dynamics. Transport coefficients, including those associated with normal pressure differences, are given for the mixture and for the species in the mixture. Non-Newtonian phenomena are observed. 20. The effect of velocity and dimension of solid nanoparticles on heat transfer in non-Newtonian nanofluid Science.gov (United States) Akbari, Omid Ali; Toghraie, Davood; Karimipour, Arash; Marzban, Ali; Ahmadi, Gholam Reza 2017-02-01 In this investigation, the behavior of non-Newtonian nanofluid hydrodynamic and heat transfer are simulated. In this study, we numerically simulated a laminar forced non-Newtonian nanofluid flow containing a 0.5 wt% carboxy methyl cellulose (CMC) solutionin water as the base fluid with alumina at volume fractions of 0.5 and 1.5 as the solid nanoparticle. Numerical solution was modelled in Cartesian coordinate system in a two-dimensional microchannel in Reynolds number range of 10≤Re≤1000. The analyzed geometrical space here was a rectangular part of whose upper and bottom walls was influenced by a constant temperature. The effect of volume fraction of the nanoparticles, Reynolds number and non-Newtonian nanofluids was studied. In this research, the changes pressure drop, the Nusselt number, dimensionless temperature and heat transfer coefficient, caused by the motion of non-Newtonian nanofluids are described. The results indicated that the increase of the volume fraction of the solid nanoparticles and a reduction in the diameter of the nanoparticles would improve heat transfer which is more significant in Reynolds number. The results of the introduced parameters in the form of graphs drawing and for different parameters are compared. 1. Non-Newtonian Study of Blood Flow in an Abdominal Aortic Aneurysm with a Stabilized Finite Element Method Science.gov (United States) Marrero, Victor; Sahni, Onkar; Jansen, Kenneth; Tichy, John; Taylor, Charles 2008-11-01 In recent years the methods of computational fluid dynamics (CFD) have been applied to the human cardiovascular system to better understand the relationship between arterial blood flow and the disease process, for example in an abdominal aortic aneurysm (AAA). Obviously, the technical challenges associated with such modeling are formidable. Among the many problems to be addressed, in this paper we add yet another complication -- the known non-Newtonian nature of blood. In this preliminary study, we used a patient-based AAA model with rigid walls. The pulsatile nature of the flow and the RCR outflow boundary condition are considered. We use the Carreau-Yasuda model to describe the non-Newtonian viscosity variation. Preliminary results for 200K, 2M, and 8M elements mesh are presented for the Newtonian and non-Newtonian cases. The broad fundamental issue we wish to eventually resolve is whether or not non-Newtonian effects in blood flow are sufficiently strong in unhealthy vessels that they must be addressed in meaningful simulations. Interesting differences during the flow cycle shed light on the problem, but further research is needed. 2. The stretching of an electrified non-Newtonian jet: A model for electrospinning Science.gov (United States) Feng, J. J. 2002-11-01 Electrospinning uses an external electrostatic field to accelerate and stretch a charged polymer jet, and may produce ultrafine "nanofibers." Many polymers have been successfully electrospun in the laboratory. Recently Hohman [et al.] [Phys. Fluids, 13, 2201 (2001)] proposed an electrohydrodynamic model for electrospinning Newtonian jets. A problem arises, however, with the boundary condition at the nozzle. Unless the initial surface charge density is zero or very small, the jet bulges out upon exiting the nozzle in a "ballooning instability," which never occurs in reality. In this paper, we will first describe a slightly different Newtonian model that avoids the instability. Well-behaved solutions are produced that are insensitive to the initial charge density, except inside a tiny "boundary layer" at the nozzle. Then a non-Newtonian viscosity function is introduced into the model and the effects of extension thinning and thickening are explored. Results show two distinct regimes of stretching. For a "mildly stretched" jet, the axial tensile force in the fiber resists stretching, so that extension thinning promotes stretching and thickening hinders stretching. For a "severely stretched" jet, on the other hand, the tensile force enhances stretching at the beginning of the jet and suppresses it farther downstream. The effects of extensional viscosity then depend on the competition between the upstream and downstream dynamics. Finally, we use an empirical correlation to simulate strain hardening typical of polymeric liquids. This generally steepens the axial gradient of the tensile stress. Stretching is more pronounced at the beginning but weakens later, and ultimately thicker fibers are produced because of strain hardening. 3. Generation of Oil Droplets in a Non-Newtonian Liquid Using a Microfluidic T-Junction Directory of Open Access Journals (Sweden) Enrico Chiarello 2015-11-01 Full Text Available We have compared the formation of oil drops in Newtonian and non-Newtonian fluids in a T-junction microfluidic device. As Newtonian fluids, we used aqueous solutions of glycerol, while as non-Newtonian fluids we prepared aqueous solutions of xanthan, a stiff rod-like polysaccharide, which exhibit strong shear-thinning effects. In the squeezing regime, the formation of oil droplets in glycerol solutions is found to scale with the ratio of the dispersed flow rate to the continuous one and with the capillary number associated to the continuous phase. Switching to xanthan solutions does not seem to significantly alter the droplet formation process. Any quantitative difference with respect to the Newtonian liquid can be accounted for by a suitable choice of the capillary number, corresponding to an effective xanthan viscosity that depends on the flow rates. We have deduced ample variations in the viscosity, on the order of 10 and more, during normal operation conditions of the T-junction. This allowed estimating the actual shear rates experienced by the xanthan solutions, which go from tens to hundreds of s−1. 4. Non--Newtonian gravity and coherence properties of light CERN Document Server Camacho, A 2001-01-01 In this work the possibility of detecting a non--Newtonian contribution to the gravitational potential by means of its effects upon the first and second--order coherence properties of light is analyzed. It will be proved that, in principle, the effects of a fifth force upon the correlation functions of electromagnetic radiation could be used to detect the existence of new forces. Some constraints upon the experimental parameters will also be deduced. 5. MHD Boundary Layer Flow of Dilatant Fluid in a Divergent Channel with Suction or Blowing Institute of Scientific and Technical Information of China (English) Krishnendu Bhattacharyya; G.C.Layek 2011-01-01 @@ An analysis is carried out to study a steady magnetohydrodynamic(MHD) boundary layer How of an electrically conducting incompressible power-law non-Newtonian fluid through a divergent channel.The channel walls are porous and subjected to either suction or blowing of equal magnitude of the same kind of fluid on both walls.The fluid is permeated by a magnetic field produced by electric current along the line of intersection of the channel walls.The governing partial differential equation is transformed into a self-similar nonlinear ordinary differential equation using similarity transformations.The possibility of boundary layer flow in a divergent channel is analyzed with the power-law fluid model.The analysis reveals that the boundary layer flow (without separation) is possible for the case of the dilatant fluid model subjected to suitable suction velocity applied through its porous walls,even in the absence of a magnetic field.Further, it is found that the boundary layer flow is possible even in the presence of blowing for a suitable value of the magnetic parameter.It is found that the velocity increases with increasing values of the power-law index for the case of dilatant fluid.The effects of suction/blowing and magnetic field on the velocity are shown graphically and discussed physical尔 6. Analysis of flow and LDL concentration polarization in siphon of internal carotid artery: Non-Newtonian effects. Science.gov (United States) Sharifi, Alireza; Niazmand, Hamid 2015-10-01 Carotid siphon is known as one of the risky sites among the human intracranial arteries, which is prone to formation of atherosclerotic lesions. Indeed, scientists believe that accumulation of low density lipoprotein (LDL) inside the lumen is the major cause of atherosclerosis. To this aim, three types of internal carotid artery (ICA) siphon have been constructed to examine variations of hemodynamic parameters in different regions of the arteries. Providing real physiological conditions, blood considered as non-Newtonian fluid and real velocity and pressure waveforms have been employed as flow boundary conditions. Moreover, to have a better estimation of risky sites, the accumulation of LDL particles has been considered, which has been usually ignored in previous relevant studies. Governing equations have been discretized and solved via open source OpenFOAM software. A new solver has been built to meet essential parameters related to the flow and mass transfer phenomena. In contrast to the common belief regarding negligible effect of blood non-Newtonian behavior inside large arteries, current study suggests that the non-Newtonian blood behavior is notable, especially on the velocity field of the U-type model. In addition, it is concluded that neglecting non-Newtonian effects underestimates the LDL accumulation up to 3% in the U-type model at the inner side of both its bends. However, in the V and C type models, non-Newtonian effects become relatively small. Results also emphasize that the outer part of the second bend at the downstream is also at risk similar to the inner part of the carotid bends. Furthermore, from findings it can be implied that the risky sites strongly depend on the ICA shape since the extension of the risky sites are relatively larger for the V-type model, while the LDL concentrations are higher for the C-type model. 7. Boundary layer flow and heat transfer to Carreau fluid over a nonlinear stretching sheet Directory of Open Access Journals (Sweden) Masood Khan 2015-10-01 Full Text Available This article studies the Carreau viscosity model (which is a generalized Newtonian model and then use it to obtain a formulation for the boundary layer equations of the Carreau fluid. The boundary layer flow and heat transfer to a Carreau model over a nonlinear stretching surface is discussed. The Carreau model, adequate for many non-Newtonian fluids, is used to characterize the behavior of the fluids having shear thinning properties and fluids with shear thickening properties for numerical values of the power law exponent n. The modeled boundary layer conservation equations are converted to non-linear coupled ordinary differential equations by a suitable transformation. Numerical solution of the resulting equations are obtained by using the Runge-Kutta Fehlberg method along with shooting technique. This analysis reveals many important physical aspects of flow and heat transfer. Computations are performed for different values of the stretching parameter (m, the Weissenberg number (We and the Prandtl number (Pr. The obtained results show that for shear thinning fluid the fluid velocity is depressed by the Weissenberg number while opposite behavior for the shear thickening fluid is observed. A comparison with previously published data in limiting cases is performed and they are in excellent agreement. 8. A Remark on the Time Decay of Non-Newtonian Flows in R3%关于非牛顿流体衰减性的一个注记 Institute of Scientific and Technical Information of China (English) 殷谷良 2009-01-01 In the study of long time asymptotic behaviors of the solutions to a class system of the incompressible non-Newtonian fluid flows in R3, it is proved that the weak solutions decay in L2 norm at (1+t) and the error of difference between non-Newtonian fluid and linear equation is also investigated. The findings are mainly based on the classic Fourier splitting methods. 9. Physical Parameters of Blood as a Non - Newtonian Fluid OpenAIRE Baieth, H. E. Abdel 2008-01-01 Do increasing doses of electromagnetic fields (EMFs) increase the hazards effect on blood? Studies on the blood of rats provide guidance for the assessment of occupational and public health significance of exposure to EMFs. Here, apparent additive viscosity of animal blood after exposing to EMFs (3,5 and 10 gauss) is examined. The results indicate that hematocrite (HCT) increased as EMF increases while the viscocity decreased with the increase of EMF. Red blood cell permeability, deformabilit... 10. Numerical simulation of pulsatile flow with newtonian and non-newtonian behavior in arterial stenosis Directory of Open Access Journals (Sweden) MM Movahedi 2008-03-01 Full Text Available Background: There is considerable evidence that vascular fluid dynamics plays an important role in the developmentand prevalence of atherosclerosis which is one of the most widespread disease in humans .The onset and prevalence of atherosclerosis hemodynamic parameter are largely affected by geometric parameters. If any obstacle interferes with the blood flow, the above parameters change dramatically. Most of the arterial diseases, such as atherosclerosis, occur in the arteries with complex patterns of fluid flow where the blood dynamics plays an important role. Arterial stenosis mostly occurs in an area with a complex pattern of fluid flow, such as coronary artery, aorta bifurcation, carotid and vessels of lower limbs. During the past three decades, many experimental studies have been performed on the hemodynamic role of the blood in forming sediment in the inner wall of the vessels. It has been shown that forming sediment in the inner wall of vessels depends on the velocity of fluid and also on the amount of wall shear stress.Methods: We have examined the effect on the blood flow of local stenosis in carotid artery in numerical form using the incompressible Navier-Stockes equations. The profile of the velocity in different parts and times in the pulsatile cycle, separation and reattachment points on the wall, the distance stability of flow and also alteration caused by the wall shear stress in entire vessel were shown and compared with two behaviors flow (Newtonian and Non-Newtonian.Finally we describe the influence of the severity of the stenosis on the separation and reattachmentpoints for a Non-Newtonian fuid. Results: In the present study, we have pointed very low and high oscillating WSS (Wall Shear Stress values play a significant role in the development of forming sediment in the inner wall of vessels. Also, we obtain this probability is higher for Newtonian than Non-Newtonian fluid behavior.Conclusion: Based on our results, the 11. 幂律非牛顿流体在偏心圆环管中流动和传热的数值计算%Numerical calculation of flow and heat transfer for non-Newtonian fluid in eccentric annular channel Institute of Scientific and Technical Information of China (English) 张钧波; 张敏 2014-01-01 This paper studies power-law non-Newtonian fluid where the shearing stress accords with the Ostwald-de Waele relational formula.It aims to solve the problem of the apparent viscosity changing with the shearing rates and the calculation which is different from Newtonian fluid.Cell-based central method is used for the non-linear viscosity coefficient discretization.It adopts FVM for flow and heat transfer numerical simulation of fully developed laminar in eccentric annular channel.The calculation results show that the power-law factor has a great impact on the fluid flow,but the impact on the heat transfer is influenced by the eccentricity .The eccentric-ity of annular channel can lead to the asymmetrical distribution of circumferential temperature and velocity.%以剪切应力符合Ostwald-de Waele关系式的幂律非牛顿流体为研究对象。针对幂律非牛顿流体的表观黏度随剪切速率变化且计算过程有别于牛顿流体的问题，运用基元中心法对非线性粘性系数进行离散。采用有限体积法对幂律非牛顿流体在偏心圆环管中的充分发展层流流动和传热进行数值计算。计算结果表明流体的幂律因子对流动的影响较大，但对传热的影响受到到偏心率的影响。流道偏心会引起圆环通道内速度和温度的周向分布不均匀，且偏心程度越严重，周向分布不均匀性越强烈。 12. Three-dimensional linear instability in pressure-driven two-layer channel flow of a Newtonian and a Herschel-Bulkley fluid Science.gov (United States) Sahu, K. C.; Matar, O. K. 2010-11-01 The three-dimensional linear stability characteristics of pressure-driven two-layer channel flow are considered, wherein a Newtonian fluid layer overlies a layer of a Herschel-Bulkley fluid. We focus on the parameter ranges for which Squire's theorem for the two-layer Newtonian problem does not exist. The modified Orr-Sommerfeld and Squire equations in each layer are derived and solved using an efficient spectral collocation method. Our results demonstrate the presence of three-dimensional instabilities for situations where the square root of the viscosity ratio is larger than the thickness ratio of the two layers; these "interfacial" mode instabilities are also present when density stratification is destabilizing. These results may be of particular interest to researchers studying the transient growth and nonlinear stability of two-fluid non-Newtonian flows. We also show that the "shear" modes, which are present at sufficiently large Reynolds numbers, are most unstable to two-dimensional disturbances. 13. In defense of a non-newtonian economic analysis OpenAIRE Filip, Diana; Piatecki, Cyrille 2014-01-01 The double-entry bookkeeping promoted by Luca Pacioli in the fifteenth century could be considered a strong argument in behalf of the multiplicative calculus which can be developed from the Grossman and Katz non-newtonian calculus concept. In order to emphasize this statement we present a brief history of the accountancy in its early time and we make the point of Ellerman's research concerning the double-entry bookkeeping.; La comptabilité en partie double présentée par Luca Pacioli au quinzi... 14. Test of non-Newtonian gravitational force at micrometer range CERN Document Server Luo, Pengshun; Guan, Shengguo; Wu, Wenjie; Tian, Zhaoyang; Yang, Shanqing; Shao, Chenggang; Luo, Jun 2016-01-01 We report an experimental test of non-Newtonian gravitational forces at mi- crometer range. To experimentally subtract off the Casimir force and the electrostatic force background, differential force measurements were performed by sensing the lateral force between a gold sphere and a density modulated source mass using a soft cantilever. The current sensitivity is limited by the patch electrostatic force, which is further improved by two dimensional (2D) force mapping. The preliminary result sets a model independent constraint on the Yukawa type force at this range. 15. Numerical simulation of pulsatile non-Newtonian flow in the carotid artery bifurcation Science.gov (United States) Fan, Yubo; Jiang, Wentao; Zou, Yuanwen; Li, Jinchuan; Chen, Junkai; Deng, Xiaoyan 2009-04-01 Both clinical and post mortem studies indicate that, in humans, the carotid sinus of the carotid artery bifurcation is one of the favored sites for the genesis and development of atherosclerotic lesions. Hemodynamic factors have been suggested to be important in atherogenesis. To understand the correlation between atherogenesis and fluid dynamics in the carotid sinus, the blood flow in artery was simulated numerically. In those studies, the property of blood was treated as an incompressible, Newtonian fluid. In fact, however, the blood is a complicated non-Newtonian fluid with shear thinning and viscoelastic properties, especially when the shear rate is low. A variety of non-Newtonian models have been applied in the numerical studies. Among them, the Casson equation was widely used. However, the Casson equation agrees well only when the shear rate is less than 10 s-1. The flow field of the carotid bifurcation usually covers a wide range of shear rate. We therefore believe that it may not be sufficient to describe the property of blood only using the Casson equation in the whole flow field of the carotid bifurcation. In the present study, three different blood constitutive models, namely, the Newtonian, the Casson and the hybrid fluid constitutive models were used in the flow simulation of the human carotid bifurcation. The results were compared among the three models. The results showed that the Newtonian model and the hybrid model had very similar distributions of the axial velocity, secondary flow and wall shear stress, but the Casson model resulted in significant differences in these distributions from the other two models. This study suggests that it is not appropriate to only use the Casson equation to simulate the whole flow field of the carotid bifurcation, and on the other hand, Newtonian fluid is a good approximation to blood for flow simulations in the carotid artery bifurcation. 16. Numerical simulation of pulsatile non-Newtonian flow in the carotid artery bifurcation Institute of Scientific and Technical Information of China (English) Yubo Fan; Wentao Jiang; Yuanwen Zou; Jinchuan Li; Junkai Chen; Xiaoyan Deng 2009-01-01 Both clinical and post mortem studies indicate that, in humans, the carotid sinus of the carotid artery bifurcation is one of the favored sites for the genesis and development of atherosclerotic lesions. Hemodynamic factors have been suggested to be important in atherogenesis. To understand the correlation between atherogenesis and fluid dynamics in the carotid sinus, the blood flow in artery was simulated numerically. In those studies, the property of blood was treated as an incompressible, Newtonian fluid. In fact,however, the blood is a complicated non-Newtonian fluid with shear thinning and viscoelastic properties, especially when the shear rate is low. A variety of non-Newtonian models have been applied in the numerical studies. Among them,the Casson equation was widely used. However, the Casson equation agrees well only when the shear rate is less than 10s-1. The flow field of the carotid bifurcation usually covers a wide range of shear rate. We therefore believe that it may not be sufficient to describe the property of blood only using the Casson equation in the whole flow field of the carotid bifurcation. In the present study, three different blood constitutive models, namely, the Newtonian, the Casson and the hybrid fluid constitutive models were used in the flow simulation of the human carotid bifurcation. The results were compared among the three models. The results showed that the Newtonian model and the hybrid model had very similar distributions of the axial velocity, secondary flow and wall shear stress, but the Casson model resulted in significant differences in these distributions from the other two models. This study suggests that it is not appropriate to only use the Casson equation to simulate the whole flow field of the carotid bifurcation, and on the other hand, Newtonian fluid is a good approximation to blood for flow simulations in the carotid artery bifurcation. 17. 3D Computer Simulations of Pulsatile Human Blood Flows in Vessels and in the Aortic Arch: Investigation of Non-Newtonian Characteristics of Human Blood CERN Document Server Sultanov, Renat A; Engelbrekt, Brent; Blankenbecler, Richard 2008-01-01 Methods of Computational Fluid Dynamics are applied to simulate pulsatile blood flow in human vessels and in the aortic arch. The non-Newtonian behaviour of the human blood is investigated in simple vessels of actual size. A detailed time-dependent mathematical convergence test has been carried out. The realistic pulsatile flow is used in all simulations. Results of computer simulations of the blood flow in vessels of two different geometries are presented. For pressure, strain rate and velocity component distributions we found significant disagreements between our results obtained with realistic non-Newtonian treatment of human blood and widely used method in literature: a simple Newtonian approximation. A significant increase of the strain rate and, as a result, wall sear stress distribution, is found in the region of the aortic arch. We consider this result as theoretical evidence that supports existing clinical observations and those models not using non-Newtonian treatment underestimate the risk of disru... 18. Break-up of a non-Newtonian jet injected downwards in a Newtonian liquid Absar M Lakdawala; Rochish Thaokar; Atul Sharma 2015-05-01 The present work on downward injection of non-Newtonian jet is an extension of our recent work (Lakdawala et al, Int. J. Multiphase Flow. 59: 206–220, 2014) on upward injection of Newtonian jet. The non-Newtonian rheology of the jet is described by a Carreau type generalized Newtonian fluid (GNF) model, which is a phenomenological constitutive equation that accounts for both rate-thinning and rate-thickening. Level set method based numerical study is done for Newtonian as well as various types of shear thinning and thickening jet fluid. Effect of average injection velocity ($V_{av,i}$) is studied at a constant Reynolds number Re = 14.15, Weber number W e = 1, Froude number F r = 0.25, density ratio $\\chi$ = 0.001 and viscosity ratio $\\eta$ = 0.01. CFD analysis of the temporal variation of interface and jet length ($L_{j}$) is done to propose different types of jet breakup regimes. At smaller, intermediate and larger values of $V_{av,i}$, the regimes found are periodic uniform drop (P-UD), quasi-periodic non-uniform drop (QP-NUD) and no breakup (NB) regimes for a shear thinning jet; and periodic along with Satellite Drop (P+S), jetting (J) and no breakup (NB) regimes for a shear thickening jet, respectively. This is presented as a drop-formation regime map. Shear thickening (thinning) is shown to produce long (short) jet length. Diameter of the primary drop increases and its frequency of release decreases, due to increase in stability of the jet for shear thickening as compared to thinning fluid. 19. The effects of non-Newtonian viscosity on the deformation of red blood cells in a shear flow Science.gov (United States) Sesay, Juldeh 2005-11-01 The analyses of the effects of non-Newtonian viscosity on the membrane of red blood cells (RBCs) suspended in a shear flow are presented. The specific objective is to investigate the mechanical deformation on the surfaces of an ellipsoidal particle model. The hydrodynamic stresses and other forces on the surface of the particle are used to determine the cell deformation. We extended previous works, which were based on the Newtonian fluid models, to the non-Newtonian case, and focus on imposed shear rate values between 1 and 100 per second. Two viscosity models are investigated, which respectively correspond to a normal person and a patient with cerebrovascular accident (CVA). The results are compared with those obtained assuming a Newtonian model. We observed that the orientation of the cell influences the deformation and the imposed shear rate drives the local shear rate distribution along the particle surface. The integral particle deformation for the non-Newtonian models in the given shear rate regime is higher than that for the Newtonian reference model. Finally, the deformation of the cell surface decreases as the dissipation ratio increases. 20. Casson fluid flow and heat transfer over a nonlinearly stretching surface Institute of Scientific and Technical Information of China (English) 2013-01-01 A boundary layer analysis is presented for non-Newtonian fluid flow and heat transfer over a nonlinearly stretching surface.The Casson fluid model is used to characterize the non-Newtonian fluid behavior.By using suitable transformations,the governing partial differential equations corresponding to the momentum and energy equations are converted into non-linear ordinary differential equations.Numerical solutions of these equations are obtained with the shooting method.The effect of increasing Casson parameter is to suppress the velocity field.However the temperature is enhanced with the increasing Casson parameter. 1. Dynamical behaviour of non newtonian spiral blood flow through arterial stenosis Science.gov (United States) Ali, Mohammad; Mahmudul Hasan, Md.; Alam Maruf, Mahbub 2017-04-01 The spiral component of blood flow has both beneficial and detrimental effects in human circulatory system. A numerical investigation is carried out to analyze the effects of spiral blood flow through an axisymmetric three dimensional artery having 75% stenosis at the center. Blood is assumed as a non-Newtonian fluid. Standard k-ω model is used for the simulation with the Reynolds number of 1000. A parabolic velocity profile with spiral flow is used as inlet boundary condition. The peak values of all velocity components are found just after stenosis. But total pressure gradually decreases at downstream. Spiral flow of blood has significant effects on tangential component of velocity. However, the effect is mild for radial and axial velocity components. The peak value of wall shear stress is at the stenosis zone and decreases rapidly in downstream. The effect of spiral flow is significant for turbulent kinetic energy. Detailed investigation and relevant pathological issues are delineated throughout the paper. 2. Rheological non-Newtonian behaviour of ethylene glycol-based Fe2O3 nanofluids Directory of Open Access Journals (Sweden) Pastoriza-Gallego María 2011-01-01 Full Text Available Abstract The rheological behaviour of ethylene glycol-based nanofluids containing hexagonal scalenohedral-shaped α-Fe2O3 (hematite nanoparticles at 303.15 K and particle weight concentrations up to 25% has been carried out using a cone-plate Physica MCR rheometer. The tests performed show that the studied nanofluids present non-Newtonian shear-thinning behaviour. In addition, the viscosity at a given shear rate is time dependent, i.e. the fluid is thixotropic. Finally, using strain sweep and frequency sweep tests, the storage modulus G', loss modulus G″ and damping factor were determined as a function of the frequency showing viscoelastic behaviour for all samples. 3. A numerical solution for the entrance region of non-newtonian flow in annuli Directory of Open Access Journals (Sweden) Maia M.C.A. 2003-01-01 Full Text Available Continuity and momentum equations applied to the entrance region of an axial, incompressible, isothermal, laminar and steady flow of a power-law fluid in a concentric annulus, were solved by a finite difference implicit method. The Newtonian case was solved used for validation of the method and then compared to reported results. For the non-Newtonian case a pseudoplastic power-law model was assumed and the equations were transformed to obtain a pseudo-Newtonian system which enabled its solution using the same technique as that used for the Newtonian case. Comparison of the results for entrance length and pressure drop with those available in the literature showed a qualitative similarity, but significant quantitative differences. This can be attributed to the differences in entrance geometries and the definition of asymptotic entrance length. 4. Steady flow and heat transfer analysis of Phan-Thein-Tanner fluid in double-layer optical fiber coating analysis with Slip Conditions Science.gov (United States) Khan, Zeeshan; Shah, Rehan Ali; Islam, Saeed; Jan, Bilal; Imran, Muhammad; Tahir, Farisa 2016-10-01 Modern optical fibers require double-layer coating on the glass fiber to provide protection from signal attenuation and mechanical damage. The most important plastic resins used in wires and optical fibers are plastic polyvinyl chloride (PVC) and low-high density polyethylene (LDPE/HDPE), nylon and Polysulfone. In this paper, double-layer optical fiber coating is performed using melt polymer satisfying PTT fluid model in a pressure type die using wet-on-wet coating process. The assumption of fully developed flow of Phan-Thien-Tanner (PTT) fluid model, two-layer liquid flows of an immiscible fluid is modeled in an annular die, where the fiber is dragged at a higher speed. The equations characterizing the flow and heat transfer phenomena are solved exactly and the effects of emerging parameters (Deborah and slip parameters, characteristic velocity, radii ratio and Brinkman numbers on the axial velocity, flow rate, thickness of coated fiber optics, and temperature distribution) are reported in graphs. It is shown that an increase in the non-Newtonian parameters increase the velocity in the absence or presence of slip parameters which coincides with related work. The comparison is done with experimental work by taking λ → 0 (non-Newtonian parameter). 5. Steady flow and heat transfer analysis of Phan-Thein-Tanner fluid in double-layer optical fiber coating analysis with Slip Conditions Science.gov (United States) Khan, Zeeshan; Shah, Rehan Ali; Islam, Saeed; Jan, Bilal; Imran, Muhammad; Tahir, Farisa 2016-01-01 Modern optical fibers require double-layer coating on the glass fiber to provide protection from signal attenuation and mechanical damage. The most important plastic resins used in wires and optical fibers are plastic polyvinyl chloride (PVC) and low-high density polyethylene (LDPE/HDPE), nylon and Polysulfone. In this paper, double-layer optical fiber coating is performed using melt polymer satisfying PTT fluid model in a pressure type die using wet-on-wet coating process. The assumption of fully developed flow of Phan-Thien-Tanner (PTT) fluid model, two-layer liquid flows of an immiscible fluid is modeled in an annular die, where the fiber is dragged at a higher speed. The equations characterizing the flow and heat transfer phenomena are solved exactly and the effects of emerging parameters (Deborah and slip parameters, characteristic velocity, radii ratio and Brinkman numbers on the axial velocity, flow rate, thickness of coated fiber optics, and temperature distribution) are reported in graphs. It is shown that an increase in the non-Newtonian parameters increase the velocity in the absence or presence of slip parameters which coincides with related work. The comparison is done with experimental work by taking λ → 0 (non-Newtonian parameter). PMID:27708412 6. Numerical Analyses of the Non-Newtonian Flow Performance and Thermal Effect on a Bearing Coated with a High Tin Content Directory of Open Access Journals (Sweden) K. Mehala 2016-12-01 Full Text Available The hydrodynamic bearings are stressed by severe workings conditions, such as speed, load, and the oil will be increasingly solicit by pressure and shear. The Newtonian behavior is far from being awarded in this case, the most loaded bearings operating at very high speeds; the shear rate of the oil is of higher order. A numerical analysis of the behavior of non-Newtonian fluid for plain cylindrical journal bearing finite dimension coated with antifriction material with a high tin content, for to facilitate the accommodation of the surfaces and save the silk of the shaft in the case of a contact. this analyses is implemented using the code-ANSYS CFX, by solving the energy equation with the finite difference method, considering that laminar regime and the fluid is non Newtonian by using the power law Ostwald model, the coefficient n is equal to 1.25 and for different model such as Bingham, cross and Hereshek-Bulkley model. This study aims to better predict the non-Newtonian behavior of the oil film in bearings operating under more severe conditions. The purpose conducted during this study is to predict the effect of non-Newtonian behavior of the film; the journal bearing operating under severe conditions, the speed of rotation varies from 1000 to 9000 rpm and the bearing working under radial load 2 to 10 kN. Temperature and the pressure within the fluid film assumed non-Newtonian are high, with a coefficient n greater than 1 that is to say for viscoelastic fluids. 7. Heat transfer enhancement in smooth tube with wire coil insert in laminar and transitional non-newtonian flow OpenAIRE García Pinar, Alberto; Solano Fernández, Juan Pedro; Viedma Robles, Antonio; Martínez Hernández, David Sebastián 2010-01-01 This work presents an experimental study on the heat transfer enhancement by means of a tube with wire-coil insert,for non-Newtonian laminar and transitional flow. The dimensionless pitch and wire diameter (based on the plain tube inner diameter) were chosen as p/D= 1 and e/D=0.09. Two pseudoplastic test fluids have been used: 1% by weight aqueous solutions of carboxymethyl cellulose (CMC) with high viscosity and medium viscosity. A wide range of flow conditions has been covered: Reynolds ... 8. Non-Newtonian Behavior of Diblock and Triblock Copolymer Solutions Science.gov (United States) Watanabe, Hiroshi 2006-03-01 Non-Newtonian flow behavior was examined for butadiene-styrene (BS) diblock and BSB triblock copolymers dissolved in a S-selective solvent, dibutyl phthalate (DBP). Spherical domains of the non-solvated B blocks were arranged on a bcc lattice in both solutions at equilibrium, as revealed from SANS. The solutions exhibited significant thinning under steady flow, which was well correlated with the disruption of the bcc lattice detected with SANS. The lattice disruption was most prominent at a shear rate comparable to the frequency of B/S concentration fluctuation. For the BS/DBP solution, the recovery of the lattice structure after cessation of flow was the slowest for the most heavily disrupted lattice, as naturally expected. In contrast, for the BSB/DBP solution, the recovery rate was insensitive to the magnitude of lattice disruption. This peculiar behavior of the BSB solution suggests that the rate-determining step of the recovery in this solution is the transient B/S mixing required for reformation of the S bridges connecting the B domains. 9. Experimental and Numerical Investigation on Non-Newtonian Nanofluids Flowing in Shell Side of Helical Baffled Heat Exchanger Combined with Elliptic Tubes Directory of Open Access Journals (Sweden) Ziye Ling 2017-01-01 Full Text Available In this paper, an aqueous solution of xanthan gum (XG at a weight fraction as high as 0.2% was elected as the non-Newtonian base liquid, the multi-walled carbon nanotubes (MWCNTs dispersed into non-Newtonian XG aqueous at different weight factions of MWCNTs was prepared. Convection heat transfer of non-Newtonian nanofluids in the shell side of helical baffled heat exchanger combined with elliptic tubes has been investigated experimentally and numerically using single-phase flow model. Results showed that the enhancement of the convective heat transfer coefficient increases with an increase in the Reynolds number and the nanoparticle concentration. For nanofluids with 0.2 wt %, 0.5 wt % and 1.0 wt % MWCNTs, the Nusselt number, respectively, increases by 11%, 21% and 35% on average at the same Reynolds number, while the comprehensive thermal performance factors are 3%–5%, 15%–17% and 24%–26% higher than that of base fluid at the same volume rate. A remarkable heat transfer enhancement can be obtained by adding MWCNTs into XG aqueous solution based on thermal resistance analysis. Correlations have been suggested for the shell-side Nusselt number and friction factor of non-Newtonian nanofluids in the helical baffled heat exchanger with elliptic tubes. Good agreements existed between corrections and experimental data. 10. The Construction of Hilbert Spaces over the Non-Newtonian Field OpenAIRE 2014-01-01 Although there are many excellent ways to present the principle of the classical calculus, the novel presentations probably lead most naturally to the development of the non-Newtonian calculi. In this paper we introduce vector spaces over real and complex non-Newtonian field with respect to the -calculus which is a branch of non-Newtonian calculus. Also we give the definitions of real and complex inner product spaces and study Hilbert spaces which are special type of normed space and complet... 11. Generalized Runge-Kutta Method with respect to the Non-Newtonian Calculus OpenAIRE 2015-01-01 Theory and applications of non-Newtonian calculus have been evolving rapidly over the recent years. As numerical methods have a wide range of applications in science and engineering, the idea of the design of such numerical methods based on non-Newtonian calculus is self-evident. In this paper, the well-known Runge-Kutta method for ordinary differential equations is developed in the frameworks of non-Newtonian calculus given in generalized form and then tested for different generating functio... 12. A Generalization on Weighted Means and Convex Functions with respect to the Non-Newtonian Calculus Directory of Open Access Journals (Sweden) 2016-01-01 Full Text Available This paper is devoted to investigating some characteristic features of weighted means and convex functions in terms of the non-Newtonian calculus which is a self-contained system independent of any other system of calculus. It is shown that there are infinitely many such useful types of weighted means and convex functions depending on the choice of generating functions. Moreover, some relations between classical weighted mean and its non-Newtonian version are compared and discussed in a table. Also, some geometric interpretations of convex functions are presented with respect to the non-Newtonian slope. Finally, using multiplicative continuous convex functions we give an application. 13. Convex functions and some inequalities in terms of the Non-Newtonian Calculus Science.gov (United States) Unluyol, Erdal; Salas, Seren; Iscan, Imdat 2017-04-01 Differentiation and integration are basic operations of calculus and analysis. Indeed, they are many versions of the subtraction and addition operations on numbers, respectively. From 1967 till 1970 Michael Grossman and Robert Katz [1] gave definitions of a new kind of derivative and integral, converting the roles of subtraction and addition into division and multiplication, and thus establish a new calculus, called Non-Newtonian Calculus. So, in this paper, it is investigated to the convex functions and some inequalities in terms of Non-Newtonian Calculus. Then we compare with the Newtonian and Non-Newtonian Calculus. 14. Stable two-layer flows at all Re; visco-plastic lubrication of shear-thinning and viscoelastic fluids NARCIS (Netherlands) Moyers-Gonzalez, M.; Frigaard, I. A.; Nouar, Cherif 2010-01-01 Multi-fluid flows are frequently thought of as being less stable than single phase flows. Consideration of different non-Newtonian models can give rise to different types of hydrodynamic instability. Here we show that with careful choice of fluid rheologies and flow paradigm, one can achieve multi-l 15. Stable two-layer flows at all Re; visco-plastic lubrication of shear-thinning and viscoelastic fluids NARCIS (Netherlands) Moyers-Gonzalez, M.; Frigaard, I. A.; Nouar, Cherif 2010-01-01 Multi-fluid flows are frequently thought of as being less stable than single phase flows. Consideration of different non-Newtonian models can give rise to different types of hydrodynamic instability. Here we show that with careful choice of fluid rheologies and flow paradigm, one can achieve multi-l 16. Numerical Investigation of Non-Newtonian Flow and Heat Transfer Characteristics in Rectangular Tubes with Protrusions Directory of Open Access Journals (Sweden) Yonghui Xie 2015-01-01 Full Text Available Flow characteristics and heat transfer performances in rectangular tubes with protrusions are numerically investigated in this paper. The thermal heat transfer enhancement of composite structures and flow resistance reduction of non-Newtonian fluid are taken advantage of to obtain a better thermal performance. Protrusion channels coupled with different CMC concentration solutions are studied, and the results are compared with that of smooth channels with water flow. The comprehensive influence of turbulence effects, structural effects, and secondary flow effects on the CMC’s flow in protrusion tubes is extensively investigated. The results indicate that the variation of flow resistance parameters of shear-thinning power-law fluid often shows a nonmonotonic trend, which is different from that of water. It can be concluded that protrusion structure can effectively enhance the heat transfer of CMC solution with low pressure penalty in specific cases. Moreover, for a specific protrusion structure and a fixed flow velocity, there exists an optimal solution concentration showing the best thermal performance. 17. Non-Newtonian unconfined flow and heat transfer over a heated cylinder using the direct-forcing immersed boundary-thermal lattice Boltzmann method. Science.gov (United States) Amiri Delouei, A; Nazari, M; Kayhani, M H; Succi, S 2014-05-01 In this study, the immersed boundary-thermal lattice Boltzmann method has been used to simulate non-Newtonian fluid flow over a heated circular cylinder. The direct-forcing algorithm has been employed to couple the off-lattice obstacles and on-lattice fluid nodes. To investigate the effect of boundary sharpness, two different diffuse interface schemes are considered to interpolate the velocity and temperature between the boundary and computational grid points. The lattice Boltzmann equation with split-forcing term is applied to consider the effects of the discrete lattice and the body force to the momentum flux, simultaneously. A method for calculating the Nusselt number based on diffuse interface schemes is developed. The rheological and thermal properties of non-Newtonian fluids are investigated under the different power-law indices and Reynolds numbers. The effect of numerical parameters on the accuracy of the proposed method has been investigated in detail. Results show that the rheological and thermal properties of non-Newtonian fluids in the presence of a heated immersed body can be suitably captured using the immersed boundary thermal lattice Boltzmann method. 18. Lubrication performances of short journal bearings operating with non-Newtonian ferrofluids Energy Technology Data Exchange (ETDEWEB) Lin, Jaw-Ren [Taoyuan Innovation Inst. of Tech., Jhongli, TW (China). Dept. of Mechanical Engineering; Li, Po-Jui [National Taipei Univ. of Technology, Taipei, TW (China). Dept. of Inst. of Mechatronic Engineering; Hung, Tzu-Chen [National Taipei Univ. of Technology, Taipei, TW (China). Dept. of Mechanical Engineering 2013-03-15 The lubrication performances of short journal bearings operating with non-Newtonian ferrofluids have been investigated in the present study. Based upon the ferrofluid model of Shliomis and the micro-continuum theory of Stokes, a two-dimensional modified Reynolds equation is derived by taking into account the effects of rotation of ferromagnetic particles and the effects of non-Newtonian properties. As an application, the short-bearing approximation is illustrated. Comparing with the conventional non-ferrofluid case, the short journal bearings with ferrofluids in the presence of magnetic fields result in a higher load capacity. Comparing with the Newtonian ferrofluid case, the non-Newtonian effects of couple stresses provide an enhancement in the load capacity, as well as a reduction in the friction parameter. The inclusion of non-Newtonian couple stresses signifies an improvement in performance characteristics of ferrofluid journal bearings. (orig.) 19. Certain Spaces of Functions over the Field of Non-Newtonian Complex Numbers Directory of Open Access Journals (Sweden) Ahmet Faruk Çakmak 2014-01-01 Full Text Available This paper is devoted to investigate some characteristic features of complex numbers and functions in terms of non-Newtonian calculus. Following Grossman and Katz, (Non-Newtonian Calculus, Lee Press, Piegon Cove, Massachusetts, 1972, we construct the field ℂ of -complex numbers and the concept of -metric. Also, we give the definitions and the basic important properties of -boundedness and -continuity. Later, we define the space C(Ω of -continuous functions and state that it forms a vector space with respect to the non-Newtonian addition and scalar multiplication and we prove that C(Ω is a Banach space. Finally, Multiplicative calculus (MC, which is one of the most popular non-Newtonian calculus and created by the famous exp function, is applied to complex numbers and functions to investigate some advance inner product properties and give inclusion relationship between C(Ω and the set of C′(Ω-differentiable functions. 20. Generalized Runge-Kutta Method with respect to the Non-Newtonian Calculus Directory of Open Access Journals (Sweden) 2015-01-01 Full Text Available Theory and applications of non-Newtonian calculus have been evolving rapidly over the recent years. As numerical methods have a wide range of applications in science and engineering, the idea of the design of such numerical methods based on non-Newtonian calculus is self-evident. In this paper, the well-known Runge-Kutta method for ordinary differential equations is developed in the frameworks of non-Newtonian calculus given in generalized form and then tested for different generating functions. The efficiency of the proposed non-Newtonian Euler and Runge-Kutta methods is exposed by examples, and the results are compared with the exact solutions. 1. Convective heat and mass transfer in a non-Newtonian-flow formation in Couette motion in magnetohydrodynamics with time-varing suction Directory of Open Access Journals (Sweden) Salama Faiza A. 2011-01-01 Full Text Available An analysis is carried out to study the effect of heat and mass transfer on a non-Newtonian-fluid between two infinite parallel walls, one of them moving with a uniform velocity under the action of a transverse magnetic field. The moving wall moves with constant velocity in the direction of fluid flow while the free stream velocity is assumed to follow the exponentially increasing small perturbation law. Time-dependent wall suction is assumed to occur at permeable surface. The governing equations for the flow are transformed into a system of nonlinear ordinary differential equations by perturbation technique and are solved numerically by using the shooting technique with fourth order Runge-Kutta integration scheme. The effect of non-Newtonian parameter, magnetic pressure parameter, Schmidt number, Grashof number and modified Grashof number on velocity, temperature, concentration and the induced magnetic field are discussed. Numerical results are given and illustrated graphically for the considered Problem. 2. Performance of a Polymer Flood with Shear-Thinning Fluid in Heterogeneous Layered Systems with Crossflow Directory of Open Access Journals (Sweden) Kun Sang Lee 2011-08-01 Full Text Available Assessment of the potential of a polymer flood for mobility control requires an accurate model on the viscosities of displacement fluids involved in the process. Because most polymers used in EOR exhibit shear-thinning behavior, the effective viscosity of a polymer solution is a highly nonlinear function of shear rate. A reservoir simulator including the model for the shear-rate dependence of viscosity was used to investigate shear-thinning effects of polymer solution on the performance of the layered reservoir in a five-spot pattern operating under polymer flood followed by waterflood. The model can be used as a quantitative tool to evaluate the comparative studies of different polymer flooding scenarios with respect to shear-rate dependence of fluids’ viscosities. Results of cumulative oil recovery and water-oil ratio are presented for parameters of shear-rate dependencies, permeability heterogeneity, and crossflow. The results of this work have proven the importance of taking non-Newtonian behavior of polymer solution into account for the successful evaluation of polymer flood processes. Horizontal and vertical permeabilities of each layer are shown to impact the predicted performance substantially. In reservoirs with a severe permeability contrast between horizontal layers, decrease in oil recovery and sudden increase in WOR are obtained by the low sweep efficiency and early water breakthrough through highly permeable layer, especially for shear-thinning fluids. An increase in the degree of crossflow resulting from sufficient vertical permeability is responsible for the enhanced sweep of the low permeability layers, which results in increased oil recovery. It was observed that a thinning fluid coefficient would increase injectivity significantly from simulations with various injection rates. A thorough understanding of polymer rheology in the reservoir and accurate numerical modeling are of fundamental importance for the exact estimation 3. A Generalization on Weighted Means and Convex Functions with respect to the Non-Newtonian Calculus OpenAIRE 2016-01-01 This paper is devoted to investigating some characteristic features of weighted means and convex functions in terms of the non-Newtonian calculus which is a self-contained system independent of any other system of calculus. It is shown that there are infinitely many such useful types of weighted means and convex functions depending on the choice of generating functions. Moreover, some relations between classical weighted mean and its non-Newtonian version are compared and discussed in a table... 4. Certain Spaces of Functions over the Field of Non-Newtonian Complex Numbers OpenAIRE Ahmet Faruk Çakmak; Feyzi Başar 2014-01-01 This paper is devoted to investigate some characteristic features of complex numbers and functions in terms of non-Newtonian calculus. Following Grossman and Katz, (Non-Newtonian Calculus, Lee Press, Piegon Cove, Massachusetts, 1972), we construct the field ${\\Bbb C}^{\\ast}$ of ${\\ast}$ -complex numbers and the concept of ${\\ast}$ -metric. Also, we give the definitions and the basic important properties of ${\\ast}$ -boundedness and ${\\ast}$ -continuity. Later, we define the space ${C}_{\\ast}(... 5. Determination of the Köthe-Toeplitz Duals over the Non-Newtonian Complex Field OpenAIRE Uğur Kadak 2014-01-01 The important point to note is that the non-Newtonian calculus is a self-contained system independent of any other system of calculus. Therefore the reader may be surprised to learn that there is a uniform relationship between the corresponding operators of this calculus and the classical calculus. Several basic concepts based on non-Newtonian calculus are presented by Grossman (1983), Grossman and Katz (1978), and Grossman (1979). Following Grossman and Katz, in the present paper, we introdu... 6. PREDICTION OF GAS HOLD-UP IN A COMBINED LOOP AIR LIFT FLUIDIZED BED REACTOR USING NEWTONIAN AND NON-NEWTONIAN LIQUIDS OpenAIRE 2011-01-01 Many experiments have been conducted to study the hydrodynamic characteristics of column reactors and loop reactors. In this present work, a novel combined loop airlift fluidized bed reactor was developed to study the effect of superficial gas and liquid velocities, particle diameter, fluid properties on gas holdup by using Newtonian and non-Newtonian liquids. Compressed air was used as gas phase. Water, 5% n-butanol, various concentrations of glycerol (60 and 80%) were used as Newtonian liqu... 7. Determination of the diffusion coefficient of salts in non-Newtonian liquids by the Taylor dispersion method Energy Technology Data Exchange (ETDEWEB) Mey, Paula; Varges, Priscilla R.; Mendes, Paulo R. de Souza [Dept. of Mechanical Engineering. Pontificia Universidade Catolica do RJ (PUC-Rio), RJ (Brazil)], e-mails: [email protected], [email protected] 2010-07-01 This research looked for a method to determine the binary diffusion coefficient D of salts in liquids (especially in drilling fluids) not only accurately, but in a reasonable time. We chose to use the Taylor Dispersion Method. This technique has been used for measuring binary diffusion coefficients in gaseous, liquid and supercritical fluids, due to its simplicity and accuracy. In the method, the diffusion coefficient is determined by the analysis of the dispersion of a pulse of soluble material in a solvent flowing laminarly through a tube. This work describes the theoretical basis and the experimental requirements for the application of the Taylor Dispersion Method, emphasizing the description of our experiment. A mathematical formulation for both Newtonian and non-Newtonian fluids is presented. The relevant sources of errors are discussed. The experimental procedure and associated analysis are validated by applying the method in well known systems, such as NaCl in water.D of salts in liquids (especially in drilling fluids) not only accurately, but in a reasonable time. We chose to use the Taylor Dispersion Method. This technique has been used for measuring binary diffusion coefficients in gaseous, liquid and supercritical fluids, due to its simplicity and accuracy. In the method, the diffusion coefficient is determined by the analysis of the dispersion of a pulse of soluble material in a solvent flowing laminarly through a tube. This work describes the theoretical basis and the experimental requirements for the application of the Taylor Dispersion Method, emphasizing the description of our experiment. A mathematical formulation for both Newtonian and non-Newtonian fluids is presented. The relevant sources of errors are discussed. The experimental procedure and associated analysis are validated by applying the method in well known systems, such as NaCl in water. (author) 8. Newtonian to non-Newtonian flow transition in lung surfactants Science.gov (United States) Sadoughi, Amir; Hirsa, Amir; Lopez, Juan 2010-11-01 The lining of normal lungs is covered by surfactants, because otherwise the surface tension of the aqueous layer would be too large to allow breathing. A lack of functioning surfactants can lead to respiratory distress syndrome, a potentially fatal condition in both premature infants and adults, and a major cause of death in the US and world-wide. We use a home-built Brewster angle microscope on an optically accessible deep channel viscometer to simultaneously observe the mesoscale structures of DPPC, the primary constituent of lung surfactant, on water surface and measure the interfacial velocity field. The measured interfacial velocity is compared to Navier-Stokes computations with the Boussinesq-Scriven surface model. Results show that DPPC monolayer behaves i) purely elastically at low surface pressures on water, ii) viscoelastically at modest surface pressures, exhibiting non-zero surface shear viscosity that is independent of the shear rate and flow inertia, and iii) at surface pressures approaching film collapse, DPPC loses its fluid characteristics, and a Newtonian surface model no longer captures its hydrodynamics. 9. Double-layer optical fiber coating analysis in MHD flow of an elastico-viscous fluid using wet-on-wet coating process Science.gov (United States) Khan, Zeeshan; Islam, Saeed; Shah, Rehan Ali; Khan, Muhammad Altaf; Bonyah, Ebenezer; Jan, Bilal; Khan, Aurangzeb Modern optical fibers require a double-layer coating on the glass fiber in order to provide protection from signal attenuation and mechanical damage. The most important plastic resins used in wires and optical fibers are plastic polyvinyl chloride (PVC) and low and high density polyethylene (LDPE/HDPE), nylon and Polysulfone. One of the most important things which affect the final product after processing is the design of the coating die. In the present study, double-layer optical fiber coating is performed using melt polymer satisfying Oldroyd 8-constant fluid model in a pressure type die with the effect of magneto-hydrodynamic (MHD). Wet-on-wet coating process is applied for double-layer optical fiber coating. The coating process in the coating die is modeled as a simple two-layer Couette flow of two immiscible fluids in an annulus with an assigned pressure gradient. Based on the assumptions of fully developed laminar and MHD flow, the Oldroyd 8-constant model of non-Newtonian fluid of two immiscible resin layers is modeled. The governing nonlinear equations are solved analytically by the new technique of Optimal Homotopy Asymptotic Method (OHAM). The convergence of the series solution is established. The results are also verified by the Adomian Decomposition Method (ADM). The effect of important parameters such as magnetic parameter Mi , the dilatant constant α , the Pseodoplastic constant β , the radii ratio δ , the pressure gradient Ω , the speed of fiber optics V , and the viscosity ratio κ on the velocity profiles, thickness of coated fiber optics, volume flow rate, and shear stress on the fiber optics are investigated. At the end the result of the present work is also compared with the experimental results already available in the literature by taking non-Newtonian parameters tends to zero. 10. Oxygenation to Bovine Blood in Artificial Heart and Lung Using Vibrating Flow Pump: Experiment and Numerical Analysis Based on Non-Newtonian Model Science.gov (United States) Shintaku, Hirofumi; Yonemura, Tsubasa; Tsuru, Kazuaki; Isoyama, Takashi; Yambe, Tomoyuki; Kawano, Satoyuki In this study, we construct an experimental apparatus for a prototype artificial heart and lung (AHL) by installing hollow fibers into the cylindrical tube of the vibrating flow pump (VFP). The oxygenation characteristics are investigated both by experiments using bovine blood and by numerical analyses based on the computational fluid dynamics. The analyses are carried out at the Reynolds numbers Re ranged from O(1) to O(103), which are determined based on the experimental conditions. The blood flow and the diffusion of oxygen gas are analyzed based on the Newtonian/non-Newtonian, unsteady, incompressible and axisymmetric Navier-Stokes equations, and the advection-diffusion equation. The results show that the oxygenation rate increases in proportion to Re1/3, where the phenomenon corresponds to the decreasing thickness of the concentration boundary layer with Re. Although the effects of the vibrating flow and the rheology of the blood are clearly appeared on the velocity field, their effects on the gas exchange are relatively small at the ranges of prescribed Reynolds numbers. Furthermore, the numerical results in terms of the oxygenation rate are compared with the experimental ones. The basic design data of VFP were accumulated for the development of AHL in the clinical applications. 11. Novel measurement of blood velocity profile using translating-stage optical method and theoretical modeling based on non-Newtonian viscosity model Science.gov (United States) Kim, Chang-Beom; Lim, Jaeho; Hong, Hyobong; Kresh, J. Yasha; Wootton, David M. 2015-07-01 Detailed knowledge of the blood velocity distribution over the cross-sectional area of a microvessel is important for several reasons: (1) Information about the flow field velocity gradients can suggest an adequate description of blood flow. (2) Transport of blood components is determined by the velocity profiles and the concentration of the cells over the cross-sectional area. (3) The velocity profile is required to investigate volume flow rate as well as wall shear rate and shear stress which are important parameters in describing the interaction between blood cells and the vessel wall. The present study shows the accurate measurement of non-Newtonian blood velocity profiles at different shear rates in a microchannel using a novel translating-stage optical method. Newtonian fluid velocity profile has been well known to be a parabola, but blood is a non-Newtonian fluid which has a plug flow region at the centerline due to yield shear stress and has different viscosities depending on shear rates. The experimental results were compared at the same flow conditions with the theoretical flow equations derived from Casson non-Newtonian viscosity model in a rectangular capillary tube. And accurate wall shear rate and shear stress were estimated for different flow rates based on these velocity profiles. Also the velocity profiles were modeled and compared with parabolic profiles, concluding that the wall shear rates were at least 1.46-3.94 times higher than parabolic distribution for the same volume flow rate. 12. Compliant model of a coupled sequential coronary arterial bypass graft: effects of vessel wall elasticity and non-Newtonian rheology on blood flow regime and hemodynamic parameters distribution. Science.gov (United States) Kabinejadian, Foad; Ghista, Dhanjoo N 2012-09-01 We have recently developed a novel design for coronary arterial bypass surgical grafting, consisting of coupled sequential side-to-side and end-to-side anastomoses. This design has been shown to have beneficial blood flow patterns and wall shear stress distributions which may improve the patency of the CABG, as compared to the conventional end-to-side anastomosis. In our preliminary computational simulation of blood flow of this coupled sequential anastomoses design, the graft and the artery were adopted to be rigid vessels and the blood was assumed to be a Newtonian fluid. Therefore, the present study has been carried out in order to (i) investigate the effects of wall compliance and non-Newtonian rheology on the local flow field and hemodynamic parameters distribution, and (ii) verify the advantages of the CABG coupled sequential anastomoses design over the conventional end-to-side configuration in a more realistic bio-mechanical condition. For this purpose, a two-way fluid-structure interaction analysis has been carried out. A finite volume method is applied to solve the three-dimensional, time-dependent, laminar flow of the incompressible, non-Newtonian fluid; the vessel wall is modeled as a linearly elastic, geometrically non-linear shell structure. In an iteratively coupled approach the transient shell equations and the governing fluid equations are solved numerically. The simulation results indicate a diameter variation ratio of up to 4% and 5% in the graft and the coronary artery, respectively. The velocity patterns and qualitative distribution of wall shear stress parameters in the distensible model do not change significantly compared to the rigid-wall model, despite quite large side-wall deformations in the anastomotic regions. However, less flow separation and reversed flow is observed in the distensible models. The wall compliance reduces the time-averaged wall shear stress up to 32% (on the heel of the conventional end-to-side model) and somewhat 13. Magnetohydrodynamic Mixed Convection Stagnation-Point Flow of a Power-Law Non-Newtonian Nanofluid towards a Stretching Surface with Radiation and Heat Source/Sink Directory of Open Access Journals (Sweden) Macha Madhu 2015-01-01 Full Text Available Two-dimensional MHD mixed convection boundary layer flow of heat and mass transfer stagnation-point flow of a non-Newtonian power-law nanofluid towards a stretching surface in the presence of thermal radiation and heat source/sink is investigated numerically. The non-Newtonian nanofluid model incorporates the effects of Brownian motion and thermophoresis. The basic transport equations are made dimensionless first and the complete nonlinear differential equations with associated boundary conditions are solved numerically by finite element method (FEM. The numerical calculations for velocity, temperature, and nanoparticles volume fraction profiles for different values of the physical parameters to display the interesting aspects of the solutions are presented graphically and discussed. The skin friction coefficient, the local Nusslet number and the Sherwood number are exhibited and examined. Our results are compatible with the existing results for a special case. 14. A modelling and experimental study of the bubble trajectory in a non-Newtonian crystal suspension Energy Technology Data Exchange (ETDEWEB) Hassan, N M S [Process Engineering and Light Metals (PELM) Centre, Faculty of Sciences, Engineering and Health, CQUniversity, Rockhampton, QLD 4702 (Australia); Khan, M M K; Rasul, M G, E-mail: [email protected] [School of Engineering and Built Environment, Faculty of Sciences, Engineering and Health, CQUniversity, Rockhampton, QLD 4702 (Australia) 2010-12-15 This paper presents an experimental and computational study of air bubbles rising in a massecuite-equivalent non-Newtonian crystal suspension. The bubble trajectory inside the stagnant liquid of a 0.05% xanthan gum crystal suspension was investigated and modelled using the computational fluid dynamics (CFD) model to gain an insight into the bubble flow characteristics. The CFD code FLUENT was used for numerical simulation, and the bubble trajectory calculations were performed through a volume of fluid (VOF) model. The influences of the Reynolds number (Re), the Weber number (We) and the bubble aspect ratio (E) on the bubble trajectory are discussed. The conditions for the bubbles' path oscillations are identified. The experimental results showed that the path instability for the crystal suspension was less rapid than in water. The trajectory analysis indicated that 5.76 mm diameter bubbles followed a zigzag motion in the crystal suspension. Conversely, the smaller bubbles (5.76 mm) followed a path of least horizontal movement and larger bubbles (21.21 mm) produced more spiral motion within the crystal suspension. Path instability occurred for bubbles of 15.63 and 21.21 mm diameter, and they induced both zigzag and spiral trajectories within the crystal suspension. At low Re and We, smaller bubbles (5.76 mm) produced a zigzag trajectory, whereas larger bubbles (15.63 and 21.21 mm) showed both zigzag and spiral trajectories at intermediate and moderately high Re and We in the crystal suspension. The simulation results illustrated that a repeating pattern of swirling vortices was created for smaller bubbles due to the unstable wake and unsteady flow of these bubbles. This is the cause of the smaller bubbles moving in a zigzag way. Larger bubbles showed two counter-rotating trailing vortices at the back of the bubble. These vortices induced a velocity component to the gas-liquid interface and caused a deformation. Hence, the larger bubbles produced a path 15. The Construction of Hilbert Spaces over the Non-Newtonian Field Directory of Open Access Journals (Sweden) Uğur Kadak 2014-01-01 the novel presentations probably lead most naturally to the development of the non-Newtonian calculi. In this paper we introduce vector spaces over real and complex non-Newtonian field with respect to the -calculus which is a branch of non-Newtonian calculus. Also we give the definitions of real and complex inner product spaces and study Hilbert spaces which are special type of normed space and complete inner product spaces in the sense of -calculus. Furthermore, as an example of Hilbert spaces, first we introduce the non-Cartesian plane which is a nonlinear model for plane Euclidean geometry. Secondly, we give Euclidean, unitary, and sequence spaces via corresponding norms which are induced by an inner product. Finally, by using the -norm properties of complex structures, we examine Cauchy-Schwarz and triangle inequalities. 16. Determination of the Köthe-Toeplitz Duals over the Non-Newtonian Complex Field Directory of Open Access Journals (Sweden) Uğur Kadak 2014-01-01 Full Text Available The important point to note is that the non-Newtonian calculus is a self-contained system independent of any other system of calculus. Therefore the reader may be surprised to learn that there is a uniform relationship between the corresponding operators of this calculus and the classical calculus. Several basic concepts based on non-Newtonian calculus are presented by Grossman (1983, Grossman and Katz (1978, and Grossman (1979. Following Grossman and Katz, in the present paper, we introduce the sets of bounded, convergent, null series and p-bounded variation of sequences over the complex field C and prove that these are complete. We propose a quite concrete approach based on the notion of Köthe-Toeplitz duals with respect to the non-Newtonian calculus. Finally, we derive some inclusion relationships between Köthe space and solidness. 17. Determination of the Köthe-Toeplitz duals over the non-Newtonian complex field. Science.gov (United States) Kadak, Uğur 2014-01-01 The important point to note is that the non-Newtonian calculus is a self-contained system independent of any other system of calculus. Therefore the reader may be surprised to learn that there is a uniform relationship between the corresponding operators of this calculus and the classical calculus. Several basic concepts based on non-Newtonian calculus are presented by Grossman (1983), Grossman and Katz (1978), and Grossman (1979). Following Grossman and Katz, in the present paper, we introduce the sets of bounded, convergent, null series and p-bounded variation of sequences over the complex field C* and prove that these are complete. We propose a quite concrete approach based on the notion of Köthe-Toeplitz duals with respect to the non-Newtonian calculus. Finally, we derive some inclusion relationships between Köthe space and solidness. 18. Stability analysis of slot-entry hybrid journal bearings operating with non-newtonian lubricant Directory of Open Access Journals (Sweden) H.C. Garg 2015-09-01 Full Text Available This paper presents theoretical investigations of rheological effects of lubricant on stability parameters of various configurations of slot-entry hybrid journal bearing system. FEM has been used to solve Reynolds equation governing flow of lubricant in bearing clearance space along with restrictor flow equation using suitable iterative technique. The non-Newtonian lubricant has been assumed to follow cubic shear stress law. The stability parameters in terms of stiffness coefficients, damping coefficients, threshold speed and whirl frequency of different configurations of slot-entry hybrid journal bearing have been computed and presented for wide range of external load while operating with Newtonian and Non-Newtonian lubricants. The computed results reveal that variation of viscosity due to non-Newtonian behavior of lubricant affects bearing stability quite significantly. The results are presented in graphical form and logical conclusions are drawn to identify best possible configuration from stability point of view. 19. Finite element analysis of heat and mass transfer by MHD mixed convection stagnation-point flow of a non-Newtonian power-law nanofluid towards a stretching surface with radiation Directory of Open Access Journals (Sweden) Macha Madhu 2016-07-01 Full Text Available Magnetohydrodynamic mixed convection boundary layer flow of heat and mass transfer stagnation-point flow of a non-Newtonian power-law nanofluid towards a stretching surface in the presence of thermal radiation is investigated numerically. The non-Newtonian nanofluid model incorporates the effects of Brownian motion and thermophoresis. The basic transport equations are made dimensionless first and the coupled non linear differential equations are solved by finite element method. The numerical calculations for velocity, temperature and concentration profiles for different values of the physical parameters presented graphically and discussed. As well as for skin friction coefficient, local Nusselt and Sherwood numbers exhibited and examined. 20. PREDICTION OF GAS HOLD-UP IN A COMBINED LOOP AIR LIFT FLUIDIZED BED REACTOR USING NEWTONIAN AND NON-NEWTONIAN LIQUIDS Directory of Open Access Journals (Sweden) Sivakumar Venkatachalam 2011-09-01 Full Text Available Many experiments have been conducted to study the hydrodynamic characteristics of column reactors and loop reactors. In this present work, a novel combined loop airlift fluidized bed reactor was developed to study the effect of superficial gas and liquid velocities, particle diameter, fluid properties on gas holdup by using Newtonian and non-Newtonian liquids. Compressed air was used as gas phase. Water, 5% n-butanol, various concentrations of glycerol (60 and 80% were used as Newtonian liquids, and different concentrations of carboxy methyl cellulose aqueous solutions (0.25, 0.6 and 1.0% were used as non-Newtonian liquids. Different sizes of spheres, Bearl saddles and Raschig rings were used as solid phases. From the experimental results, it was found that the increase in superficial gas velocity increases the gas holdup, but it decreases with increase in superficial liquid velocity and viscosity of liquids. Based on the experimental results a correlation was developed to predict the gas hold-up for Newtonian and non-Newtonian liquids for a wide range of operating conditions at a homogeneous flow regime where the superficial gas velocity is approximately less than 5 cm/s 1. A fractional model for time-variant non-Newtonian flow Directory of Open Access Journals (Sweden) Yang Xu 2017-01-01 Full Text Available This work applies a fractional flow model to describe a time-variant behavior of non-Newtonian substances. Specifically, we model the physical mechanism underlying the thixotropic and anti-thixotropic phenomena of non-Newtonian flow. This study investigates the behaviors of cellulose suspensions and SMS pastes under constant shear rate. The results imply that the presented model with only two parameters is adequate to fit experimental data. Moreover, the parameter of fractional order is an appropriate index to characterize the state of given substances. Its value indicates the extent of thixotropy and anti-thixotropy with positive and negative order respectively. 2. Entropy generation in non-Newtonian ﬂuid ﬂow in a slider bearing Indian Academy of Sciences (India) M Pakdemirli; B S Yilbas; M Yurusoy 2004-12-01 In the present study, entropy production in ﬂow ﬁelds due to slider bearings is formulated. The rate of entropy generation is computed for different ﬂuid properties and geometric conﬁgurations of the slider bearing. In order to account for the non-Newtonian effect, a special type of third-grade ﬂuid is considered. It is found that the rate of entropy generation is inﬂuenced signiﬁcantly by the height of the bearing clearance and the non-Newtonian parameter of the ﬂuid. 3. Non-Newtonian behavior and molecular structure of Cooee bitumen under shear flow: a non-equilibrium molecular dynamics study CERN Document Server Lemarchand, Claire A; Todd, Billy D; Daivis, Peter J; Hansen, Jesper S 2015-01-01 The rheology and molecular structure of a model bitumen (Cooee bitumen) under shear is investigated in the non-Newtonian regime using non-equilibrium molecular dynamics simulations. The shear viscosity and normal stress differences of the bitumen mixture are computed at different shear rates and different temperatures. The model bitumen is shown to be a shear-thinning fluid. The corresponding molecular structure is studied at the same shear rates and temperatures. The Cooee bitumen is able to reproduce experimental results showing the formation of nanoaggregates composed of stacks of flat aromatic molecules. These nanoaggregates are immersed in a solvent of saturated hydrocarbon molecules. The nanoaggregates are shown to break up at very high shear rates, leading only to a minor effect on the viscosity of the mixture. At low shear rates, bitumen can be seen as a colloidal suspension of nanoaggregates in a solvent. The slight anisotropy of the whole sample due to the nanoaggregates is considered and quantified... 4. Experimental investigation of heat transfer and pressure drop characteristics of non-Newtonian nanofluids flowing in the shell-side of a helical baffle heat exchanger with low-finned tubes Science.gov (United States) Tan, Yunkai; He, Zhenbin; Xu, Tao; Fang, Xiaoming; Gao, Xuenong; Zhang, Zhengguo 2017-09-01 An aqueous solution of Xanthan Gum (XG) at a weight fraction as high as 0.2% was used as the base liquid, the stable MWCNTs-dispersed non-Newtonian nanofluids at different weight factions of MWCNTs was prepared. The base fluid and all nanofluids show pseudoplastic (shear-thinning) rheological behavior. Experiments were performed to compare the shell-side forced convective heat transfer coefficient and pressure drop of non-Newtonian nanofluids to those of non-Newtonian base fluid in an integrally helical baffle heat exchanger with low-finned tubes. The experimental results showed that the enhancement of the convective heat transfer coefficient increases with an increase in the Peclet number and the nanoparticle concentration. For nanofluids with 1.0, 0.5 and 0.2 wt% of multi-walled carbon nanotubes (MWCNTs), the heat transfer coefficients respectively augmented by 24.3, 13.2 and 4.7% on average and the pressure drops become larger than those of the base fluid. The comprehensive thermal performance factor is higher than one and increases with an increasing weight fraction of MWCNTs. A remarkable heat transfer enhancement in the shell side of helical baffle heat exchanger with low-finned tubes can be obtained by adding MWCNTs into XG aqueous solution based on thermal resistance analysis. New correlations have been suggested for the shell-side friction coefficient and the Nusselt numbers of non-Newtonian nanofluids and give very good agreement with experimental data. 5. Simultaneous pulsatile flow and oscillating wall of a non-Newtonian liquid Science.gov (United States) Herrera-Valencia, E. E.; Sánchez-Villavicencio, M. L.; Calderas, F.; Pérez-Camacho, M.; Medina-Torres, L. 2016-11-01 In this work, analytical predictions of the rectilinear flow of a non-Newtonian liquid are given. The fluid is subjected to a combined flow: A pulsatile time-dependent pressure gradient and a random longitudinal vibration at the wall acting simultaneously. The fluctuating component of the combined pressure gradient and oscillating flow is assumed to be of small amplitude and can be adequately represented by a weakly stochastic process, for which a quasi-static perturbation solution scheme is suggested, in terms of a small parameter. This flow is analyzed with the Tanner constitutive equation model with the viscosity function represented by the Ellis model. According to the coupled Tanner-Ellis model, the flow enhancement can be separated in two contributions (pulsatile and oscillating mechanisms) and the power requirement is always positive and can be interpreted as the sum of a pulsatile, oscillating, and the coupled systems respectively. Both expressions depend on the amplitude of the oscillations, the perturbation parameter, the exponent of the Ellis model (associated to the shear thinning or thickening mechanisms), and the Reynolds and Deborah numbers. At small wall stress values, the flow enhancement is dominated by the axial wall oscillations whereas at high wall stress values, the system is governed by the pulsating noise perturbation. The flow transition is obtained for a critical shear stress which is a function of the Reynolds number, dimensionless frequency and the ratio of the two amplitudes associated with the pulsating and oscillating perturbations. In addition, the flow enhancement is compared with analytical and numerical predictions of the Reiner-Phillipoff and Carreau models. Finally, the flow enhancement and power requirement are predicted using biological rheometric data of blood with low cholesterol content. 6. The numerical analysis of non-Newtonian blood flow in human patient-specific left ventricle. Science.gov (United States) Doost, Siamak N; Zhong, Liang; Su, Boyang; Morsi, Yosry S 2016-04-01 Recently, various non-invasive tools such as the magnetic resonance image (MRI), ultrasound imaging (USI), computed tomography (CT), and the computational fluid dynamics (CFD) have been widely utilized to enhance our current understanding of the physiological parameters that affect the initiation and the progression of the cardiovascular diseases (CVDs) associated with heart failure (HF). In particular, the hemodynamics of left ventricle (LV) has attracted the attention of the researchers due to its significant role in the heart functionality. In this study, CFD owing its capability of predicting detailed flow field was adopted to model the blood flow in images-based patient-specific LV over cardiac cycle. In most published studies, the blood is modeled as Newtonian that is not entirely accurate as the blood viscosity varies with the shear rate in non-linear manner. In this paper, we studied the effect of Newtonian assumption on the degree of accuracy of intraventricular hemodynamics. In doing so, various non-Newtonian models and Newtonian model are used in the analysis of the intraventricular flow and the viscosity of the blood. Initially, we used the cardiac MRI images to reconstruct the time-resolved geometry of the patient-specific LV. After the unstructured mesh generation, the simulations were conducted in the CFD commercial solver FLUENT to analyze the intraventricular hemodynamic parameters. The findings indicate that the Newtonian assumption cannot adequately simulate the flow dynamic within the LV over the cardiac cycle, which can be attributed to the pulsatile and recirculation nature of the flow and the low blood shear rate. 7. The wall shear rate in non-Newtonian turbulent pipe flow CERN Document Server Trinh, K T 2010-01-01 This paper presents a method for calculating the wall shear rate in pipe turbulent flow. It collapses adequately the data measured in laminar flow and turbulent flow into a single flow curve and gives the basis for the design of turbulent flow viscometers. Key words: non-Newtonian, wall shear rate, turbulent, rheometer 8. Application of the homotopy method for analytical solution of non-Newtonian channel flows Energy Technology Data Exchange (ETDEWEB) Roohi, Ehsan [Department of Aerospace Engineering, Sharif University of Technology, PO Box 11365-8639, Azadi Avenue, Tehran (Iran, Islamic Republic of); Kharazmi, Shahab [Department of Mechanical Engineering, Sharif University of Technology, PO Box 11365-8639, Azadi Avenue, Tehran (Iran, Islamic Republic of); Farjami, Yaghoub [Department of Computer Engineering, University of Qom, Qom (Iran, Islamic Republic of)], E-mail: [email protected] 2009-06-15 This paper presents the homotopy series solution of the Navier-Stokes and energy equations for non-Newtonian flows. Three different problems, Couette flow, Poiseuille flow and Couette-Poiseuille flow have been investigated. For all three cases, the nonlinear momentum and energy equations have been solved using the homotopy method and analytical approximations for the velocity and the temperature distribution have been obtained. The current results agree well with those obtained by the homotopy perturbation method derived by Siddiqui et al (2008 Chaos Solitons Fractals 36 182-92). In addition to providing analytical solutions, this paper draws attention to interesting physical phenomena observed in non-Newtonian channel flows. For example, it is observed that the velocity profile of non-Newtonian Couette flow is indistinctive from the velocity profile of the Newtonian one. Additionally, we observe flow separation in non-Newtonian Couette-Poiseuille flow even though the pressure gradient is negative (favorable). We provide physical reasoning for these unique phenomena. 9. On line and double integrals in the non-Newtonian sense Science.gov (United States) ćakmak, Ahmet Faruk; Başar, Feyzi 2014-08-01 This paper is devoted to line and double integrals in the sense of non-Newtonian calculus (-calculus). Moreover, in the sense of -calculus, the fundamental theorem of calculus for line integrals and double integrals are stated and proved, and some applications are presented. 10. HARNACK ESTIMATES FOR WEAK SOLUTIONSOFEQUATIONS OF NON-NEWTONIAN POLYTROPIC FILTRATION Institute of Scientific and Technical Information of China (English) 2001-01-01 An intrinsic Harnack estimate and some sup-estimates are established for nonnegative weak solutions of equations of non-Newtonian polytropic filtration ut -div(｜Dum ｜p-2Dum) =0, m(p- 1) ＜ 1, m＞0, p＞ 1. 11. Entropy generation in a pipe due to non-Newtonian ﬂuid ﬂow: Constant viscosity case Indian Academy of Sciences (India) M Pakdemirli; B S Yilbas 2006-02-01 Non-Newtonian ﬂuid ﬂow in a pipe system is considered and a third grade non-Newtonian ﬂuid is employed in the analysis. The velocity and temperature distributions across the pipe are presented. Entropy generation number due to heat transfer and ﬂuid friction is formulated. The inﬂuences of non-Newtonian parameter and Brinkman number on entropy generation number are examined. It is found that increasing the non-Newtonian parameter reduces the ﬂuid friction in the region close to the pipe wall. This in turn results in low entropy generation with increasing non-Newtonian parameter. Increasing Brinkman number enhances the ﬂuid friction and heat transfer rates; in which case, entropy number increases with increasing Brinkman number. 12. Generalized solution for 1-D non-Newtonian flow in a porous domain due to an instantaneous mass injection Science.gov (United States) Di Federico, V.; Ciriello, V. 2011-12-01 Non-Newtonian fluid flow in porous media is of considerable interest in hydrology, chemical and petroleum engineering, and biofluid mechanics. We consider an infinite porous domain of plane (d=1), cylindrical (d=2) or semi-spherical geometry (d=3), having uniform permeability k and porosity Φ, initially at uniform pressure and saturated by a weakly compressible non-Newtonian fluid, and analyze the dynamics of the pressure variation generated within the domain by an instantaneous mass injection m0 in its origin. The fluid is described by a rheological power-law model of given consistency index H and flow behavior index n; the flow law is a modified Darcy's law depending on H, Φ, n. Coupling flow law and mass balance equations yields the nonlinear partial differential equation governing the pressure field; an analytical solution is derived in space r and time t as a function of a self-similar variable η=r/tβ(n). We revisit and expand the work in previous papers by providing a dimensionless general formulation and solution to the problem for d=1,2,3. When a shear-thinning fluid (nfluids; the front velocity is proportional to t(n-2)/2 in plane geometry, t(2n-3)/(3-n) in cylindrical geometry, and t(3n-4)/(4-2n) in semi-spherical geometry. The front position is a markedly increasing function of n and is inversely dependent on d; the pressure front advances at a slower rate for larger values of compressibility, higher injected mass and lower porosity. When pressure is considered, it is seen that an increase in d from 1 to 3 brings about an order of magnitude reduction. An increase in compressibility implies a significant decrease in pressure, especially at early times. To reflect the uncertainty inherent in values of the problem parameters, we then consider selected properties of fluid (flow behavior index n) and porous domain (permeability k, porosity Φ, and medium compressibility cp) as independent random variables with uniform probability distribution. The 13. Mass Transport in a Thin Layer of Bi-Viscous Mud Under Surface Waves Institute of Scientific and Technical Information of China (English) NG Chiu-on; FU Sau-chung; BAI Yu-chuan(白玉川) 2002-01-01 The mass transport in a thin layer of non-Newtonian bed mud under surface waves is examined with a two-fluidStokes boundary layer model. The mud is assumed to be a bi-viscous fluid, which tends to resist motion for small-appliedstresses, but flows readily when the yield stress is exceeded. Asymptotic expansions suitable for shallow fluid layers areapplied, and the second-order solutions for the mass transport induced by surface progressive waves are obtained numeri-cally. It is found that the stronger the non-Newtonian behavior of the mud, the more pronounced intermittency of theflow. Consequently, the mass transport velocity is diminished in magnitude, and can even become negative (i. e., oppo-site to wave propagation) for a certain range of yield stress. 14. Blasius flow and heat transfer of fourth-grade fluid with slip Institute of Scientific and Technical Information of China (English) B SAHOO; S PONCET 2013-01-01 This investigation deals with the effects of slip, magnetic field, and non-Newtonian flow parameters on the flow and heat transfer of an incompressible, electrically conducting fourth-grade fluid past an infinite porous plate. The heat transfer analysis is carried out for two heating processes. The system of highly non-linear differential equations is solved by the shooting method with the fourth-order Runge-Kutta method for moderate values of the parameters. The effective Broyden technique is adopted in order to improve the initial guesses and to satisfy the boundary conditions at infinity. An exceptional cross-over is obtained in the velocity profile in the presence of slip. The fourth-grade fluid parameter is found to increase the momentum boundary layer thickness, whereas the slip parameter substantially decreases it. Similarly, the non-Newtonian fluid parameters and the slip have opposite effects on the thermal boundary layer thickness. 15. The effect of the expansion ratio on a turbulent non-Newtonian recirculating flow Energy Technology Data Exchange (ETDEWEB) Pereira, A.S. [Departamento de Engenharia Quimica Instituto Superior de Engenharia do Porto (Portugal); Pinho, F.T. [Centro de Estudos de Fenomenos de Transporte, DEMEGI, Faculdade de Engenharia, Universidade do Porto (Portugal) 2002-04-01 Measurements of the mean and turbulent flow characteristics of shear-thinning moderately elastic 0.1% and 0.2% xanthan gum aqueous solutions were carried out in a sudden expansion having a diameter ratio of 2. The inlet flow was turbulent and fully developed, and the results were compared with data for water in the same geometry and with previous published Newtonian and non-Newtonian data in a smaller expansion of diameter ratio equal to 1.538. An increase in expansion ratio led to an increase in the recirculation length and in the axial normal Reynolds stress at identical normalised locations, but the difference between Newtonian and non-Newtonian characteristics was less intense than in the smaller expansion. An extensive comparison of mean and turbulent flow characteristics was carried out in order to understand the variation of flow features. (orig.) 16. Non-Newtonian ﬂuid ﬂow in annular pipes and entropy generation: Temperature-dependent viscosity Indian Academy of Sciences (India) M Yürüsoy; B S Yilbaş; M Pakdemirli 2006-12-01 Non-Newtonian ﬂuid ﬂow in annular pipes is considered and the entropy generation due to ﬂuid friction and heat transfer in them is formulated. A third-grade ﬂuid is employed to account for the non-Newtonian effect, while the Reynolds model is accommodated for temperature-dependent viscosity. Closed-form solutions for velocity, temperature, and entropy ﬁelds are presented. It is found that entropy generation number increases with reducing non-Newtonian parameter, while it is the reverse for the viscosity parameter, which is more pronounced in the region close to the annular pipe inner wall. 17. Matrix Transformations between Certain Sequence Spaces over the Non-Newtonian Complex Field OpenAIRE Uğur Kadak; Hakan Efe 2014-01-01 In some cases, the most general linear operator between two sequence spaces is given by an infinite matrix. So the theory of matrix transformations has always been of great interest in the study of sequence spaces. In the present paper, we introduce the matrix transformations in sequence spaces over the field C* and characterize some classes of infinite matrices with respect to the non-Newtonian calculus. Also we give the necessary and sufficient conditions on an infinite matrix ... 18. BLOW-UP ESTIMATES FOR A NON-NEWTONIAN FILTRATION SYSTEM Institute of Scientific and Technical Information of China (English) 杨作东; 陆启韶 2001-01-01 The prior estimate and decay property of positive solutions are derived for a system of quasi-linear elliptic differential equations first. Hence, the result of non-existence for differential equation system of radially nonincreasing positive solutions is implied. By using this non-existence result, blow-up estimates for a class quasi-linear reaction-diffusion systems (non-Newtonian filtration systems ) are established, which extends the result of semi- linear reaction- diffusion ( Fujita type ) systems . 19. Travelling-Wave Solutions and Interfaces for Non-Newtonian Diffusion Equations with Strong Absorption Institute of Scientific and Technical Information of China (English) Zhongping LI; Wanjuan DU; Chunlai MU 2013-01-01 In this paper,we first find finite travelling-wave solutions,and then investigate the short time development of interfaces for non-Newtonian diffusion equations with strong absorption.We show that the initial behavior of the interface depends on the concentration of the mass of u(x,0) near x =0.More precisely,we find a critical value of the concentration,which separates the heating front of interfaces from the cooling front of them. 20. Mixed convection in fluid superposed porous layers CERN Document Server Dixon, John M 2017-01-01 This Brief describes and analyzes flow and heat transport over a liquid-saturated porous bed. The porous bed is saturated by a liquid layer and heating takes place from a section of the bottom. The effect on flow patterns of heating from the bottom is shown by calculation, and when the heating is sufficiently strong, the flow is affected through the porous and upper liquid layers. Measurements of the heat transfer rate from the heated section confirm calculations. General heat transfer laws are developed for varying porous bed depths for applications to process industry needs, environmental sciences, and materials processing. Addressing a topic of considerable interest to the research community, the brief features an up-to-date literature review of mixed convection energy transport in fluid superposed porous layers. 1. Non-Newtonian versus numerical rheology: Practical impact of shear-thinning on the prediction of stable and unstable flows in intracranial aneurysms. Science.gov (United States) Khan, M O; Steinman, D A; Valen-Sendstad, K 2017-07-01 Computational fluid dynamics (CFD) shows promise for informing treatment planning and rupture risk assessment for intracranial aneurysms. Much attention has been paid to the impact on predicted hemodynamics of various modelling assumptions and uncertainties, including the need for modelling the non-Newtonian, shear-thinning rheology of blood, with equivocal results. Our study clarifies this issue by contextualizing the impact of rheology model against the recently demonstrated impact of CFD solution strategy on the prediction of aneurysm flow instabilities. Three aneurysm cases were considered, spanning a range of stable to unstable flows. Simulations were performed using a high-resolution/accuracy solution strategy with Newtonian and modified-Cross rheology models and compared against results from a so-called normal-resolution strategy. Time-averaged and instantaneous wall shear stress (WSS) distributions, as well as frequency content of flow instabilities and dome-averaged WSS metrics, were minimally affected by the rheology model, whereas numerical solution strategy had a demonstrably more marked impact when the rheology model was fixed. We show that point-wise normalization of non-Newtonian by Newtonian WSS values tended to artificially amplify small differences in WSS of questionable physiological relevance in already-low WSS regions, which might help to explain the disparity of opinions in the aneurysm CFD literature regarding the impact of non-Newtonian rheology. Toward the goal of more patient-specific aneurysm CFD, we conclude that attention seems better spent on solution strategy and other likely "first-order" effects (eg, lumen segmentation and choice of flow rates), as opposed to "second-order" effects such as rheology. Copyright © 2016 John Wiley & Sons, Ltd. 2. A fast algorithm for non-Newtonian flow. An enhanced particle-tracking finite element code for solving boundary-valve problems in viscoelastic flow Science.gov (United States) Malkus, David S. 1989-01-01 This project concerned the development of a new fast finite element algorithm to solve flow problems of non-Newtonian fluids such as solutions or melts of polymers. Many constitutive theories for such materials involve single integrals over the deformation history of the particle at the stress evaluation point; examples are the Doi-Edwards and Curtiss-Bird molecular theories and the BKZ family derived from continuum arguments. These theories are believed to be among the most accurate in describing non-Newtonian effects important to polymer process design, effects such as stress relaxation, shear thinning, and normal stress effects. This research developed an optimized version of the algorithm which would run a factor of two faster than the pilot algorithm on scalar machines and would be able to take full advantage of vectorization on machines. Significant progress was made in code vectorization; code enhancement and streamlining; adaptive memory quadrature; model problems for the High Weissenberg Number Problem; exactly incompressible projection; development of multimesh extrapolation procedures; and solution of problems of physical interest. A portable version of the code is in the final stages of benchmarking and testing. It interfaces with the widely used FIDAP fluid dynamics package. 3. Uniqueness of self-similar very singular solution for non-Newtonian polytropic filtration equations with gradient absorption Directory of Open Access Journals (Sweden) Hailong Ye 2015-04-01 Full Text Available Uniqueness of self-similar very singular solutions with compact support are proved for the non-Newtonian polytropic filtration equation with gradient absorption$\\frac{\\partial u}{\\partial t} =\\hbox{div}(\|\ 4. Deposition Velocities of Newtonian and Non-Newtonian Slurries in Pipelines Energy Technology Data Exchange (ETDEWEB) Poloski, Adam P. [Pacific Northwest National Lab. (PNNL), Richland, WA (United States); Adkins, Harold E. [Pacific Northwest National Lab. (PNNL), Richland, WA (United States); Abrefah, John [Pacific Northwest National Lab. (PNNL), Richland, WA (United States); Casella, Andrew M. [Pacific Northwest National Lab. (PNNL), Richland, WA (United States); Hohimer, Ryan E. [Pacific Northwest National Lab. (PNNL), Richland, WA (United States); Nigl, Franz [Pacific Northwest National Lab. (PNNL), Richland, WA (United States); Minette, Michael J. [Pacific Northwest National Lab. (PNNL), Richland, WA (United States); Toth, James J. [Pacific Northwest National Lab. (PNNL), Richland, WA (United States); Tingey, Joel M. [Pacific Northwest National Lab. (PNNL), Richland, WA (United States); Yokuda, Satoru T. [Pacific Northwest National Lab. (PNNL), Richland, WA (United States) 2009-03-01 correlation used in the WTP design guide has been shown to be inaccurate for Hanford waste feed materials. The use of the Thomas (1979) correlation in the design guide is not conservative—In cases where 100% of the particles are smaller than 74 μm or particles are considered to be homogeneous due to yield stress forces suspending the particles the homogeneous fraction of the slurry can be set to 100%. In such cases, the predicted critical velocity based on the conservative Oroskar and Turian (1980) correlation is reduced to zero and the design guide returns a value from the Thomas (1979) correlation. The measured data in this report show that the Thomas (1979) correlation predictions often fall below that measured experimental values. A non-Newtonian deposition velocity design guide should be developed for the WTP— Since the WTP design guide is limited to Newtonian fluids and the WTP expects to process large quantities of such materials, the existing design guide should be modified address such systems. A central experimental finding of this testing is that the flow velocity required to reach turbulent flow increases with slurry rheological properties due to viscous forces dampening the formation of turbulent eddies. The flow becomes dominated by viscous forces rather than turbulent eddies. Since the turbulent eddies necessary for particle transport are not present, the particles will settle when crossing this boundary called the transitional deposition boundary. This deposition mechanism should be expected and designed for in the WTP. 5. Non-Newtonian behavior and molecular structure of Cooee bitumen under shear flow: A non-equilibrium molecular dynamics study Science.gov (United States) Lemarchand, Claire A.; Bailey, Nicholas P.; Todd, Billy D.; Daivis, Peter J.; Hansen, Jesper S. 2015-06-01 The rheology and molecular structure of a model bitumen (Cooee bitumen) under shear are investigated in the non-Newtonian regime using non-equilibrium molecular dynamics simulations. The shear viscosity, normal stress differences, and pressure of the bitumen mixture are computed at different shear rates and different temperatures. The model bitumen is shown to be a shear-thinning fluid at all temperatures. In addition, the Cooee model is able to reproduce experimental results showing the formation of nanoaggregates composed of stacks of flat aromatic molecules in bitumen. These nanoaggregates are immersed in a solvent of saturated hydrocarbon molecules. At a fixed temperature, the shear-shinning behavior is related not only to the inter- and intramolecular alignments of the solvent molecules but also to the decrease of the average size of the nanoaggregates at high shear rates. The variation of the viscosity with temperature at different shear rates is also related to the size and relative composition of the nanoaggregates. The slight anisotropy of the whole sample due to the nanoaggregates is considered and quantified. Finally, the position of bitumen mixtures in the broad literature of complex systems such as colloidal suspensions, polymer solutions, and associating polymer networks is discussed. 6. Flow of polymer fluids through porous media OpenAIRE Zami-Pierre, Frédéric; Davit, Yohan; Loubens, Romain de; Quintard, Michel 2016-01-01 Non-Newtonian fluids are extensively used in enhanced oil recovery. However, understanding the flow of such fluids in complex porous media remains a challenging problem. In the presented study, we use computational fluid dynamics to investigate the creeping flow of a particular non-Newtonian fluid through porous media, namely a power-law fluid with a newtonian behavior below a critical shear rate. We show that the nonlinear effects induced by the rheology only weakly impact the topological st... 7. Flow pattern-based mass and heat transfer and frictional drag of gas-non-Newtonian liquid flow in helical coil: two- and three-phase systems Science.gov (United States) Thandlam, Anil Kumar; Das, Chiranjib; Majumder, Subrata Kumar 2016-08-01 Investigation of wall-liquid mass transfer and heat transfer phenomena with gas-Newtonian and non-Newtonian fluids in vertically helical coil reactor have been reported in this article. Experiments were conducted to investigate the effect of various dynamic and geometric parameters on mass and heat transfer coefficients in the helical coil reactor. The flow pattern-based heat and mass transfer phenomena in the helical coil reactor are highlighted at different operating conditions. The study covered a wide range of geometric parameters such as diameter of the tube (d t ), diameter of the coil (D c ), diameter of the particle (d p ), pitch difference (p/D c ) and concentrations of non-Newtonian liquid. The correlation models for the heat and mass transfer coefficient based on the flow pattern are developed which may be useful in process scale-up of the helical coil reactor for industrial application. The frictional drag coefficient was also estimated and analyzed by mass transfer phenomena based on the electrochemical method. 8. Non-Newtonian Couette-Poiseuille flow of a dilute gas OpenAIRE Tij, Mohamed; Santos, Andrés 2010-01-01 The steady state of a dilute gas enclosed between two infinite parallel plates in relative motion and under the action of a uniform body force parallel to the plates is considered. The Bhatnagar-Gross-Krook model kinetic equation is analytically solved for this Couette-Poiseuille flow to first order in the force and for arbitrary values of the Knudsen number associated with the shear rate. This allows us to investigate the influence of the external force on the non-Newtonian properties of the... 9. Matrix transformations between certain sequence spaces over the non-Newtonian complex field. Science.gov (United States) Kadak, Uğur; Efe, Hakan 2014-01-01 In some cases, the most general linear operator between two sequence spaces is given by an infinite matrix. So the theory of matrix transformations has always been of great interest in the study of sequence spaces. In the present paper, we introduce the matrix transformations in sequence spaces over the field ℂ() and characterize some classes of infinite matrices with respect to the non-Newtonian calculus. Also we give the necessary and sufficient conditions on an infinite matrix transforming one of the classical sets over ℂ() to another one. Furthermore, the concept for sequence-to-sequence and series-to-series methods of summability is given with some illustrated examples. 10. An overview on the non-newtonian calculus and its potential applications to economics OpenAIRE Filip, Diana; Piatecki, Cyrille 2014-01-01 20; Until now, non-newtonian calculus, multiplicative calculus in particular, has been presented as a curiosity and is nearly ignored for the social scientists field. In this paper, after a brief presentation of this calculus, we try to show how it could be used to re-explore from another perspective classical economic theory, more particularly the economic growth and in the maximum likelihood method from statistics.; Jusqu'à présent, le calcul non-newtonien, calcul multiplicatif en particuli... 11. Communication: Non-Newtonian rheology of inorganic glass-forming liquids: Universal patterns and outstanding questions Science.gov (United States) Zhu, W.; Aitken, B. G.; Sen, S. 2017-02-01 All families of inorganic glass-forming liquids display non-Newtonian rheological behavior in the form of shear thinning at high shear rates. Experimental evidence is presented to demonstrate the existence of remarkable universality in this behavior, irrespective of chemical composition, structure, topology, and viscosity. However, contrary to intuition, in all cases the characteristic shear rates that mark the onset of shear thinning in these liquids are orders of magnitude slower than the global shear relaxation rates. Attempt is made to reconcile such differences within the framework of the cooperative structural relaxation model of glass-forming liquids. 12. LOADING CAPACITY ANALYSIS OF THE MISALIGNED CONICAL-CYLINDRICAL BEARING WITH NON-NEWTONIAN LUBRICANTS Institute of Scientific and Technical Information of China (English) Yang Yongkuang; Yang Rongtai; Ho Minghsiung; Jheng Mingchang 2004-01-01 A novel numerical method to lubricate a conventional finite diameter conical(cylindrical bearing with a non-Newtonian lubricant, while adhering to the power-law model, is presented. The elastic deformation of bearing and varied viscosity of lubrication due to the pressure distribution of film thickness are also considered. Simulation results indicate that the normal load carrying capacity is more pronounced for higher values of flow behavior index n, higher eccentricity ratios and larger misalignment factors. It is found that the viscosity-pressure to the effect of lubricant viscosity is significant. 13. Nonlinear shear wave in a non Newtonian visco-elastic medium Energy Technology Data Exchange (ETDEWEB) Banerjee, D.; Janaki, M. S.; Chakrabarti, N. [Saha Institute of Nuclear Physics, 1/AF Bidhannagar, Calcutta 700 064 (India); Chaudhuri, M. [Max-Planck-Institut fuer extraterrestrische Physik, 85741 Garching (Germany) 2012-06-15 An analysis of nonlinear transverse shear wave has been carried out on non-Newtonian viscoelastic liquid using generalized hydrodynamic model. The nonlinear viscoelastic behavior is introduced through velocity shear dependence of viscosity coefficient by well known Carreau-Bird model. The dynamical feature of this shear wave leads to the celebrated Fermi-Pasta-Ulam problem. Numerical solution has been obtained which shows that initial periodic solutions reoccur after passing through several patterns of periodic waves. A possible explanation for this periodic solution is given by constructing modified Korteweg de Vries equation. This model has application from laboratory to astrophysical plasmas as well as in biological systems. 14. Nonlinear Shear Wave in a Non Newtonian Visco-elastic Medium CERN Document Server Janaki, D Banerjee M S; Chaudhuri, M 2013-01-01 An analysis of nonlinear transverse shear wave has been carried out on non-Newtonian viscoelastic liquid using generalized hydrodynamic(GH) model. The nonlinear viscoelastic behavior is introduced through velocity shear dependence of viscosity coefficient by well known Carreau -Bird model. The dynamical feature of this shear wave leads to the celebrated Fermi-Pasta-Ulam (FPU) problem. Numerical solution has been obtained which shows that initial periodic solutions reoccur after passing through several patterns of periodic waves. A possible explanation for this periodic solution is given by constructing modified Korteweg de Vries (mKdV) equation. This model has application from laboratory to astrophysical plasmas as well as biological systems. 15. OPTIMAL MIXED H- P FINITE ELEMENT METHODS FOR STOKES AND NON-NEWTONIAN FLOW Institute of Scientific and Technical Information of China (English) Ping-bing Ming; Zhong-ci Shi 2001-01-01 Based upon a new mixed variational formulation for the three-field Stokes equations and linearized Non-Newtonian flow, an h -p finite element method is presented with or without a stabilization. As to the variational formulation without stabilization, optimal error bounds in h as well as in p are obtained. As with stabilization, optimal error bounds are obtained which is optimal in h and one order deterioration in p for the pressure, that is consistent with numerical results in [9, 12] and therefore solved the problem therein.Moreover, we proposed a stabilized formulation which is optimal in both h and p. 16. Non-newtonian flow and pressure drop of pineapple juice in a plate heat exchanger OpenAIRE CABRAL, R. A. F.; GUT, J. A. W.; V. R. N. Telis; Telis-Romero, J. [UNESP 2010-01-01 The study of non-Newtonian flow in plate heat exchangers (PHEs) is of great importance for the food industry. The objective of this work was to study the pressure drop of pineapple juice in a PHE with 50º chevron plates. Density and flow properties of pineapple juice were determined and correlated with temperature (17.4 < T < 85.8ºC) and soluble solids content (11.0 < Xs < 52.4 ºBrix). The Ostwald-de Waele (power law) model described well the rheological behavior. The friction factor for non-... 17. Non-Darcy Free Convection of Power-Law Fluids Over a Two-Dimensional Body Embedded in a Porous Medium KAUST Repository El-Amin, Mohamed 2010-11-27 A boundary layer analysis was presented to study the non-Darcy-free convection of a power-law fluid over a non-isothermal two-dimensional body embedded in a porous medium. The Ostwald-de Waele power-law model was used to characterize the non-Newtonian fluid behavior. Similarity solutions were obtained with variations in surface temperature or surface heat flux. In view of the fact that most of the non-Newtonian fluids have large Prandtl numbers, this study was directed toward such fluids. The effects of the porous medium parameters, k1 and k2, body shape parameter, m, and surface thermal variations parameter, p, as well as the power-law index, n, were examined. © 2010 Springer Science+Business Media B.V. 18. Numerical Well Test Analysis for Polymer Flooding considering the Non-Newtonian Behavior Directory of Open Access Journals (Sweden) Jia Zhichun 2015-01-01 Full Text Available Well test analysis for polymer flooding is different from traditional well test analysis because of the non-Newtonian properties of underground flow and other mechanisms involved in polymer flooding. Few of the present works have proposed a numerical approach of pressure transient analysis which fully considers the non-Newtonian effect of real polymer solution and interprets the polymer rheology from details of pressure transient response. In this study, a two-phase four-component fully implicit numerical model incorporating shear thinning effect for polymer flooding based on PEBI (Perpendicular Bisection grid is developed to study transient pressure responses in polymer flooding reservoirs. Parametric studies are conducted to quantify the effect of shear thinning and polymer concentration on the pressure transient response. Results show that shear thinning effect leads to obvious and characteristic nonsmoothness on pressure derivative curves, and the oscillation amplitude of the shear-thinning-induced nonsmoothness is related to the viscosity change decided by shear thinning effect and polymer concentration. Practical applications are carried out with shut-in data obtained in Daqing oil field, which validates our findings. The proposed method and the findings in this paper show significant importance for well test analysis for polymer flooding and the determination of the polymer in situ rheology. 19. Gas holdup in a reciprocating plate bioreactor: Non-Newtonian - liquid phase Directory of Open Access Journals (Sweden) Naseva Olivera S. 2002-01-01 Full Text Available The gas holdup was studied in non-newtonian liquids in a gas-liquid and gas-liquid-solid reciprocating plate bioreactor. Aqueous solutions of carboxy methyl cellulose (CMC; Lucel, Lučane, Yugoslavia of different degrees of polymerization (PP 200 and PP 1000 and concentration (0,5 and 1%, polypropylene spheres (diameter 8.3 mm; fraction of spheres: 3.8 and 6.6% by volume and air were used as the liquid, solid and gas phase. The gas holdup was found to be dependent on the vibration rate, the superficial gas velocity, volume fraction of solid particles and Theological properties of the liquid ohase. Both in the gas-liquid and gas-liquid-solid systems studied, the gas holdup increased with increasing vibration rate and gas flow rate. The gas holdup was higher in three-phase systems than in two-phase ones under otter operating conditions being the same. Generally the gas holdup increased with increasing the volume fraction of solid particles, due to the dispersion action of the solid particles, and decreased with increasing non-Newtonian behaviour (decreasing flow index i.e. with increasing degree of polymerization and solution concentration of CMC applied, as a result of gas bubble coalescence. 20. Non-Newtonian steady shear flow characteristics of waxy crude oil Institute of Scientific and Technical Information of China (English) 黄树新; 陈鑫; 鲁传敬; 侯磊; 范毓润 2008-01-01 The experimental research on the non-Newtonian flow characteristic of a waxy crude oil was conducted through a rotational parallel-plates rheometer system.The test temperature is about 6.5 ℃ higher than its gel point.The shear stress and viscosity of the waxy crude oil show sophisticate non-Newtonian characteristics in the shear rate of 10-4-102 s-1,in which the shear stress can be divided into three parts qualitatively,i.e.stress-up region,leveling-off region,and stress-up region.This indicates that there is a yielding process in shearing for the waxy crude oil at the experimental temperature,which is similar to the yield phenomenon in thixotropy-loop test discussed by CHANG and BOGER.Furthermore,the steady shear experiment after the pre-shear process shows that the stress leveling-off region at low shear rate disappears for the waxy crude oil and the stress curve becomes a monotonic climbing one,which demonstrates that the internal structure property presenting through yielding stress at low shear rate can be changed by shearing.The experimental results also show that the internal structure of waxy crude oil presenting at low shear rate has no influence on the shear viscosity obtained at the shear rate higher than 0.1 s-1.The generalized Newtonian model is adopted to describe the shear-thinning viscosity property of the waxy crude oil at high shear rate. 1. The Future of Aerospace Propulsion: Visco-elastic non-Newtonian liquids Directory of Open Access Journals (Sweden) Nicole Arockiam 2011-01-01 Full Text Available Aerospace propulsion often involves the spray and combustion of liquids. When a liquid is sprayed, large drops form first, in a process known as primary atomization. Then, each drop breaks up into smaller droplets, in a process known as secondary atomization. This determines final drop sizes, which affect the liquid’s evaporation and mixing rates and ultimately influence combustor efficiency. Little has been published concerning the secondary atomization of visco-elastic non-Newtonian liquids, such as gels. These substances have special potential as aerospace propellants, because they are safer to handle than their Newtonian liquid counterparts, such as water. Additionally, they can be injected at varying rates, allowing for more control than solid propellants. To learn more about the atomization process of these liquids, a liquid drop generator and a high-speed camera were used to create and measure the conditions at which different breakup modes occurred, as well as the time required for the process. These results were compared to experimental and theoretical results for Newtonian liquids. Based on the data, one can conclude that solutions that are more elastic require higher shear forces to break up. In addition, while Newtonian liquids form droplets as they atomize, visco-elastic non-Newtonian solutions form ligaments. As a result, a combustion system utilizing these types of propellants must be capable of generating these forces. It may also be necessary to find a way to transform the ligaments into more spherically-shaped droplets to increase combustion efficiency. 2. Non-Newtonian flow of an ultralow-melting chalcogenide liquid in strongly confined geometry Energy Technology Data Exchange (ETDEWEB) Wang, Siyuan; Jain, Chhavi; Wondraczek, Katrin; Kobelke, Jens [Leibniz Institute of Photonic Technology, Albert-Einstein-Str. 9, 07745 Jena (Germany); Wondraczek, Lothar [Otto Schott Institute of Material Research (OSIM), Friedrich Schiller University Jena, Fraunhoferstr. 6, 07743 Jena (Germany); Troles, Johann; Caillaud, Celine [Université de Rennes I, Equipe Verres et Céramiques, UMR 6226 Sciences Chimiques de Rennes, Campus de Beaulieu, 35042 Rennes (France); Schmidt, Markus A., E-mail: [email protected] [Leibniz Institute of Photonic Technology, Albert-Einstein-Str. 9, 07745 Jena (Germany); Otto Schott Institute of Material Research (OSIM), Friedrich Schiller University Jena, Fraunhoferstr. 6, 07743 Jena (Germany) 2015-05-18 The flow of high-viscosity liquids inside micrometer-size holes can be substantially different from the flow in the bulk, non-confined state of the same liquid. Such non-Newtonian behavior can be employed to generate structural anisotropy in the frozen-in liquid, i.e., in the glassy state. Here, we report on the observation of non-Newtonian flow of an ultralow melting chalcogenide glass inside a silica microcapillary, leading to a strong deviation of the shear viscosity from its value in the bulk material. In particular, we experimentally show that the viscosity is radius-dependent, which is a clear indication that the microscopic rearrangement of the glass network needs to be considered if the lateral confinement falls below a certain limit. The experiments have been conducted using pressure-assisted melt filling, which provides access to the rheological properties of high-viscosity melt flow under previously inaccessible experimental conditions. The resulting flow-induced structural anisotropy can pave the way towards integration of anisotropic glasses inside hybrid photonic waveguides. 3. Experiments on densely-loaded non-Newtonian slurries in laminar and turbulent pipe flows Science.gov (United States) Park, Joel T.; Mannheimer, Richard J.; Grimley, Terrence A.; Morrow, Thomas B. 1989-06-01 An experimental description of the flow structure of non-Newtonian slurries in the laminar, transitional, and fully-developed turbulent pipe flow regimes was the primary objective of this research. Experiments were conducted in a large-scale pipe slurry flow facility with an inside diameter of 51 mm (2 inches). Approximately, 550 liters (145 gal) of slurry were necessary in the operation of the loop. Detailed velocity profile measurements by a two-color, two-component laser Doppler anemometer (LDA) were accomplished in a transparent test section with an optically transparent slurry. These velocity measurements were apparently the first ever reported for a non-Newtonian slurry with a yield value. The transparent slurry was formulated for these experiments from silica with a particle size of one to two microns, mineral oil, and Stoddard solvent. From linear regression analysis of concentric-cylinder viscometer data, the slurry exhibited yield-power-law behavior with a yield stress of 100 dynes/cm(sup 2), and an exponent of 0.630 for a solids concentration of 5.65 percent by weight. Good agreement was attained with rheological data derived from the pressure drop data in the flow loop under laminar flow conditions. The rheological properties of the transparent slurry were similar to many industrial slurries, including coal slurries, which have a yield value. 4. Casson fluid flow and heat transfer past an exponentially porous stretching surface in presence of thermal radiation Directory of Open Access Journals (Sweden) S. Pramanik 2014-03-01 Full Text Available The present paper aims at investigating the boundary layer flow of a non-Newtonian fluid accompanied by heat transfer toward an exponentially stretching surface in presence of suction or blowing at the surface. Casson fluid model is used to characterize the non-Newtonian fluid behavior. Thermal radiation term is incorporated into the equation for the temperature field. With the help of similarity transformations, the governing partial differential equations corresponding to the momentum and heat transfer are reduced to a set of non-linear ordinary differential equations. Numerical solutions of these equations are then obtained. The effect of increasing values of the Casson parameter is seen to suppress the velocity field. But the temperature is enhanced with increasing Casson parameter. Thermal radiation enhances the effective thermal diffusivity and the temperature increases. It is found that the skin-friction coefficient increases with the increase in suction parameter. 5. Rheo-NMR of the secondary flow of non-Newtonian fluids in square ducts. Science.gov (United States) Schroeder, Christian B; Jeffrey, Kenneth R 2011-01-28 We report the first real-time observations of the entire fully developed laminar secondary flow field of aqueous 2% Viscarin GP-209NF (a λ-carrageenan polysaccharide) in a square duct as made using a modest rheological NMR imaging (rheo-NMR) apparatus. Simulations using the Reiner-Rivlin constitutive equation verify the results. An included rheo-NMR flow rate quantification study assesses the measurement precision. Rheo-NMR resolves slow flows superimposed on primary flows about 300 times greater, making it a universally accessible technique by which full secondary flow field data may be systematically gathered. 6. Shear History Extensional Rheology Experiment II (SHERE II) Microgravity Rheology with Non-Newtonian Polymeric Fluids Science.gov (United States) Jaishankar, Aditya; Haward, Simon; Hall, Nancy Rabel; Magee, Kevin; McKinley, Gareth 2012-01-01 The primary objective of SHERE II is to study the effect of torsional preshear on the subsequent extensional behavior of filled viscoelastic suspensions. Microgravity environment eliminates gravitational sagging that makes Earth-based experiments of extensional rheology challenging. Experiments may serve as an idealized model system to study the properties of lunar regolith-polymeric binder based construction materials. Filled polymeric suspensions are ubiquitous in foods, cosmetics, detergents, biomedical materials, etc. 7. Effects of Flow and Non-Newtonian Fluids on Nonspherical Cavitation Bubbles, Science.gov (United States) 1983-04-10 which incLudes stress accumulation with fading memory was employed by Fogler and Goddard (1970. 1971), who specified a relaxation modulus (memory... Fogler and Goddard present large elastic effects, i.e. changes in the R(t) profiles, but for parameter values which minimize surface 1’ .1 23 tension...also on its appropriate time derivative in a differential model or on the pertinent past values for an Integral equation. Follow- ing Fogler and 8. Numerical Modeling of the Side Flow in Tape Casting of a Non-Newtonian Fluid DEFF Research Database (Denmark) Jabbari, Masoud; Hattel, Jesper Henri 2013-01-01 in the tape casting process is modeled numerically with ANSYS FLUENT in combination with an Ostwald-de Waele power law constitutive equation. Based on rheometer experiments, the constants in the Ostwald-de Waele power law are identified for the considered LSM material and applied in the numerical modeling... 9. Elastically driven surface plumes in rimming flow of a non-Newtonian fluid. Science.gov (United States) Seiden, Gabriel; Steinberg, Victor 2012-11-01 A polymer solution partially filling a rotating horizontal drum undergoes an elastically driven instability at low Reynolds numbers. This instability manifests itself through localized plumelike bursts, perturbing the free liquid surface. Here we present an expanded experimental account regarding the dynamics of individual plumes and the statistics pertaining to the complex collective interaction between plumes, which leads to plume coagulation. We also present a detailed description of an optical technique that enables the visualization and measurement of surface perturbations in coating flows within a rotating horizontal drum. 10. Numerical Modelling of Non-Newtonian Fluid in a Rotational Cross-Flow MBR DEFF Research Database (Denmark) Bentzen, Thomas Ruby; Ratkovich, Nicolas Rios; Rasmussen, Michael R. 2011-01-01 Fouling is the main bottleneck of the widespread of MBR systems. One way to decrease and/or control fouling is by process hydrodynamics. This can be achieved by the increase of liquid crossflow velocity. In rotational cross-flow MBR systems, this is attained by the spinning of e.g. impellers. Val...... as function of the angular velocity and the total suspended solids concentration.... 11. Bouncing, Helical and Buckling Instabilities During Droplet Collision: Newtonian and Non-Newtonian Liquids CERN Document Server Chen, Xiaodong 2012-01-01 In this video, Ray-tracing data visualization technique was used to obtain realistic and detailed flow motions during droplet collision. The differences of collision outcome between Newtonian and non-Newtonian were compared. Various types of droplet collision were presented, including bouncing, coalescence, and stretching separation. Because of the reducing of equivalent viscosity caused by shear stress, the gas film between shear-thinning droplet is thinner than Newtonian liquid. Since thinner gas film promotes coalescence, shear thinning liquid has smaller area of bouncing regime in the diagram of Weber number and impact parameter. During the ligament/thread breakup process of stretching separation, two kinds of instabilities are identified, helical and buckling instabilities. Helical instability is analogous to a viscous rotating liquid jet, while the buckling instability is analogous to electrically charged liquid jets of polymer solutions. 12. Magnetohydrodynamic third-grade non-Newtonian nanofluid flow through a porous coaxial cylinder Science.gov (United States) Sadikin, Zubaidah; Kechil, Seripah Awang 2015-10-01 The convective flow of third grade non-Newtonian nanofluid through porous coaxial cylinders with inclined magnetic field is investigated. The governing partial differential equations are transformed to a system of nonlinear ordinary differential equations using the non-dimensional quantities. The transformed system of nonlinear ordinary differential equations is solved numerically using the fourth-order Runge-Kutta method. The viscosity of the nanofluid is considered as a function of temperature in form of Vogel's model. Numerical solutions are obtained for the velocity, temperature and nanoparticles concentration. The effects of the some physical parameters particularly the angle of inclination, the magnetic, Brownian motion and thermophoresis parameters on non-dimensional velocity, temperature and nanoparticles concentration are analyzed. It is found that as the angle of inclination of magnetic field increases, the velocity decreases. The results also show that increasing the thermophoresis parameter and Brownian motion, the temperature increases. By increasing the Brownian motion or decreasing the thermophoresis parameter, nanoparticles concentration increases. 13. Exact Solutions for Stokes' Flow of a Non-Newtonian Nanofluid Model: A Lie Similarity Approach Science.gov (United States) Aziz, Taha; Aziz, A.; Khalique, C. M. 2016-07-01 The fully developed time-dependent flow of an incompressible, thermodynamically compatible non-Newtonian third-grade nanofluid is investigated. The classical Stokes model is considered in which the flow is generated due to the motion of the plate in its own plane with an impulsive velocity. The Lie symmetry approach is utilised to convert the governing nonlinear partial differential equation into different linear and nonlinear ordinary differential equations. The reduced ordinary differential equations are then solved by using the compatibility and generalised group method. Exact solutions for the model equation are deduced in the form of closed-form exponential functions which are not available in the literature before. In addition, we also derived the conservation laws associated with the governing model. Finally, the physical features of the pertinent parameters are discussed in detail through several graphs. 14. The Non-Newtonian Rheology of Real Magmas: insights into 3D microstructures Science.gov (United States) Pistone, M.; Caricchi, L.; Ulmer, P.; Reusser, E.; Marone, F.; Burlini, L. 2010-12-01 We present high-resolution 3D microstructures of three-phase magmas composed of melt, bubbles and crystals in different proportions deformed at magmatic pressure and temperature conditions. This study aims to constrain the dependence of rheological and physical properties of magmas on the viscosity of the silicate melt, the applied deformation rate, the relative contents of crystals and bubbles and on the interactions between these phases. The starting material is composed of a hydrous haplogranitic melt containing H2O (2.26 wt%) and CO2 (624 ppm) and different proportions of quartz crystals (between 24 and 65 vol%; 63-125 μm in diameter) and bubbles (between 9 and 12 vol%; 5-150 μm in diameter). Experiments were performed in simple shear using a HT-HP internally-heated Paterson-type rock deformation apparatus (Paterson and Olgaard, 2000) at strain rates ranging between 5×10-5 s-1 and 4×10-3 s-1, at a constant pressure of 200 MPa and temperatures ranging between 723 and 1023 K. Synchrotron based X-ray tomographic microscopy performed at the TOMCAT beamline (Stampanoni et al., 2006) at the Swiss Light Source enabled quantitative evaluation of the 3D microstructure. At high temperature and low strain rate conditions the silicate melt behaves as a Newtonian liquid (Webb and Dingwell, 1990). Higher deformation rates and the contemporary presence of gas bubbles and solid crystals make magma rheology more complex and non-Newtonian behaviour occurs. In all experimental runs two different non-Newtonian effects were observed: shear thinning (decrease of viscosity with increasing strain rate) in high crystal-content magmas (55-65 vol% crystals; 9-10 vol% bubbles) and shear thickening (increase of viscosity with increasing strain rate) in magmas at lower degree of crystallinity (24 vol% crystals; 12 vol% bubbles). Both behaviours were observed at intermediate crystal-content (44 vol% crystals; 12 vol% bubbles), with an initial thickening that subsequently gives way to 15. Non-newtonian flow and pressure drop of pineapple juice in a plate heat exchanger Directory of Open Access Journals (Sweden) R. A. F. Cabral 2010-12-01 Full Text Available The study of non-Newtonian flow in plate heat exchangers (PHEs is of great importance for the food industry. The objective of this work was to study the pressure drop of pineapple juice in a PHE with 50º chevron plates. Density and flow properties of pineapple juice were determined and correlated with temperature (17.4 < T < 85.8ºC and soluble solids content (11.0 < Xs < 52.4 ºBrix. The Ostwald-de Waele (power law model described well the rheological behavior. The friction factor for non-isothermal flow of pineapple juice in the PHE was obtained for diagonal and parallel/side flow. Experimental results were well correlated with the generalized Reynolds number (20 < Re g < 1230 and were compared with predictions from equations from the literature. The mean absolute error for pressure drop prediction was 4% for the diagonal plate and 10% for the parallel plate. 16. Two-Fluid Mathematical Models for Blood Flow in Stenosed Arteries: A Comparative Study Directory of Open Access Journals (Sweden) Sankar DS 2009-01-01 Full Text Available The pulsatile flow of blood through stenosed arteries is analyzed by assuming the blood as a two-fluid model with the suspension of all the erythrocytes in the core region as a non-Newtonian fluid and the plasma in the peripheral layer as a Newtonian fluid. The non-Newtonian fluid in the core region of the artery is assumed as a (i Herschel-Bulkley fluid and (ii Casson fluid. Perturbation method is used to solve the resulting system of non-linear partial differential equations. Expressions for various flow quantities are obtained for the two-fluid Casson model. Expressions of the flow quantities obtained by Sankar and Lee (2006 for the two-fluid Herschel-Bulkley model are used to get the data for comparison. It is found that the plug flow velocity and velocity distribution of the two-fluid Casson model are considerably higher than those of the two-fluid Herschel-Bulkley model. It is also observed that the pressure drop, plug core radius, wall shear stress and the resistance to flow are significantly very low for the two-fluid Casson model than those of the two-fluid Herschel-Bulkley model. Hence, the two-fluid Casson model would be more useful than the two-fluid Herschel-Bulkley model to analyze the blood flow through stenosed arteries. 17. Modulation Equations for Roll Waves of a liquid film Down an Inclined Plane as a Power-Law Fluid Institute of Scientific and Technical Information of China (English) Kan.ZHU; Abdelaziz.Boudlal; Gilmar.Mompean.Mompean 2014-01-01 Roll waves of finite amplitude on a thin layer of non-Newtonian fluid modeled as a power-law fluid are considered. In the long wave approximation, the flow is governed by a non-homogeneous hyperbolic system of equations. As the linearized instability analysis of a uniform flow delivers only a diagnosis of instability, the nonlinear stability is investigated and the criterion for roll waves based on the hyperbolicity of the modulation equation is suggested. The main problem in defining the roll wave stability region on a roll wave diagram is due to the singularities of functions for the mean values and their derivatives near the boundaries of roll wave existence. Asymptotic formulae for nonlinear stability of roll waves of small and maximal amplitudes are derived. Numerical calculation reveals that for a Newtonian fluid, as the bottom inclination decreases downwardly the amplitude of admissible waves diminishes, and the stability domain reduces until it disappears. These results remain valid for a slightly non-Newtonian fluid. For highly non-Newtonian fluid, a transition in the nature of stability is observed. 18. Droplets bouncing over a vibrating fluid layer CERN Document Server Cabrera-Garcia, Pablo 2012-01-01 This is an entry for the Gallery of Fluid Motion of the 65st Annual Meeting of the APS-DFD (fluid dynamics video). This video shows the motion of levitated liquid droplets. The levitation is produced by the vertical vibration of a liquid container. We made visualizations of the motion of many droplets to study the formation of clusters and their stability. 19. On the Application of Viscoelastic & Viscoplastic Constitutive Relations in the CFD Bio-Fluid Simulations CERN Document Server Akherat, S M Javid Mahmoudzadeh 2016-01-01 Considerations on implementation of the stress-strain constitutive relations applied in Computational Fluid dynamics (CFD) simulation of cardiovascular flows have been addressed extensively in the literature. However, the matter is yet controversial. The author suggests that the choice of non-Newtonian models and the consideration of non-Newtonian assumption versus the Newtonian assumption is very application oriented and cannot be solely dependent on the vessel size. In the presented work, where a renal disease patient-specific geometry is used, the non-Newtonian effects manifest insignificant, while the vessel is considered to be medium to small which, according to the literature, suggest a strict use of non-Newtonian formulation. The insignificance of the non-Newtonian effects specially manifests in Wall Shear Stress (WSS) along the walls of the numerical domain, where the differences between Newtonian calculated WSS and non-Newtonian calculated WSS is barely visible. 20. Experimental investigation of non-Newtonian/Newtonian liquid-liquid flow in microchannel Science.gov (United States) Roumpea, Eynagelia-Panagiota; Weheliye, Weheliye; Chinaud, Maxime; Angeli, Panagiota; Lyes Kahouadji Collaboration; Omar. K. Matar Collaboration 2015-11-01 Plug flow of an organic phase and an aqueous non-Newtonian solution was investigated experimentally in a quartz microchannel with I.D. 200 μm. The aqueous phase was a glycerol solution where 1000 and 2000 ppm of xanthan gum was added while the organic phase was silicon oil with 155 and 5 cSt viscosity. The two phases were brought together in a T-junction and their flowrates varied from 0.3 to 6 ml/hr. High speed imaging was used to study the characteristics of the plugs and the effect of the liquid properties on the flow patterns while a two-colour micro-PIV technique was used to investigate velocity profiles and circulation patterns within the plugs. The experimental results revealed that plug length was affected by both flowrate and viscosity. In all cases investigated, a film of the continuous phase always surrounded the plugs and its thickness was compared with existing literature models. Circulation patterns inside plugs were obtained by subtracting the plug velocity and found to be depended on the plug length and the amount of xanthan gum in the aqueous phase. Finally, the dimensionless circulation time was calculated and plotted as a function of the plug length. Department of Chemical Engineering South Kensington Campus Imperial College London SW7 2AZ. 1. Entropy Generation in Flow of Highly Concentrated Non-Newtonian Emulsions in Smooth Tubes Directory of Open Access Journals (Sweden) Rajinder Pal 2014-10-01 Full Text Available Entropy generation in adiabatic flow of highly concentrated non-Newtonian emulsions in smooth tubes of five different diameters (7.15–26.54 mm was investigated experimentally. The emulsions were of oil-in-water type with dispersed-phase concentration (Φ ranging from 59.61–72.21% vol. The emulsions exhibited shear-thinning behavior in that the viscosity decreased with the increase in shear rate. The shear-stress (τ versus shear rate (˙γ data of emulsions could be described well by the power-law model: τ=K˙γn. The flow behavior index n was less than 1 and it decreased sharply with the increase in Φ whereas the consistency index K increased rapidly with the increase in Φ . For a given emulsion and tube diameter, the entropy generation rate per unit tube length increased linearly with the increase in the generalized Reynolds number ( Re_n on a log-log scale. For emulsions with Φ ≤65.15 % vol., the entropy generation rate decreased with the increase in tube diameter. A reverse trend in diameter-dependence was observed for the emulsion with Φ of 72.21% vol. New models are developed for the prediction of entropy generation rate in flow of power-law emulsions in smooth tubes. The experimental data shows good agreement with the proposed models. 2. Influence of Droplet Size on Exergy Destruction in Flow of Concentrated Non-Newtonian Emulsions Directory of Open Access Journals (Sweden) Rajinder Pal 2016-04-01 Full Text Available The influence of droplet size on exergy destruction rate in flow of highly concentrated oil-in-water emulsions was investigated experimentally in a cone and plate geometry. The oil concentration was fixed at 74.5% by volume. At this dispersed-phase (oil concentration, two different droplet size emulsions were prepared: fine and coarse emulsions. The fine and coarse emulsions were mixed in different proportions to vary the droplet size distribution. Although the dispersed and matrix phases of the emulsions were Newtonian in nature, the emulsions exhibited a non-Newtonian (shear-thinning behavior due to the high droplet concentration. The shear stress—shear rate data of the emulsions could be described adequately by a power law model. At low shear rates, the exergy destruction rate per unit volume of emulsion exhibited a minimum at a fine emulsion proportion of 35%. The results from the cone and plate geometry were used to simulate exergy loss in pipeline flow of emulsions. The pumping of emulsions becomes more efficient thermodynamically upon mixing of fine and coarse emulsions provided that the flow regime is maintained to be laminar and that the Reynolds number is kept at a low to moderate value. In the turbulent regime, the exergy loss generally increases upon mixing the fine and coarse emulsions. 3. Natural convection in superposed fluid-porous layers CERN Document Server Bagchi, Aniruddha 2013-01-01 Natural Convection in Composite Fluid-Porous Domains provides a timely overview of the current state of understanding on the phenomenon of convection in composite fluid-porous layers. Natural convection in horizontal fluid-porous layers has received renewed attention because of engineering problems such as post-accident cooling of nuclear reactors, contaminant transport in groundwater, and convection in fibrous insulation systems. Because applications of the problem span many scientific domains, the book serves as a valuable resource for a wide audience. 4. Effect of shim configuration on internal die flows for non-Newtonian coating liquids in slot coating process Science.gov (United States) Jin, Guang Lin; Ahn, Won-Gi; Kim, See Jo; Nam, Jaewook; Jung, Hyun Wook; Hyun, Jae Chun 2016-05-01 In this study, a strategy for designing optimal shim configuration inside a slot die is suggested to assure the uniform coating flow distribution of various non-Newtonian shear-thinning liquids at the die exit in a slot coating system. Flow patterns of non-Newtonian liquids inside the slot die, via three-dimensional computations, have been compared using various shim geometries which can adjust the flow region in a slot manifold. The rather non-uniform (parabolic) velocity distributions of shear-thinning liquids at the die exit under the basic shim condition could be effectively flattened by the modification of shim geometry without the change of die manifold structure. Dimensions of hybrid shims for controlling flow features at edge and center regions within slit channel are positively tuned, according to the shear-thinning level of coating liquids. 5. Effect of Non-Newtonian Viscosity for Surfactant Solutions on Vortex Characteristics in a Swirling Pipe Flow Institute of Scientific and Technical Information of China (English) Mizue MUNEKATA; Hidefumi TAKAKI; Hideki OHBA; Kazuyoshi MATSUZAKI 2005-01-01 Effects of non-Newtonian viscosity for surfactant solution on the vortex characteristics and drag-reducing rate in a swirling pipe flow are investigated by pressure drop measurements, velocity profile measurements and viscosity measurements. Non-Newtonian viscosity is represented by power-law model (τ = kD n). Surfactant solution used has shear-thinning viscosity with n ＜ 1.0. The swirling flow in this study has decay of swirl and vortex-type change from Rankin's combined vortex to forced vortex. It is shown that the effect of shear-thinning viscosity on the decay of swirl intensity is different by vortex category and the critical swirl number with the vortex-type change depends on shear-thinning viscosity. 6. Physics of Life: A Model for Non-Newtonian Properties of Living Systems Science.gov (United States) Zak, Michail 2010-01-01 This innovation proposes the reconciliation of the evolution of life with the second law of thermodynamics via the introduction of the First Principle for modeling behavior of living systems. The structure of the model is quantum-inspired: it acquires the topology of the Madelung equation in which the quantum potential is replaced with the information potential. As a result, the model captures the most fundamental property of life: the progressive evolution; i.e. the ability to evolve from disorder to order without any external interference. The mathematical structure of the model can be obtained from the Newtonian equations of motion (representing the motor dynamics) coupled with the corresponding Liouville equation (representing the mental dynamics) via information forces. All these specific non-Newtonian properties equip the model with the levels of complexity that matches the complexity of life, and that makes the model applicable for description of behaviors of ecological, social, and economical systems. Rather than addressing the six aspects of life (organization, metabolism, growth, adaptation, response to stimuli, and reproduction), this work focuses only on biosignature ; i.e. the mechanical invariants of life, and in particular, the geometry and kinematics of behavior of living things. Living things obey the First Principles of Newtonian mechanics. One main objective of this model is to extend the First Principles of classical physics to include phenomenological behavior on living systems; to develop a new mathematical formalism within the framework of classical dynamics that would allow one to capture the specific properties of natural or artificial living systems such as formation of the collective mind based upon abstract images of the selves and non-selves; exploitation of this collective mind for communications and predictions of future expected characteristics of evolution; and for making decisions and implementing the corresponding corrections if 7. A Qualitative Investigation of Deposition Velocities of a Non-Newtonian Slurry in Complex Pipeline Geometries Energy Technology Data Exchange (ETDEWEB) Yokuda, Satoru T.; Poloski, Adam P.; Adkins, Harold E.; Casella, Andrew M.; Hohimer, Ryan E.; Karri, Naveen K.; Luna, Maria; Minette, Michael J.; Tingey, Joel M. 2009-05-11 The External Flowsheet Review Team (EFRT) has identified the issues relating to the Waste Treatment and Immobilization Plant (WTP) pipe plugging. Per the review’s executive summary, “Piping that transports slurries will plug unless it is properly designed to minimize this risk. This design approach has not been followed consistently, which will lead to frequent shutdowns due to line plugging.” To evaluate the potential for plugging, testing was performed to determine critical velocities for the complex WTP piping layout. Critical velocity is defined as the point at which a moving bed of particles begins to form on the pipe bottom during slurry-transport operations. Pressure drops across the fittings of the test pipeline were measured with differential pressure transducers, from which the critical velocities were determined. A WTP prototype flush system was installed and tested upon the completion of the pressure-drop measurements. We also provide the data for the overflow relief system represented by a WTP complex piping geometry with a non-Newtonian slurry. A waste simulant composed of alumina (nominally 50 μm in diameter) suspended in a kaolin clay slurry was used for this testing. The target composition of the simulant was 10 vol% alumina in a suspending medium with a yield stress of 3 Pa. No publications or reports are available to confirm the critical velocities for the complex geometry evaluated in this testing; therefore, for this assessment, the results were compared to those reported by Poloski et al. (2008) for which testing was performed for a straight horizontal pipe. The results of the flush test are compared to the WTP design guide 24590-WTP-GPG-M-0058, Rev. 0 (Hall 2006) in an effort to confirm flushing-velocity requirements. 8. Sinking, wedging, spreading - viscous spreading on a layer of fluid Science.gov (United States) Bergemann, Nico; Juel, Anne; Heil, Matthias 2016-11-01 We study the axisymmetric spreading of a sessile drop on a pre-existing layer of the same fluid in a regime where the drop is sufficiently large so that the spreading is driven by gravity while capillary and inertial effects are negligible. Experiments performed with 5 ml drops and layer thicknesses in the range 0.1 mm drop evolves as R tn , where the spreading exponent n increases with the layer thickness h. Numerical simulations, based on the axisymmetric free-surface Navier-Stokes equations, reveal three distinct spreading regimes depending on the layer thickness. For thick layers the drop sinks into the layer, accompanied by significant flow in the layer. By contrast, for thin layers the layer ahead of the propagating front is at rest and the spreading behaviour resembles that of a gravity-driven drop spreading on a dry substrate. In the intermediate regime the spreading is characterised by an advancing wedge, which is sustained by fluid flow from the drop into the layer. 9. Stagnation-point flow of the Walters' B' fluid with slip OpenAIRE Labropulu, F.; Husain, I; Chinichian, M. 2004-01-01 The steady two-dimensional stagnation point flow of a non-Newtonian Walters' B' fluid with slip is studied. The fluid impinges on the wall either orthogonally or obliquely. A finite difference technique is employed to obtain solutions. 10. Electrocaloric cooler combining ceramic multi-layer capacitors and fluid Directory of Open Access Journals (Sweden) Daniele Sette 2016-09-01 Full Text Available In this paper, an electrocaloric (EC cooler prototype made of 150 ceramic-based Multi-Layer Capacitors (MLCs has been detailed. This cooler involves a column of dielectric fluid where heat exchange with the MLCs takes place. The maximum variation of temperature in the fluid column due to the EC effect reaches 0.13 K whereas the heat exchanged during one stroke is 0.28 J. Although this prototype requires improvements with respect to heat exchange, the basic principle of creating a temperature gradient in a column of fluid has been validated. 11. Ultrasonic characterization of a fluid layer using a broadband transducer. Science.gov (United States) Samet, Naïm; Maréchal, Pierre; Duflo, Hugues 2012-03-01 A measurement method is proposed for the ultrasonic characterization of a fluid layer, corresponding to the resin transfer molding (RTM) manufacturing process. The ultrasonic velocity and attenuation of the silicone oil are measured in three samples having different viscosities. The measurement method is established on the basis of the attenuation of ultrasonic waves in fluids. A correction of the beam diffraction is implemented to improve measurement precision. A single element transducer with central frequency of 15 MHz is used. The tested fluids simulate the industrial resin used to manufacture composite materials. When injecting this resin, its viscosity increases until it reaches a critical state of polymerization. In this paper we focus on ultrasonic characterization of three fluids representing three intermediate cases of fluid resin during its injection before reaching the polymerization state. 12. Drag characteristics of power law fluids on an upstream moving surface Institute of Scientific and Technical Information of China (English) Liancun Zheng; Xinxin Zhang; Jicheng He 2005-01-01 The specific problem to be considered here concerns the boundary layer problem of a non-Newtonian fluid on a flat plate in length, whose surface has a constant velocity opposite in the direction to that of the mainstream with Uw ＞＞ U∞, or alternatively when the plate surface velocity is kept fixed but the stream speed is reduced to zero. A theoretical analysis for a boundary layer flow is made and the self-similar equation is determined. Solutions are presented numerically for special power index and the associated transfer behavior is discussed. 13. The Viscosity of Polymeric Fluids. Science.gov (United States) Perrin, J. E.; Martin, G. C. 1983-01-01 To illustrate the behavior of polymeric fluids and in what respects they differ from Newtonian liquids, an experiment was developed to account for the shear-rate dependence of non-Newtonian fluids. Background information, procedures, and results are provided for the experiment. Useful in transport processes, fluid mechanics, or physical chemistry… 14. Linear waves in two-layer fluids over periodic bottoms NARCIS (Netherlands) Yu, Jie; Maas, L.R.M. 2016-01-01 A new, exact Floquet theory is presented for linear waves in two-layer fluids over a periodic bottom of arbitrary shape and amplitude. A method of conformal transformation is adapted. The solutions are given, in essentially analytical form, for the dispersion relation between wave frequency and gene 15. Ultrasonic wave's interaction at fluid-porous piezoelectric layered interface. Science.gov (United States) Vashishth, Anil K; Gupta, Vishakha 2013-02-01 The complete description of acoustic propagation in a multilayered system is of great interest in a variety of applications such as non-destructive evaluation and acoustic design and there is need for a flexible model that can describe the reflection and transmission of ultrasonic waves in these media. The reflection and transmission of ultrasonic waves from a fluid loaded porous piezoelectric layered structure is studied analytically. The layered structure is considered to be consisting of n number of layers of porous piezoelectric materials. Transfer matrix technique is used to study the layered materials. The analytical expressions for the reflected, transmitted, interaction energy ratios and surface impedance are obtained. The effects of frequency, porosity, angle of incidence, layer thickness and number of layers on the energy ratios and surface impedance are studied for different configurations of the layered materials. The results obtained are deduced for the poro-elastic and fluid loaded porous piezoelectric half space case, which are in agreement with earlier established results. A comparison of the results, obtained by alternate numerical techniques, is made. 16. Locomotion in complex fluids: Integral theorems CERN Document Server Lauga, Eric 2014-01-01 The biological fluids encountered by self-propelled cells display complex microstructures and rheology. We consider here the general problem of low-Reynolds number locomotion in a complex fluid. {Building on classical work on the transport of particles in viscoelastic fluids,} we demonstrate how to mathematically derive three integral theorems relating the arbitrary motion of an isolated organism to its swimming kinematics {in a non-Newtonian fluid}. These theorems correspond to three situations of interest, namely (1) squirming motion in a linear viscoelastic fluid, (2) arbitrary surface deformation in a weakly non-Newtonian fluid, and (3) small-amplitude deformation in an arbitrarily non-Newtonian fluid. Our final results, valid for a wide-class of {swimmer geometry,} surface kinematics and constitutive models, at most require mathematical knowledge of a series of Newtonian flow problems, and will be useful to quantity the locomotion of biological and synthetic swimmers in complex environments. 17. The carreau-yasuda fluids: a skin friction equation for turbulent flow in pipes and kolmogorov dissipative scales OpenAIRE ANDRADE, Luiz Claudio Fialho; PETRONÍLIO, Jamilson A.; MANESCHY, Carlos Edilson de Almeida; CRUZ, Daniel Onofre de Almeida 2007-01-01 In this work the turbulent flow of the Non-Newtonian Carreau-Yasuda fluid will be studied. A skin friction equation for the turbulent flow of Carreau-Yasuda fluids will be derived assuming a logarithmic behavior of the turbulent mean velocity for the near wall flow out of the viscous sub layer. An alternative near wall characteristic length scale which takes into account the effects of the relaxation time will be introduced. The characteristic length will be obtained through the analysis of v... 18. The Stokes boundary layer for a thixotropic or antithixotropic fluid KAUST Repository McArdle, Catriona R. 2012-10-01 We present a mathematical investigation of the oscillatory boundary layer in a semi-infinite fluid bounded by an oscillating wall (the so-called \\'Stokes problem\\'), when the fluid has a thixotropic or antithixotropic rheology. We obtain asymptotic solutions in the limit of small-amplitude oscillations, and we use numerical integration to validate the asymptotic solutions and to explore the behaviour of the system for larger-amplitude oscillations. The solutions that we obtain differ significantly from the classical solution for a Newtonian fluid. In particular, for antithixotropic fluids the velocity reaches zero at a finite distance from the wall, in contrast to the exponential decay for a thixotropic or a Newtonian fluid.For small amplitudes of oscillation, three regimes of behaviour are possible: the structure parameter may take values defined instantaneously by the shear rate, or by a long-term average; or it may behave hysteretically. The regime boundaries depend on the precise specification of structure build-up and breakdown rates in the rheological model, illustrating the subtleties of complex fluid models in non-rheometric settings. For larger amplitudes of oscillation the dominant behaviour is hysteretic. We discuss in particular the relationship between the shear stress and the shear rate at the oscillating wall. © 2012 Elsevier B.V. 19. Phase Distribution of Nitrogen/Non-Newtonian Flow at Micro-T-Junction%T形微通道内氮气/非牛顿流体两相流相分配 Institute of Scientific and Technical Information of China (English) 周云龙; 刘博; 孙科 2013-01-01 以氮气作为气相工质,以纯水和浓度分别为0.10wt％、0.25wt％、0.50wt％的PAAm水溶液分别为液相工质,在当量直径为178μm的T形微通道内进行气液两相流相分配的实验.实验结果表明:非牛顿流体的性质对两相流相分配特性有着重要的影响.当液体的有效黏度增加时,弹状流和环状流的液相采出分率增加.通过对弹状流与环状流两种流型的对比,发现环状流的液相采出分率受非牛顿流体性质影响较大,而弹状流受其影响较小.%With air as the gas test fluid and both pure water and PAAm aqueous solution with 0.01wt％,0.25wt％ and 0.50wt％ as the test fluid,an experimental investigation into phase distribution of gas-liquid flow through a T-junction with 178μm hydraulic diameter was conducted.The experimental results show that the property of non-Newtonian fluid in micro-T-junction can influence the phase distribution,and the decrease in liquid effective viscosity makes liquid' s taking off of both slug and annular flow reduced.Comparing the flow pattern of slug and annular flow shows that the property of non-Newtonian fluid can influence liquid' s taking-off much of the annular flow and does little on that of slug flow. 20. Nuclear Equation of State, Yukawa-type Non-Newtonian Gravity and Pulsating Frequencies of Neutron Stars CERN Document Server Lin, Weikang; Chen, Lie-Wen; Wen, De-Hua; Xu, Jun 2013-01-01 A thorough understanding of many astrophysical phenomena and objects requires reliable knowledge about both the equation of state (EOS) of super-dense nuclear matter and the theory of ultra-strong gravity simultaneously because of the EOS-gravity degeneracy. Currently, deviations of the neutron star (NS) mass-radius correlation predicted by various gravity theories are larger than its uncertainties due to the poorly known NS matter content and its EOS. At least two independent observables are required to break the EOS-gravity degeneracy. Using model EOSs for hybrid stars and a Yukawa-type non-Newtonian gravity, we investigate both the mass-radius correlation and pulsating frequencies of NSs. While the maximum mass of NSs increases with increasing strength of the Yukawa-type non-Newtonian gravity, the frequencies of thef$,$p_1$,$p_2$, and$w_I\$ pulsating modes are found to decrease with it, providing a useful reference for future determination simultaneously of both the gravitational theory and the supranu... 1. Numerical analysis of mixed convection in lid-driven cavity using non-Newtonian ferrofluid with rotating cylinder inside Science.gov (United States) Rabbi, Khan Md.; Shuvo, Moinuddin; Kabir, Rabiul Hasan; Mojumder, Satyajit; Saha, Sourav 2016-07-01 Mixed convection in a lid-driven square enclosure with a rotating cylinder inside has been analyzed using non-Newtonian ferrofluid (Fe3O4-water). Left vertical wall is heated while the right vertical wall is kept cold. Bottom wall and cylinder surface are assumed to be adiabatic. Top wall has a moving lid with a constant velocity U0. Galerkin method of finite element analysis has been used to solve the governing equations. Numerical accuracy of solution is ensured by the grid independency test. A variety of Richardson number (Ri = 0.1 - 10) at a governing Reynolds number (Re = 100), power law index (n = 0.5 - 1.5), rotational speed (Ω = 0 - 15) and solid volume fraction of ferrous particles (φ = 0 - 0.05) are employed for this present problem. To illustrate flow and thermal field, streamline and isotherms are included. Average Nusselt number plots are shown to show overall heat transfer rate. It is observed that better heat transfer is achieved at higher rotational speed (Ω), Richardson number (Ri) and power law index (n). This paper also concludes significant variation in streamline and isotherm patterns for higher solid volume fraction (φ) of non-Newtonian ferrofluid. 2. Boundary conditions at the cartilage-synovial fluid interface for joint lubrication and theoretical verifications. Science.gov (United States) Hou, J S; Holmes, M H; Lai, W M; Mow, V C 1989-02-01 The objective of this study is to establish and verify the set of boundary conditions at the interface between a biphasic mixture (articular cartilage) and a Newtonian or non-Newtonian fluid (synovial fluid) such that a set of well-posed mathematical problems may be formulated to investigate joint lubrication problems. A "pseudo-no-slip" kinematic boundary condition is proposed based upon the principle that the conditions at the interface between mixtures or mixtures and fluids must reduce to those boundary conditions in single phase continuum mechanics. From this proposed kinematic boundary condition, and balances of mass, momentum and energy, the boundary conditions at the interface between a biphasic mixture and a Newtonian or non-Newtonian fluid are mathematically derived. Based upon these general results, the appropriate boundary conditions needed in modeling the cartilage-synovial fluid-cartilage lubrication problem are deduced. For two simple cases where a Newtonian viscous fluid is forced to flow (with imposed Couette or Poiseuille flow conditions) over a porous-permeable biphasic material of relatively low permeability, the well known empirical Taylor slip condition may be derived using matched asymptotic analysis of the boundary layer at the interface. 3. Non-Newtonian flow between concentric cylinders calculated from thermophysical properties obtained from simulations Energy Technology Data Exchange (ETDEWEB) Narayan, A.P. [Univ. of Colorado, Boulder, CO (United States); Rainwater, J.C. [National Institute of Standards and Technology, Boulder, CO (United States); Hanley, H.J.M. [Univ. of Colorado, Boulder, CO (United States)]\|[National Institute of Standards and Technology, Boulder, CO (United States) 1995-03-01 A study of the Weissenberg effect (rod climbing in a stirred system) based on nonequilibrium molecular dynamics (NEMD) is reported. Simulation results from a soft-sphere fluid are used to obtain a self-consistent free-surface profile of the fluid of finite compressibility undergoing Couette flow between concentric cylinders. A numerical procedure is then applied to calculate the height profile for a hypothetical fluid with thermophysical properties of the soft-sphere liquid and of a dense colloidal suspension. The height profile calculated is identified with shear thickening and the forms of the viscometric functions. The maximum climb occurs between the cylinders rather than at the inner cylinder. 4. Parameter determination for the Cross rheology equation and its application to modeling non-Newtonian flows using the WC-MPS method Directory of Open Access Journals (Sweden) Jun Xie 2016-01-01 Full Text Available A weakly compressible moving particle semi-implicit (WC-MPS method is utilized to simulate non-Newtonian free surface flows due to the advantages of particle methods with respect to handling large deformation and fragmentation. The Cross rheology equation was selected in order to capture the viscous features of the mixture flows. To numerically implement the Cross equation, an experiment-based method was proposed to determine the four rheology parameters in the equation. The method of using a WC-MPS model to study non-Newtonian dam break flow problems was then adopted. The capabilities of the proposed method were tested by simulating different materials with the proposed method in modeling non-Newtonian free surface flows. Significant viscous features were reproduced by the proposed model. 5. In-vitro interferometric characterization of dynamic fluid layers on contact lenses Science.gov (United States) Primeau, Brian C.; Greivenkamp, John E.; Sullivan, John J. 2011-08-01 The anterior refracting surface of the eye when wearing a contact lens is the thin fluid layer that forms on the surface of the contact lens. Under normal conditions, this fluid layer is less than 10 microns thick. The fluid layer thickness and topography change over time and are affected by the material properties of the contact lens, and may affect vision quality and comfort. An in vitro method of characterizing dynamic fluid layers applied to contact lenses mounted on mechanical substrates has been developed using a phase-shifting Twyman-Green interferometer. This interferometer continuously measures light reflected from the surface of the fluid layer, allowing precision analysis of the dynamic fluid layer. Movies showing this fluid layer behavior can be generated. The fluid behavior on the contact lens surface is measured, allowing quantitative analysis beyond what typical contact angle or visual inspection methods provide. The interferometer system has measured the formation and break up of fluid layers. Different fluid and contact lens material combinations have been used, and significant fluid layer properties have been observed in some cases. The interferometer is capable of identifying features in the fluid layer less than a micron in depth with a spatial resolution of about ten microns. An area on the contact lens approximately 6 mm wide can be measured with the system. This paper will discuss the interferometer design and analysis methods used. Measurement results of different material and fluid combinations are presented. 6. Effects of unsteadiness and non-Newtonian rheology on blood flow through a tapered time-variant stenotic artery Directory of Open Access Journals (Sweden) A. Zaman 2015-03-01 Full Text Available A two-dimensional model is used to analyze the unsteady pulsatile flow of blood through a tapered artery with stenosis. The rheology of the flowing blood is captured by the constitutive equation of Carreau model. The geometry of the time-variant stenosis has been used to carry out the present analysis. The flow equations are set up under the assumption that the lumen radius is sufficiently smaller than the wavelength of the pulsatile pressure wave. A radial coordinate transformation is employed to immobilize the effect of the vessel wall. The resulting partial differential equations along with the boundary and initial conditions are solved using finite difference method. The dimensionless radial and axial velocity, volumetric flow rate, resistance impedance and wall shear stress are analyzed for normal and diseased artery with particular focus on variation of these quantities with non-Newtonian parameters. 7. Group method analysis of mixed convection stagnation-point flow of non-Newtonian nanofluid over a vertical stretching surface Science.gov (United States) Nabwey, Hossam A.; Boumazgour, Mohamed; Rashad, A. M. 2017-03-01 The group method analysis is applied to study the steady mixed convection stagnation-point flow of a non-Newtonian nanofluid towards a vertical stretching surface. The model utilized for the nanofluid incorporates the Brownian motion and thermophoresis effects. Applying the one-parameter transformation group which reduces the number of independent variables by one and thus, the system of governing partial differential equations has been converted to a set of nonlinear ordinary differential equations, and these equations are then computed numerically using the implicit finite-difference scheme. Comparison with previously published studies is executed and the results are found to be in excellent agreement. Results for the velocity, temperature, and the nanoparticle volume fraction profiles as well as the local skin-friction coefficient and local Nusselt number are presented in graphical and tabular forms, and discussed for different values of the governing parameters to show interesting features of the solutions. 8. DETERMINATION OF THE EFFECTIVE RADIAL THERMAL DIFFUSIVITY FOR EVALUATING ENHANCED HEAT TRANSFER IN TUBES UNDER NON-NEWTONIAN LAMINAR FLOW Directory of Open Access Journals (Sweden) A. O. Morais 2015-06-01 Full Text Available AbstractEnhanced heat transfer in tubes under laminar flow conditions can be found in coils or corrugated tubes or in the presence of high wall relative roughness, curves, pipe fittings or mechanical vibration. Modeling these cases can be complex because of the induced secondary flow. A modification of the Graetz problem for non-Newtonian power-law flow is proposed to take into account the augmented heat transfer by the introduction of an effective radial thermal diffusivity. The induced mixing was modeled as an increased radial heat transfer in a straight tube. Three experiments using a coiled tube and a tubular heat exchanger with high relative wall roughness are presented in order to show how this parameter can be obtained. Results were successfully correlated with Reynolds number. This approach can be useful for modeling laminar flow reactors (LFR and tubular heat exchangers available in the chemical and food industries. 9. Perturbation of a Multiple Eigenvalue in the Benard Problem for Two Fluid Layers. Science.gov (United States) 1984-12-01 EIGENVAWUE IN THlE BENARtD PROBLEM FOR TWO FLUID LAYERS Ca O~ Yuriko Renardy and Michael Renardy MUathematics Research Center University of Wisconsin...OF WISCONSIN - MADISON MATHEMATICS RESEARCH CENTER PERTUBBATION OF A MULTIPLE EIGENVALUE IN THE BENARD PROBLEM FOR TWO FLUID LAYERS Yuriko Renardy and...PROBLEM FOR TWO FLUID LAYERS Yuriko Renardy and Michael Renardy 1. INTRODUCTION In the B6nard problem for one fluid, "exchange of stabilities" holds 10. An active particle in a complex fluid Science.gov (United States) Datt, Charu; Natale, Giovanniantonio; Hatzikiriakos, Savvas G.; Elfring, Gwynn J. 2016-11-01 Active particles are self-driven units capable of converting stored or ambient free-energy into systematic movement. We discuss here the case when such particles move through non-Newtonian fluids. Neglecting inertial forces, we employ the reciprocal theorem to calculate the propulsion velocity of a single swimmer in a weakly non-Newtonian fluid with background flow. We also derive a general expression for the velocity of an active particle modelled as a squirmer in a second-order fluid. We then discuss how active colloids are affected by the medium rheology, namely viscoelasticity and shear-thinning. 11. Algebraically explicit analytical solutions for the unsteady non-Newtonian swirling flow in an annular pipe Institute of Scientific and Technical Information of China (English) CAI; Ruixian; GOU; Chenhua 2006-01-01 This paper presents two algebraically explicit analytical solutions for the incompressible unsteady rotational flow of Oldroyd-B type in an annular pipe. The first solution is derived with the common method of separation of variables. The second one is deduced with the method of separation of variables with addition developed in recent years. The first analytical solution is of clear physical meaning and both of them are fairly simple and valuable for the newly developing computational fluid dynamics. They can be used as the benchmark solutions to verify the applicability of the existing numerical computational methods and to inspire new differencing schemes, grid generation ways, etc. Moreover, a steady solution for the generalized second grade rheologic fluid flow is also presented. The correctness of these solutions can be easily proven by substituting them into the original governing equation. 12. Non-Newtonian model study for blood flow through a tapered artery with a stenosis Directory of Open Access Journals (Sweden) Noreen Sher Akbar 2016-03-01 Full Text Available The blood flow through a tapered artery with a stenosis is analyzed, assuming the blood as tangent hyperbolic fluid model. The resulting nonlinear implicit system of partial differential equations is solved analytically with the help of perturbation method. The expressions for shear stress, velocity, flow rate, wall shear stress and longitudinal impedance are obtained. The variations of power law index m, Weissenberg number We, shape of stenosis n and stenosis size δ are discussed different type of tapered arteries. 13. Turbulent characteristics of shear-thinning fluids in recirculating flows Energy Technology Data Exchange (ETDEWEB) Pereira, A.S. [Inst. Superior de Engenharia do Porto (Portugal). Dept. de Engenharia Quimica; Pinho, F.T. [Centro de Estudos de Fenomenos de Transporte, Departamento de Engenharia Mecanica e Gestao Industrial, Faculdade de Engenharia da Universidade do Porto, Rua dos Bragas, 4050-123 Porto (Portugal) 2000-03-01 A miniaturised fibre optic laser-Doppler anemometer was used to carry out a detailed hydrodynamic investigation of the flow downstream of a sudden expansion with 0.1-0.2% by weight shear-thinning aqueous solutions of xanthan gum. Upstream of the sudden expansion the pipe flow was fully-developed and the xanthan gum solutions exhibited drag reduction with corresponding lower radial and tangential normal Reynolds stresses, but higher axial Reynolds stress near the wall and a flatter axial mean velocity profile in comparison with Newtonian flow. The recirculation bubble length was reduced by more than 20% relative to the high Reynolds number Newtonian flow, and this was attributed to the occurrence further upstream of high turbulence for the non-Newtonian solutions, because of advection of turbulence and earlier high turbulence production in the shear layer. Comparisons with the measurements of Escudier and Smith (1999) with similar fluids emphasized the dominating role of inlet turbulence. The present was less anisotropic, and had lower maximum axial Reynolds stresses (by 16%) but higher radial turbulence (20%) than theirs. They reported considerably longer recirculating bubble lengths than we do for similar non-Newtonian fluids and Reynolds numbers. (orig.) 14. An Analytical Approach for Analysis of Slider Bearings with Non-Newtonian Lubricants Directory of Open Access Journals (Sweden) Li-Ming Chu 2014-01-01 Full Text Available In this study, a regular perturbation technique is utilized to derive the modified Reynolds equation which is applicable to power-law lubricant. The performance of slider bearings including pressure distributions, velocity distributions, film thickness, load capacity, flow rate, shear force, and friction coefficient is also derived analytically for various ξ, flow indices (n, and outlet film thicknesses (H0. These analytical solutions are clear to find the effects of the operation parameters rather than numerical methods. It can be simply and fast used for engineers. Subsequently, these proposed analytical solutions are used to analyze the lubrication performance of slider bearing with the power-law fluids. 15. Free Convective Nonaligned Non-Newtonian Flow with Non-linear Thermal Radiation Science.gov (United States) Rana, S.; Mehmood, R.; Narayana, PV S.; Akbar, N. S. 2016-12-01 The present study explores the free convective oblique Casson fluid over a stretching surface with non-linear thermal radiation effects. The governing physical problem is modelled and transformed into a set of coupled non-linear ordinary differential equations by suitable similarity transformation, which are solved numerically with the help of shooting method keeping the convergence control of 10-5 in computations. Influence of pertinent physical parameters on normal, tangential velocity profiles and temperature are expressed through graphs. Physical quantities of interest such as skin friction coefficients and local heat flux are investigated numerically. 16. Numerical modelling of thermochemically driven fluid flow with non-Newtonian rheology : applied to the earth's lithosphere and mantle NARCIS (Netherlands) van Keken, P.E. 1993-01-01 In the 25 years after the general acceptance of the concept of plate tectonics we have witnessed large progress in observational, laboratory, forward modelling and inversion techniques. These provide a clear view of the immense complexities that are facing us when studying the dynamics of the interi 17. The effect of shear and extensional viscosities on atomization of Newtonian and non-Newtonian fluids in ultrasonic inhaler. Science.gov (United States) Broniarz-Press, L; Sosnowski, T R; Matuszak, M; Ochowiak, M; Jabłczyńska, K 2015-05-15 The paper contains results of the experimental study on atomization process of aqueous solutions of glycerol and aqueous solutions of glycerol-polyacrylamide (Rokrysol WF1) in an ultrasonic inhaler. In experiments the different concentration aqueous solutions of glycerol and glycerol-polyacrylamide have been tested. The results have been obtained by the use of laser diffraction technique. The differences between characteristics of ultrasonic atomization for test liquids have been observed. The analysis of drop size histograms shows that the different sizes of drops have been formed during atomization process. The present study confirmed the previous reports which suggested that the drops size changes with the increase in viscosity of solution changes in spray characteristics were also observed. It has been shown that the shear and extensional viscosities affect the process of atomization. 18. MHD Flow of a Non-Newtonian Power Law Fluid over a Vertical Stretching Sheet with the Convective Boundary Condition Directory of Open Access Journals (Sweden) 2013-02-01 Full Text Available In this article, we study the power law model of steady state, viscous, incompressible MHD flow over a vertically stretching sheet. Furthermore, heat transfer is also addressed by using the convective boundary conditions. The coupled partial differential equations are transformed into ordinary differential equations (ODEs using similarity transformations. The transformed highly non-linear ODEs are solved by using the Homotopy Analysis Method (HAM. The influence of different parameters on the velocity and temperature fields are analyzed and discussed. 19. Reynolds Number Effects in the Flow of a Vočadlo Electrorheological Fluid in a Curved Gap Science.gov (United States) Walicka, A.; Falicki, J. 2017-08-01 Many electrorheological fluids (ERFs) as fluids with micro-structure demonstrate a non-Newtonian behaviour. Rheometric measurements indicate that some flows of these fluids may by modelled as the flows of a Vočadlo ER fluid. In this paper, the flow of a Vočadlo fluid - with a fractional index of non-linearity - in a narrow gap between two fixed surfaces of revolution with a common axis of symmetry is considered. The flow is externally pressurized and it is considered with inertia effect. In order to solve this problem the boundary layer equations are used. The Reynolds number effects (the effects of inertia forces) on the pressure distribution are examined by using the method of averaged inertia terms of the momentum equation. Numerical examples of externally pressurized flows in the gap between parallel disks and concentric spherical surfaces are presented. 20. Preconditioned iterative methods for unsteady non-Newtonian flow between eccentrically rotating cylinders Energy Technology Data Exchange (ETDEWEB) Gwynllyw, D.Rh.; Phillips, T.N. [Univ. of Wales, Aberystwyth (United Kingdom) 1994-12-31 The journal bearing is an essential part of all internal combustion engines as a means of transferring the energy from the piston rods to the rotating crankshaft. It consists essentially of an inner cylinder (the journal), which is part of the crankshaft, and an outer cylinder (the bearing), which is at the end of the piston rod. In general, the two cylinders are eccentric and there is a lubricating film of oil separating the two surfaces. The addition of polymers to mineral (Newtonian) oils to minimize the variation of viscosity with temperature has the added effect of introducing strain-dependent viscosity and elasticity. The physical problem has many complicating features which need to be modelled. It is a fully three-dimensional problem which means that significant computational effort is required to solve the problem numerically. The system is subject to dynamic loading in which the journal is allowed to move under the forces the fluid imparts on it and also any other loads such as that imparted by the engine force. The centre of the journal traces out a nontrivial locus in space. In addition, there is significant deformation of the bearing and journal and extensive cavitation of the oil lubricant. In the present study the authors restrict themselves to the two-dimensional statically loaded problem. In previous work a single domain spectral method was used which employed a bipolar coordinate transformation to map the region between the journal and the bearing onto a rectangle. The flow variables were then approximated on this rectangle using Fourier-Chebyshev expansions. However, to allow for future possible deformation of the journal and bearing surfaces due to increased load in the dynamically loaded case they have decided to use a more versatile spectral element formulation. 1. NMR imaging and hydrodynamic analysis of neutrally buoyant non-Newtonian slurry flows Science.gov (United States) Bouillard, J. X.; Sinton, S. W. The flow of solids loaded suspension in cylindrical pipes has been the object of intense experimental and theoretical investigations in recent years. These types of flows are of great interest in chemical engineering because of their important use in many industrial manufacturing processes. Such flows are for example encountered in the manufacture of solid-rocket propellants, advanced ceramics, reinforced polymer composites, in heterogeneous catalytic reactors, and in the pipeline transport of liquid-solids suspensions. In most cases, the suspension microstructure and the degree of solids dispersion greatly affect the final performance of the manufactured product. For example, solid propellant pellets need to be extremely-well dispersed in gel matrices for use as rocket engine solid fuels. The homogeneity of pellet dispersion is critical to allow good uniformity of the burn rate, which in turn affects the final mechanical performance of the engine. Today's manufacturing of such fuels uses continuous flow processes rather than batch processes. Unfortunately, the hydrodynamics of such flow processes is poorly understood and is difficult to assess because it requires the simultaneous measurements of liquid/solids phase velocities and volume fractions. Due to the recent development in pulsed Fourier Transform NMR imaging, NMR imaging is now becoming a powerful technique for the non intrusive investigation of multi-phase flows. This paper reports and exposes a state-of-the-art experimental and theoretical methodology that can be used to study such flows. The hydrodynamic model developed for this study is a two-phase flow shear thinning model with standard constitutive fluid/solids interphase drag and solids compaction stresses. this model shows good agreement with experimental data and the limitations of this model are discussed. 2. Linking the fractional derivative and the Lomnitz creep law to non-Newtonian time-varying viscosity Science.gov (United States) Pandey, Vikash; Holm, Sverre 2016-09-01 Many of the most interesting complex media are non-Newtonian and exhibit time-dependent behavior of thixotropy and rheopecty. They may also have temporal responses described by power laws. The material behavior is represented by the relaxation modulus and the creep compliance. On the one hand, it is shown that in the special case of a Maxwell model characterized by a linearly time-varying viscosity, the medium's relaxation modulus is a power law which is similar to that of a fractional derivative element often called a springpot. On the other hand, the creep compliance of the time-varying Maxwell model is identified as Lomnitz's logarithmic creep law, making this possibly its first direct derivation. In this way both fractional derivatives and Lomnitz's creep law are linked to time-varying viscosity. A mechanism which yields fractional viscoelasticity and logarithmic creep behavior has therefore been found. Further, as a result of this linking, the curve-fitting parameters involved in the fractional viscoelastic modeling, and the Lomnitz law gain physical interpretation. 3. Explaining the non-newtonian character of aggregating monoclonal antibody solutions using small-angle neutron scattering. Science.gov (United States) Castellanos, Maria Monica; Pathak, Jai A; Leach, William; Bishop, Steven M; Colby, Ralph H 2014-07-15 A monoclonal antibody solution displays an increase in low shear rate viscosity upon aggregation after prolonged incubation at 40°C. The morphology and interactions leading to the formation of the aggregates responsible for this non-Newtonian character are resolved using small-angle neutron scattering. Our data show a weak repulsive barrier before proteins aggregate reversibly, unless a favorable contact with high binding energy occurs. Two types of aggregates were identified after incubation at 40°C: oligomers with radius of gyration ∼10 nm and fractal submicrometer particles formed by a slow reaction-limited aggregation process, consistent with monomers colliding many times before finding a favorable strong interaction site. Before incubation, these antibody solutions are Newtonian liquids with no increase in low shear rate viscosity and no upturn in scattering at low wavevector, whereas aggregated solutions under the same conditions have both of these features. These results demonstrate that fractal submicrometer particles are responsible for the increase in low shear rate viscosity and low wavevector upturn in scattered intensity of aggregated antibody solutions; both are removed from aggregated samples by filtering. 4. Convective stability of a vertical layer of magnetizable fluid in a uniform magnetic field Energy Technology Data Exchange (ETDEWEB) Bashtovoy, V.G.; Pavlinov, M.I. 1978-01-01 An infinitely large plane vertical layer of magnetizable fluid is considered, this layer being heated from below and bounded on both lateral surfaces by ferromagnetic half-spaces. The fluid and the ferromagnetic material on both sides have the same pyromagnetic coefficient. The possibility of overcoming a convective instability of such a fluid layer in a uniform magnetic field is demonstrated by a solution of the equilibrium equation. The result indicates that such a magnetic field raises the stability threshold to full stabilization of the fluid layer, with the instability range in terms of the Rayleigh number now having both a lower and an upper limit. 3 references. 5. Deposition Velocities of Non-Newtonian Slurries in Pipelines: Complex Simulant Testing Energy Technology Data Exchange (ETDEWEB) Poloski, Adam P.; Bonebrake, Michael L.; Casella, Andrew M.; Johnson, Michael D.; Toth, James J.; Adkins, Harold E.; Chun, Jaehun; Denslow, Kayte M.; Luna, Maria; Tingey, Joel M. 2009-07-01 One of the concerns expressed by the External Flowsheet Review Team (EFRT) is about the potential for pipe plugging at the Waste Treatment and Immobilization Plant (WTP). Per the review’s executive summary, “Piping that transports slurries will plug unless it is properly designed to minimize this risk. This design approach has not been followed consistently, which will lead to frequent shutdowns due to line plugging.” To evaluate the potential for plugging, deposition-velocity tests were performed on several physical simulants to determine whether the design approach is conservative. Deposition velocity is defined as the velocity below which particles begin to deposit to form a moving bed of particles on the bottom of a straight horizontal pipe during slurry-transport operations. The deposition velocity depends on the system geometry and the physical properties of the particles and fluid. An experimental program was implemented to test the stability-map concepts presented in WTP-RPT-175 Rev. 01. Two types of simulant were tested. The first type of simulant was similar to the glass-bead simulants discussed in WTP-RPT-175 Rev. 0 ; it consists of glass beads with a nominal particle size of 150 µm in a kaolin/water slurry. The initial simulant was prepared at a target yield stress of approximately 30 Pa. The yield stress was then reduced, stepwise, via dilution or rheological modifiers, ultimately to a level of <1 Pa. At each yield-stress step, deposition-velocity testing was performed. Testing over this range of yield-stress bounds the expected rheological operating window of the WTP and allows the results to be compared to stability-map predictions for this system. The second simulant was a precipitated hydroxide that simulates HLW pretreated sludge from Hanford waste tank AZ-101. Testing was performed in a manner similar to that for the first simulant over a wide range of yield stresses; however, an additional test of net-positive suction-head required (NPSHR 6. Interferometer and analysis methods for the in vitro characterization of dynamic fluid layers on contact lenses Science.gov (United States) Primeau, Brian C.; Greivenkamp, John E. 2012-06-01 The anterior refracting surface of the eye when wearing a contact lens is the thin fluid layer that forms on the surface of the contact lens. Under normal conditions, this fluid layer is less than 10 μm thick. The fluid layer thickness and topography change over time and are affected by the material properties of the contact lens and may affect vision quality and comfort. An in vitro method of characterizing dynamic fluid layers applied to contact lenses mounted on mechanical substrates has been developed by use of a phase-shifting Twyman-Green interferometer. This interferometer continuously measures light reflected from the surface of the fluid layer, allowing precision analysis of the dynamic fluid layer. Movies showing this fluid layer behavior can be generated. Quantitative analysis beyond typical contact angle or visual inspection methods is provided. Different fluid and contact lens material combinations have been evaluated, and variations in fluid layer properties have been observed. This paper discusses the interferometer design and analysis methods used. Example measurement results of different contact lens are presented. 7. Asymptotic Laws of Thermovibrational Convecton in a Horizontal Fluid Layer Science.gov (United States) Smorodin, B. L.; Myznikova, B. I.; Keller, I. O. 2017-02-01 Theoretical study of convective instability is applied to a horizontal layer of incompressible single-component fluid subjected to the uniform steady gravity, longitudinal vibrations of arbitrary frequency and initial temperature difference. The mathematical model of thermovibrational convection has the form of initial boundary value problem for the Oberbeck-Boussinesq system of equations. The problems are solved using different simulation strategies, like the method of averaging, method of multiple scales, Galerkin approach, Wentzel-Kramers-Brillouin method and Floquet technique. The numerical analysis has shown that the effect of vibrations on the stability threshold is complex: vibrations can either stabilize or destabilize the basic state depending on values of the parameters. The influence of the Prandtl number on the instability thresholds is investigated. The asymptotic behaviour of critical values of the parameters is studied in two limiting cases: (i) small amplitude and (ii) low frequency of vibration. In case (i), the instability is due to the influence of thermovibrational mechanism on the classical Rayleigh-Benard convective instability. In case (ii), the nature of the instability is related to the instability of oscillating counter-streams with a cubic profile. 8. Drop Characteristics of non-Newtonian Impinging Jets at High Generalized Bird-Carreau Jet Reynolds Numbers Science.gov (United States) Sojka, Paul E.; Rodrigues, Neil S. 2015-11-01 The current study investigates the drop characteristics of three Carboxymethylcellulose (CMC) sprays produced by the impingement of two liquid jets. The three water-based solutions used in this work (0.5 wt.-% CMC-7MF, 0.8 wt.-% CMC-7MF, and 1.4 wt.-% CMC-7MF) exhibited strong shear-thinning, non-Newtonian behavior - characterized by the Bird-Carreau rheological model. A generalized Bird-Carreau jet Reynolds number was used as the primary parameter to characterize the drop size and the drop velocity, which were measured using Phase Doppler Anemometry (PDA). PDA optical configuration enabled a drop size measurement range of approximately 2.3 to 116.2 μm. 50,000 drops were measured at each test condition to ensure statistical significance. The arithmetic mean diameter (D10) , Sauter mean diameter (D32) , and mass median diameter (MMD) were used as representative diameters to characterize drop size. The mean axial drop velocity Uz -mean along with its root-mean square Uz -rms were used to characterize drop velocity. Incredibly, measurements for all three CMC liquids and reference DI water sprays seemed to follow a single curve for D32 and MMD drop diameters in the high generalized Bird-Carreau jet Reynolds number range considered in this work (9.21E +03 9. Thermal convection of viscoelastic shear-thinning fluids Science.gov (United States) Albaalbaki, Bashar; Khayat, Roger E.; Ahmed, Zahir U. 2016-12-01 The Rayleigh-Bénard convection for non-Newtonian fluids possessing both viscoelastic and shear-thinning behaviours is examined. The Phan-Thien-Tanner (PTT) constitutive equation is implemented to model the non-Newtonian character of the fluid. It is found that while the shear-thinning and viscoelastic effects could annihilate one another for the steady roll flow, presence of both behaviours restricts the roll stability limit significantly compared to the cases when the fluid is either inelastic shear-thinning or purely viscoelastic with constant viscosity. 10. Proteins at fluid interfaces: adsorption layers and thin liquid films. Science.gov (United States) Yampolskaya, Galina; Platikanov, Dimo 2006-12-21 A review in which many original published results of the authors as well as many other papers are discussed. The structure and some properties of the globular proteins are shortly presented, special accent being put on the alpha-chymotrypsin (alpha-ChT), lysozyme (LZ), human serum albumin (HSA), and bovine serum albumin (BSA) which have been used in the experiments with thin liquid films. The behaviour of protein adsorption layers (PAL) is extensively discussed. The dynamics of PAL formation, including the kinetics of adsorption as well as the time evolution of the surface tension of protein aqueous solutions, are considered. A considerable place is devoted to the surface tension and adsorption isotherms of the globular protein solutions, the simulation of PAL by interacting hard spheres, the experimental surface tension isotherms of the above mentioned proteins, and the interfacial tension isotherms for the protein aqueous solution/oil interface. The rheological properties of PAL at fluid interfaces are shortly reviewed. After a brief information about the experimental methods for investigation of protein thin liquid (foam or emulsion) films, the properties of the protein black foam films are extensively discussed: the conditions for their formation, the influence of the electrolytes and pH on the film type and stability, the thermodynamic properties of the black foam films, the contact angles film/bulk and their dynamic hysteresis. The next center of attention concerns some properties of the protein emulsion films: the conditions for formation of emulsion black films, the formation and development of a dimpling in microscopic, circular films. The protein-phospholipid mixed foam films are also briefly considered. 11. The impact of lipid composition on the stability of the tear fluid lipid layer DEFF Research Database (Denmark) Kulovesi, P.; Telenius, J.; Koivuniemi, A. 2012-01-01 The tear fluid protects the corneal epithelium from drying and pathogens and it also provides nutrients to these cells. Tear fluid is composed of an aqueous layer as well as a lipid layer that resides at the air-tear interface. The function of the lipid layer is to lower the surface tension of th......-neutral lipid ratio. The results provide a plausible rationale for the development of dry eye syndrome in blepharitis patients.... 12. An Analysis of the Characteristics of the Thermal Boundary Layer in Power Law Fluid Institute of Scientific and Technical Information of China (English) 2008-01-01 This paper presents a theoretical analysis of the heat transfer for the boundary layer flow on a continuous moving surface in power law fluid. The expressions of the thermal boundary layer thickness with the different heat conductivity coefficients are obtained according to the theory of the dimensional analysis of fluid dynamics and heat transfer. And the numerical results of CFD agree well with the proposed expressions. The estimate formulas can be successfully applied to giving the thermal boundary layer thickness. 13. Casson fluid flow over an Directory of Open Access Journals (Sweden) 2013-12-01 Full Text Available The unsteady two-dimensional flow of a non-Newtonian fluid over a stretching surface having a prescribed surface temperature is investigated. The Casson fluid model is used to characterise the non-Newtonian fluid behaviour. Similarity transformations are employed to transform the governing partial differential equations into ordinary differential equations. The transformed equations are then solved numerically by shooting method. Exact solution corresponding to momentum equation for steady case is obtained. The flow features and heat transfer characteristics for different values of the governing parameters viz. unsteadiness parameter, Casson parameter and Prandtl number are analysed and discussed in detail. Fluid velocity initially decreases with increasing unsteadiness parameter and temperature decreases significantly due to unsteadiness. The effect of increasing values of the Casson parameter is to suppress the velocity field. But the temperature is enhanced with increasing Casson parameter. 14. Autowaves in near-surface layer of magnetic fluid Energy Technology Data Exchange (ETDEWEB) Chekanov, V.V. [Stavropol State University, 1 Pushkin st., Stavropol 355009 (Russian Federation)]. E-mail: [email protected]; Iljuch, P.M. [Stavropol Branch of Moskow Apparatus Bilding and Informatics Academy, 25 Kulakova st., Stavropol 355000 (Russian Federation); Kandaurova, N.V. [Stavropol Branch of Moskow Apparatus Bilding and Informatics Academy, 25 Kulakova st., Stavropol 355000 (Russian Federation); Bondarenko, E.A. [Stavropol State University, 1 Pushkin st., Stavropol 355009 (Russian Federation) 2005-03-15 Autowaves processes in a magnetic fluids in an electric and magnetic fields has been investigated. Different type of autowaves in near-surface region of electrochemical cell has been found. Equation of autowave process has been described. 15. Gelled propellant flow: Boundary layer theory for power-law fluids in a converging planar channel Science.gov (United States) Kraynik, Andrew M.; Geller, A. S.; Glick, J. H. 1989-10-01 A boundary layer theory for the flow of power-law fluids in a converging planar channel has been developed. This theory suggests a Reynolds number for such flows, and following numerical integration, a boundary layer thickness. This boundary layer thickness has been used in the generation of a finite element mesh for the finite element code FIDAP. FIDAP was then used to simulate the flow of power-law fluids through a converging channel. Comparison of the analytic and finite element results shows the two to be in very good agreement in regions where entrance and exit effects (not considered in the boundary layer theory) can be neglected. 16. Oscillatory and Steady Flows in the Annular Fluid Layer inside a Rotating Cylinder Directory of Open Access Journals (Sweden) Veronika Dyakova 2016-01-01 Full Text Available The dynamics of a low-viscosity fluid inside a rapidly rotating horizontal cylinder were experimentally studied. In the rotating frame, the force of gravity induces azimuthal fluid oscillations at a frequency equal to the velocity of the cylinder’s rotation. This flow is responsible for a series of phenomena, such as the onset of centrifugal instability in the Stokes layer and the growth of the relief at the interface between the fluid and the granular medium inside the rotating cylinder. The phase inhomogeneity of the oscillatory fluid flow in the viscous boundary layers near the rigid wall and the free surface generates the azimuthal steady streaming. We studied the relative contribution of the viscous boundary layers in the generation of the steady streaming. It is revealed that the velocity of the steady streaming can be calculated using the velocity of the oscillatory fluid motion. 17. Symmetries of boundary layer equations of power-law fluids of second grade Institute of Scientific and Technical Information of China (English) Mehmet Pakdemirli; Yi(g)it Aksoy; Muhammet Y(u)r(u)soy; Chaudry Masood Khalique 2008-01-01 A modified power-law fluid of second grade is considered. The model is a combination of power-law and second grade fluid in which the fluid may exhibit normal stresses, shear thinning or shear thickening behaviors. The equations of motion are derived for two dimensional incom-pressible flows, and from which the boundary layer equations are derived. Symmetries of the boundary layer equations are found by using Lie group theory, and then group classifica-tion with respect to power-law index is performed. By using one of the symmetries, namely the scaling symmetry, the partial differential system is transformed into an ordinary differential system, which is numerically integrated under the classical boundary layer conditions. Effects of power-law index and second grade coefficient on the boundary layers are shown and solutions are contrasted with the usual second grade fluid solutions. 18. Absorbing Backside Anti-reflecting Layers for high contrast imaging in fluid cells CERN Document Server Ausserré, Dominique; Amra, Claude; Zerrad, Myriam 2014-01-01 The single Anti-Reflecting (AR) layer is a classical problem in optics. When all materials are pure dielectrics, the solution is the so-called lambda/4 layer. Here we examine the case of absorbing layers between non absorbing media. We find a solution for any layer absorption coefficient provided that the light goes from the higher towards the lower index medium, which characterizes backside layers. We describe these AR absorbing (ARA) layers through generalized index and thickness conditions. They are most often ultrathin, and have important applications for high contrast imaging in fluid cells. 19. An active particle in a complex fluid CERN Document Server Datt, Charu; Hatzikiriakos, Savvas; Elfring, Gwynn J 2016-01-01 In this work, we study active particles with prescribed surface velocities in non-Newtonian fluids. We employ the reciprocal theorem to derive a general form of the propulsion velocity of a single active particle (or swimmer) in a weakly non-Newtonian background flow in the absence of inertia. Using this formulation, we obtain the velocity of an active spherical particle with an arbitrary axisymmetric slip-velocity in an otherwise quiescent second-order fluid. Finally, we determine how the motion of a diffusiophoretic Janus particle is affected by complex fluid rheology, namely viscoelasticity and shear-thinning. We find that a Janus particle may go faster or slower in a viscoelastic fluid, but is always slower in a shear-thinning fluid as compared to a Newtonian fluid. 20. Enhanced active swimming in viscoelastic fluids CERN Document Server Riley, Emily E 2014-01-01 Swimming microorganisms often self propel in fluids with complex rheology. While past theoretical work indicates that fluid viscoelasticity should hinder their locomotion, recent experiments on waving swimmers suggest a possible non-Newtonian enhancement of locomotion. We suggest a physical mechanism, based on fluid-structure interaction, leading to swimming in a viscoelastic fluid at a higher speed than in a Newtonian one. Using Taylor's two-dimensional swimming sheet model, we solve for the shape of an active swimmer as a balance between the external fluid stresses, the internal driving moments, and the passive elastic resistance. We show that this dynamic balance leads to a generic transition from hindered rigid swimming to enhanced flexible locomotion. The results are physically interpreted as due to a viscoelastic suction increasing the swimming amplitude in a non-Newtonian fluid and overcoming viscoelastic damping. 1. Membranes having aligned 1-D nanoparticles in a matrix layer for improved fluid separation Energy Technology Data Exchange (ETDEWEB) Revanur, Ravindra; Lulevich, Valentin; Roh, Il Juhn; Klare, Jennifer E.; Kim, Sangil; Noy, Aleksandr; Bakajin, Olgica 2015-12-22 Membranes for fluid separation are disclosed. These membranes have a matrix layer sandwiched between an active layer and a porous support layer. The matrix layer includes 1-D nanoparticles that are vertically aligned in a porous polymer matrix, and which substantially extend through the matrix layer. The active layer provides species-specific transport, while the support layer provides mechanical support. A matrix layer of this type has favorable surface morphology for forming the active layer. Furthermore, the pores that form in the matrix layer tend to be smaller and more evenly distributed as a result of the presence of aligned 1-D nanoparticles. Improved performance of separation membranes of this type is attributed to these effects. 2. Curvilinear Squeeze Film Bearing with Porous Wall Lubricated by a Rabinowitsch Fluid Science.gov (United States) Walicka, A.; Walicki, E.; Jurczak, P.; Falicki, J. 2017-05-01 The present theoretical analysis is to investigate the effect of non-Newtonian lubricant modelled by a Rabinowitsch fluid on the performance of a curvilinear squeeze film bearing with one porous wall. The equations of motion of a Rabinowitsch fluid are used to derive the Reynolds equation. After general considerations on the flow in a bearing clearance and in a porous layer using the Morgan-Cameron approximation the modified Reynolds equation is obtained. The analytical solution of this equation for the case of a squeeze film bearing is presented. As a result one obtains the formulae expressing pressure distribution and load-carrying capacity. Thrust radial bearing and spherical bearing with a squeeze film are considered as numerical examples. 3. THE WAVE-MAKING CHARACTERISTICS OF A MOVING BODY IN A TWO-LAYER FLUID Institute of Scientific and Technical Information of China (English) ZHU Wei 2005-01-01 The Wave-making characteristics of a moving body in a two-layer fluid with free surface is investigated numerically and experimentally. The numerical analysis is based on the modified layered boundary integral equation system. The wave characteristics on the free surface and interface generated by a moving sphere and an ellipsoid is numerically simulated in both finite depth and infinite depth of lower layer model. The numerical results of the sphere are compared with the analytical results for a dipole with the same velocity in a two-layer fluid of finite depth. The dependence of the wave systems and structures on the characteristic quantities is discussed. Three kinds of measurement techniques are used in model experiments on the internal waves generated by a sphere advancing in a two-layer fluid. The effects of the varying velocity and stratification on the wavelength, wave amplitudes and the maximum half angles of internal waves are analyzed qualitatively. 4. Mathematical modeling for laminar flow of power law fluid in porous media Energy Technology Data Exchange (ETDEWEB) Silva, Renato A.; Mesquita, Maximilian S. [Universidade Federal do Espirito Santo (UFES), Sao Mateus, ES (Brazil). Centro Universitario Norte do Espirito Santo. Dept. de Engenharias e Computacao 2010-07-01 In this paper, the macroscopic equations for laminar power-law fluid flow is obtained for a porous medium starting from traditional equations (Navier-Stokes). Then, the volume averaging is applied in traditional transport equations with the power-law fluid model. This procedure leads to macroscopic transport equations set for non-Newtonian fluid. (author) 5. Global chaotization of fluid particle trajectories in a sheared two-layer two-vortex flow Energy Technology Data Exchange (ETDEWEB) Ryzhov, Evgeny A., E-mail: [email protected] [Pacific Oceanological Institute of FEB RAS, 43, Baltiyskaya Street, Vladivostok 690041 (Russian Federation); Koshel, Konstantin V., E-mail: [email protected] [Pacific Oceanological Institute of FEB RAS, 43, Baltiyskaya Street, Vladivostok 690041 (Russian Federation); Far Eastern Federal University, 8, Sukhanova Street, Vladivostok 690950 (Russian Federation) 2015-10-15 In a two-layer quasi-geostrophic approximation, we study the irregular dynamics of fluid particles arising due to two interacting point vortices embedded in a deformation flow consisting of shear and rotational components. The two vortices are arranged within the bottom layer, but an emphasis is on the upper-layer fluid particle motion. Vortices moving in one layer induce stirring of passive scalars in the other layer. This is of interest since point vortices induce singular velocity fields in the layer they belong to; however, in the other layer, they induce regular velocity fields that generally result in a change in passive particle stirring. If the vortices are located at stagnation points, there are three different types of the fluid flow. We examine how properties of each flow configuration are modified if the vortices are displaced from the stagnation points and thus circulate in the immediate vicinity of these points. To that end, an analysis of the steady-state configurations is presented with an emphasis on the frequencies of fluid particle oscillations about the elliptic stagnation points. Asymptotic relations for the vortex and fluid particle zero–oscillation frequencies are derived in the vicinity of the corresponding elliptic points. By comparing the frequencies of fluid particles with the ones of the vortices, relations between the parameters that lead to enhanced stirring of fluid particles are established. It is also demonstrated that, if the central critical point is elliptic, then the fluid particle trajectories in its immediate vicinity are mostly stable making it harder for the vortex perturbation to induce stirring. Change in the type of the central point to a hyperbolic one enhances drastically the size of the chaotic dynamics region. Conditions on the type of the central critical point also ensue from the derived asymptotic relations. 6. A source-sink model of the generation of plate tectonics from non-Newtonian mantle flow Science.gov (United States) Bercovici, David 1995-01-01 A model of mantle convection which generates plate tectonics requires strain rate- or stress-dependent rheology in order to produce strong platelike flows with weak margins as well as strike-slip deformation and plate spin (i.e., toroidal motion). Here, we employ a simple model of source-sink driven surface flow to determine the form of such a rheology that is appropriate for Earth's present-day plate motions. In this model, lithospheric motion is treated as shallow layer flow driven by sources and sinks which correspond to spreading centers and subduction zones, respectively. Two plate motion models are used to derive the source sink field. As originally implied in the simpler Cartesian version of this model, the classical power law rheologies do not generate platelike flows as well as the hypothetical Whitehead-Gans stick-slip rheology (which incorporates a simple self-lubrication mechanism). None of the fluid rheologies examined, however, produce more than approximately 60% of the original maximum shear. For either plate model, the viscosity fields produced by the power law rheologies are diffuse, and the viscosity lows over strike-slip shear zones or pseudo-margins are not as small as over the prescribed convergent-divergent margins. In contrast, the stick-slip rheology generates very platelike viscosity fields, with sharp gradients at the plate boundaries, and margins with almost uniformly low viscosity. Power law rheologies with high viscosity contrasts, however, lead to almost equally favorable comparisons, though these also yield the least platelike viscosity fields. This implies that the magnitude of toroidal flow and platelike strength distributions are not necessarily related and thus may present independent constraints on the determination of a self-consistent plate-mantle rheology. 7. Computational Fluid Dynamics model of stratified atmospheric boundary-layer flow DEFF Research Database (Denmark) Koblitz, Tilman; Bechmann, Andreas; Sogachev, Andrey; 2015-01-01 For wind resource assessment, the wind industry is increasingly relying on computational fluid dynamics models of the neutrally stratified surface-layer. So far, physical processes that are important to the whole atmospheric boundary-layer, such as the Coriolis effect, buoyancy forces and heat... 8. Patterns of gravity induced aggregate migration during casting of fluid concretes Energy Technology Data Exchange (ETDEWEB) Spangenberg, J. [Department of Mechanical Engineering, Technical University of Denmark (DTU) (Denmark); Roussel, N., E-mail: [email protected] [Universite Paris Est, Laboratoire Central des Ponts et Chaussees (LCPC) (France); Hattel, J.H. [Department of Mechanical Engineering, Technical University of Denmark (DTU) (Denmark); Sarmiento, E.V.; Zirgulis, G. [Department of Structural Engineering, Norwegian University of Science and Technology (NTNU) (Norway); Geiker, M.R. [Department of Structural Engineering, Norwegian University of Science and Technology (NTNU) (Norway); Department of Civil Engineering, Technical University of Denmark (DTU) (Denmark) 2012-12-15 In this paper, aggregate migration patterns during fluid concrete castings are studied through experiments, dimensionless approach and numerical modeling. The experimental results obtained on two beams show that gravity induced migration is primarily affecting the coarsest aggregates resulting in a decrease of coarse aggregates volume fraction with the horizontal distance from the pouring point and in a puzzling vertical multi-layer structure. The origin of this multi layer structure is discussed and analyzed with the help of numerical simulations of free surface flow. Our results suggest that it finds its origin in the non Newtonian nature of fresh concrete and that increasing casting rate shall decrease the magnitude of gravity induced particle migration. 9. Steady internal waves in an exponentially stratified two-layer fluid Science.gov (United States) Makarenko, Nikolay; Maltseva, Janna; Ivanova, Kseniya 2016-04-01 The problem on internal waves in a weakly stratified two-layered fluid is studied analytically. We suppose that the fluid possess exponential stratification in both the layers, and the fluid density has discontinuity jump at the interface. By that, we take into account the influence of weak continuous stratification outside of sharp pycnocline. The model equation of strongly nonlinear interfacial waves propagating along the pycnocline is considered. This equation extends approximate models [1-3] suggested for a two-layer fluid with one homogeneous layer. The derivation method uses asymptotic analysis of fully nonlinear Euler equations. The perturbation scheme involves the long wave procedure with a pair of the Boussinesq parameters. First of these parameters characterizes small density slope outside of pycnocline and the second one defines small density jump at the interface. Parametric range of solitary wave solutions is characterized, including extreme regimes such as plateau-shape solitary waves. This work was supported by RFBR (grant No 15-01-03942). References [1] N. Makarenko, J. Maltseva. Asymptotic models of internal stationary waves, J. Appl. Mech. Techn. Phys, 2008, 49(4), 646-654. [2] N. Makarenko, J. Maltseva. Phase velocity spectrum of internal waves in a weakly-stratified two-layer fluid, Fluid Dynamics, 2009, 44(2), 278-294. [3] N. Makarenko, J. Maltseva. An analytical model of large amplitude internal solitary waves, Extreme Ocean Waves, 2nd ed. Springer 2015, E.Pelinovsky and C.Kharif (Eds), 191-201. 10. PIV measurement of a droplet impact on a thin fluid layer Science.gov (United States) Ninomiya, Nao; Iwamoto, Kazuya 2012-03-01 Upon the impact of a droplet onto a thin fluid layer, the fluid is pushed away around the impact point until it reaches a certain radius to go upward to form a thin liquid wall. At the tip of the liquid wall, the circumferential instability creates a several droplets, which is commonly known as "milk crown". The size and the height of the crown and the number of the tip droplets are affected by the conditions of a droplet and of a fluid layer. Presently, the fundamental characteristics of the milk crown have been extensively investigated based on the flow visualization by the high-speed camera. Moreover, the motions of the fluid layer and the liquid film have been measured with the aid of PIV. The result reveals several interesting features of the formation of a milk crown. 11. Free surface simulation of a two-layer fluid by boundary element method Directory of Open Access Journals (Sweden) Weoncheol Koo 2010-09-01 Full Text Available A two-layer fluid with free surface is simulated in the time domain by a two-dimensional potential-based Numerical Wave Tank (NWT. The developed NWT is based on the boundary element method and a leap-frog time integration scheme. A whole domain scheme including interaction terms between two layers is applied to solve the boundary integral equation. The time histories of surface elevations on both fluid layers in the respective wave modes are verified with analytic results. The amplitude ratios of upper to lower elevation for various density ratios and water depths are also compared. 12. Plasma interfacial mixing layers: Comparisons of fluid and kinetic models Science.gov (United States) Vold, Erik; Yin, Lin; Taitano, William; Albright, B. J.; Chacon, Luis; Simakov, Andrei; Molvig, Kim 2016-10-01 We examine plasma transport across an initial discontinuity between two species by comparing fluid and kinetic models. The fluid model employs a kinetic theory approximation for plasma transport in the limit of small Knudsen number. The kinetic simulations include explicit particle-in-cell simulations (VPIC) and a new implicit Vlasov-Fokker-Planck code, iFP. The two kinetic methods are shown to be in close agreement for many aspects of the mixing dynamics at early times (to several hundred collision times). The fluid model captures some of the earliest time dynamic behavior seen in the kinetic results, and also generally agrees with iFP at late times when the total pressure gradient relaxes and the species transport is dominated by slow diffusive processes. The results show three distinct phases of the mixing: a pressure discontinuity forms across the initial interface (on times of a few collisions), the pressure perturbations propagate away from the interfacial mixing region (on time scales of an acoustic transit) and at late times the pressure relaxes in the mix region leaving a non-zero center of mass flow velocity. The center of mass velocity associated with the outward propagating pressure waves is required to conserve momentum in the rest frame. Work performed under the auspices of the U.S. DOE by the LANS, LLC, Los Alamos National Laboratory under Contract No. DE-AC52-06NA25396. Funding provided by the Advanced Simulation and Computing (ASC) Program. 13. Steady flow for shear thickening fluids in domains with unbounded sections Science.gov (United States) Dias, Gilberlandio J. 2017-02-01 We solve the stationary Stokes and Navier-Stokes equations for non-Newtonian incompressible fluids with shear dependent viscosity in domains with outlets containing unbounded cross sections, in the case of shear thickening viscosity. The flux assumes arbitrary given values and the growth of the cross sections are analyzed under different convergence hypotheses, inclusive the growth of Dirichlet's integral of the velocity field is deeply related the convergence hypotheses of such sections. We extend the results of the section 4 of [12, Ladyzhenskaya and Solonnikov] (for Newtonian fluids) to non-Newtonian fluids using the techniques found in [3, Dias and Santos]. 14. Diffusion of chemically reactive species in Casson fluid flow over an unsteady permeable stretching surface Institute of Scientific and Technical Information of China (English) 2013-01-01 In this paper we investigate the two-dimensional flow of a non-Newtonian fluid over an unsteady stretching permeable surface.The Casson fluid model is used to characterize the non-Newtonian fluid behavior.First-order constructive/destructive chemical reaction is considered.With the help of a shooting method,numerical solutions for a class of nonlinear coupled differential equations subject to appropriate boundary conditions are obtained.For the steady flow,the exact solution is obtained.The flow features and the mass transfer characteristics for different values of the governing parameters are analyzed and discussed in detail. 15. Financial Brownian Particle in the Layered Order-Book Fluid and Fluctuation-Dissipation Relations Science.gov (United States) Yura, Yoshihiro; Takayasu, Hideki; Sornette, Didier; Takayasu, Misako 2014-03-01 We introduce a novel description of the dynamics of the order book of financial markets as that of an effective colloidal Brownian particle embedded in fluid particles. The analysis of comprehensive market data enables us to identify all motions of the fluid particles. Correlations between the motions of the Brownian particle and its surrounding fluid particles reflect specific layering interactions; in the inner layer the correlation is strong and with short memory, while in the outer layer it is weaker and with long memory. By interpreting and estimating the contribution from the outer layer as a drag resistance, we demonstrate the validity of the fluctuation-dissipation relation in this nonmaterial Brownian motion process. 16. Simulation of Variable Viscosity and Jeffrey Fluid Model for Blood Flow Through a Tapered Artery with a Stenosis Institute of Scientific and Technical Information of China (English) 2012-01-01 Non-Newtonian fluid model for blood flow through a tapered artery with a stenosis and variable viscosity by modeling blood as Jeffrey fluid has been studied in this paper. The Jeffrey fluid has two parameters, the relaxation time A1 and retardation time A2. The governing equations are simplified using the case of mild stenosis. Perturbation method is used to solve the resulting equations. The effects of non-Newtonian nature of blood on velocity profile, temperature profile, wall shear stress, shearing stress at the stenotsis throat and impedance of the artery are discussed. The results for Newtonian fluid are obtained as special case from this model. 17. Linear waves in two-layer fluids over periodic bottoms NARCIS (Netherlands) Yu, J.; Maas, L.R.M. 2016-01-01 A new, exact Floquet theory is presented for linear waves in two-layer fluidsover a periodic bottom of arbitrary shape and amplitude. A method of conformaltransformation is adapted. The solutions are given, in essentially analytical form, forthe dispersion relation between wave frequency and general 18. Phase-separation models for swimming enhancement in complex fluids CERN Document Server Man, Yi 2015-01-01 Swimming cells often have to self-propel through fluids displaying non-Newtonian rheology. While past theoretical work seems to indicate that stresses arising from complex fluids should systematically hinder low-Reynolds number locomotion, experimental observations suggest that locomotion enhancement is possible. In this paper we propose a physical mechanism for locomotion enhancement of microscopic swimmers in a complex fluid. It is based on the fact that micro-structured fluids will generically phase-separate near surfaces, leading to the presence of low-viscosity layers which promote slip and decrease viscous friction near the surface of the swimmer. We use two models to address the consequence of this phase separation: a nonzero apparent slip length for the fluid and then an explicit modeling of the change of viscosity in a thin layer near the swimmer. Considering two canonical setups for low-Reynolds number locomotion, namely the waving locomotion of a two-dimensional sheet and that of a three-dimensiona... 19. Simulating the Dynamic Behavior of Shear Thickening Fluids CERN Document Server Ozgen, Oktar; Brown, Eric 2015-01-01 While significant research has been dedicated to the simulation of fluids, not much attention has been given to exploring new interesting behavior that can be generated with the different types of non-Newtonian fluids with non-constant viscosity. Going in this direction, this paper introduces a computational model for simulating the interesting phenomena observed in non-Newtonian shear thickening fluids, which are fluids where the viscosity increases with increased stress. These fluids have unique and unconventional behavior, and they often appear in real world scenarios such as when sinking in quicksand or when experimenting with popular cornstarch and water mixtures. While interesting behavior of shear thickening fluids can be easily observed in the real world, the most interesting phenomena of these fluids have not been simulated before in computer graphics. The fluid exhibits unique phase changes between solid and liquid states, great impact resistance in its solid state and strong hysteresis effects. Our... 20. VIBRATION CONTROL OF FLUID- FILLED PRISMATIC SHELL WITH ACTIVE CONSTRAINED LAYER DAMPING TREATMENTS Institute of Scientific and Technical Information of China (English) LIU Lijun; ZHANG Zhiyi; HUA Hongxing; ZHANG Yi 2008-01-01 Active constrained layer damping (ACLD) combines the simplicity and reliability of passive damping with the light weight and high efficiency of active actuators to obtain high damping over a wide frequency band. A fluid-filled prismatic shell is set up to investigate the validity and efficiency of ACLD treatments in the case of fluid-structure interaction. By using state subspace identification method, modal parameters of the ACLD system are identified and a state space model is established subsequently for the design of active control laws. Experiments are conducted to the fluid-filled prismatic shell subjected to random and impulse excitation, respectively. For comparison, the shell model without fluid interaction is experimented as well. Experimental results have shown that the ACLD treatments can suppress vibration of the fluid-free and fluid-filled prismatic shell effectively. Under the same control gain, vibration attenuation is almost the same in both cases.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8113422989845276, "perplexity": 2059.8411530129556}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891815560.92/warc/CC-MAIN-20180224112708-20180224132708-00477.warc.gz"}
https://www.physicsforums.com/threads/3d-rectangular-scattering-potential.546188/	# Homework Help: 3D Rectangular Scattering Potential 1. Nov 1, 2011 ### qmphysics 1. The problem statement, all variables and given/known data There is a constant potential, V0, in the region -a<x<a, -a<y<a, -b<z<b, and V=0 otherwise. Particles of mass m are incident on the scatterer with wave vector k in the x direction with a flux at the origin of one particle per second per cm^2. There is a detector with cross-sectional area A is located at a large distance R from the origin, in the direction with polar coordinates (theta, phi). Assumptions: a<<b, ka<<1, V0<<hbar^2/(mb^2), k*b not necessarily small. We want to find the counting rate in the detector. The problem also asks how the answer would change if the k vector was parallel to z instead. 2. Relevant equations It seems that due to the small potential, we can use the Born approximation, which gives the scattering amplitude $f(k\prime,k)=-\frac{m}{2 \pi \hbar^2}\int exp(i (k-k\prime)\cdot r\prime) V(r\prime) d^{3}r\prime)$. 3. The attempt at a solution In spherical coordinates, the Born approximation isn't bad to calculate. However, in this rectangular case, I can't get a nice looking result out of the integration above. I'm interested to see if anyone thinks there is a different way to do the problem, or if I should just keep trying the integral. As for the second part of the problem, it doesn't seem like the Born approximation would necessarily hold when the particle sees a longer range potential. But hopefully this part will become more clear when I can figure out the first part. Thanks for your help! Share this great discussion with others via Reddit, Google+, Twitter, or Facebook Can you offer guidance or do you also need help? Draft saved Draft deleted	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9323148131370544, "perplexity": 493.72288010662714}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-34/segments/1534221215404.70/warc/CC-MAIN-20180819224020-20180820004020-00583.warc.gz"}
https://support.mozilla.org/zh-TW/questions/1365858	## A Very Important Parent Bookmark Disappeared Again! • 5 回覆 • 1 有這個問題 • 25 次檢視 • 最近回覆由 A very important parent Bookmark folder disappeared again! When I do a Search, I can find Bookmarks that are in the Parent Bookmarks folder but as I have reported before, the HUGE issue is that there is NO way to find where the parent Bookmarks folder or the Bookmarks in the parent Bookmarks folder using the Search function! How can I find the parent Bookmarks folder and where the Bookmarks are that were in the Parent Bookmarks folder? I have added many Bookmarks since yesterday so I do not want to try to do a Restore. I need to find the Bookmarks that disappeared soon. #### 被選擇的解決方法 Note that in Firefox 96 you can simply right-click a bookmarks search result and click "Show in Folder", so you no longer need to use an extension. ### 所有回覆 (5) One way is to use the bookmark to open that page. Then click the bookmarks star on the toolbar. Search for both bookmarks and folders. Find exact location in bookmark tree (show parent folder, go parent folder). Advanced filters and searches, use regex. #### 有幫助嗎？ Using the bookmark to open that page, then clicking the bookmarks star on the toolbar does not show the actual location where the Bookmark is. I added Bookmark Search Plus 2 but and ran it neither does it show the actual location of the searched Bookmark. All I see is the the tree below with no indication where the Searched Bookmark is actually located. How can I find the actual location where the Bookmarks are that disappeared? It appears that regex is complicated to use. There should be an easy way to find Bookmarks that disappear! In the first place, Bookmarks should NOT disappear. This has happened too many times. ### 選擇的解決方法 Note that in Firefox 96 you can simply right-click a bookmarks search result and click "Show in Folder", so you no longer need to use an extension. #### 有幫助嗎？ I was able to find and move the Bookmark folder to the correct place using the new feature in Firefox 96! This is a very useful feature. Thanks #### 有幫助嗎？ An important Bookmark disappeared again! I found it using this feature. This problem should NOT be occurring, especially so frequently. The missing Bookmark was "Equipment" which was in the root of "From Internet Explorer" and was found in Government/U.S./COVID-19/Churches/.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8364282250404358, "perplexity": 3530.0422685647873}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662530553.34/warc/CC-MAIN-20220519235259-20220520025259-00787.warc.gz"}
https://www.physicsforums.com/threads/is-the-energy-generated-from-matter-antimatter-annihilation-useful.730672/	Is the energy generated from matter-antimatter annihilation useful? 1. Dec 31, 2013 anj16 I understand that when matter and antimatter come in contact with each other, they annihilate to generate either gamma rays or other particles such as pions, electrons, etc. So my question is, how do we use the gamma rays or the particles to say move a spaceship from one point to another? (?generate momentum?) 2. Dec 31, 2013 DennisN Before that question is considered, I'd suggest you consider these questions: 1. How is antimatter produced? 2. How much antimatter has been produced by scientists up until today? 3. How is antimatter stored? http://en.wikipedia.org/wiki/Antimatter Check out e.g. "Preservation" and "Cost". Last edited: Dec 31, 2013 Draft saved Draft deleted Similar Discussions: Is the energy generated from matter-antimatter annihilation useful?	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9759762287139893, "perplexity": 1426.7186803857182}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-40/segments/1474738661768.23/warc/CC-MAIN-20160924173741-00264-ip-10-143-35-109.ec2.internal.warc.gz"}
http://clay6.com/qa/18243/the-solution-set-of-the-equation-begin1-4-20-1-2-5-1-2x-5x-2-end-0-is	Browse Questions # The solution set of the equation $\begin{vmatrix}1 &4&20\\1 &-2&5\\1 &2x&5x^2\end{vmatrix}=0$ is (A) 1,2 (B) 1,0 (C) -1,2 (D) None of these Given equation is $\begin{vmatrix}1 &4&20\\1&-2&5\\1&2x&5x^2\end{vmatrix}=0$ On expanding the determinant we get a quadratic equation in $x$. It has two roots. We observe that $R_3$ becomes identical to $R_1$ if $x=2$ thus at $x=2$ $\Rightarrow \Delta =0$ $\therefore x=2$ is a root of the equation . Similarly $R_3$ becomes identical to $R_2$ if $x=-1$ At $x=-1$ is a root of the given equation . Hence equation has roots as -1 and 2 Hence (c) is the correct answer.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9892510771751404, "perplexity": 530.7356549725652}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-04/segments/1484560280086.25/warc/CC-MAIN-20170116095120-00399-ip-10-171-10-70.ec2.internal.warc.gz"}
http://quant.stackexchange.com/questions/14630/how-can-i-break-down-the-change-in-value-for-an-inflation-linked-bond	# How can I break down the change in value for an inflation-linked bond I am trying to decompose the change in value of an inflation-linked bond into two constituent parts: 1) That due to changing nominal rates on the issuer's non-linked bonds 2) That due to changing inflation	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9082878828048706, "perplexity": 1799.8138778024033}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-22/segments/1432207928078.25/warc/CC-MAIN-20150521113208-00000-ip-10-180-206-219.ec2.internal.warc.gz"}
http://faculty.washington.edu/lind/recent-publications/	# Recent Papers #### These papers can be downloaded from the arXiv by clicking their link • ###### A Survey of Algebraic Actions of the Discrete Heisenberg Group (with Klaus Schmidt) Abstract: The study of actions of countable groups by automorphisms of compact abelian groups has recently undergone intensive development, revealing deep connections with operator algebras and other areas. The discrete Heisenberg group is the simplest noncommutative example, where dynamical phenomena related to its noncommutativity already illustrate many of these connections. The explicit structure of this group means that these phenomena have concrete descriptions, which are not only instances of the general theory but are also testing grounds for further work. We survey here what is known about such actions of the discrete Heisenberg group, providing numerous examples and emphasizing many of the open problems that remain. • ###### Homoclinic Points, Atoral Polynomials, and Periodic Points of Algebraic $$\mathbb{Z}^d$$-actions (with Klaus Schmidt and Evgeny Verbitskiy) Abstract: Cyclic algebraic $$\mathbb{Z}^d$$-actions are defined by ideals of Laurent polynomials in $$d$$ commuting variables. Such an action is expansive precisely when the complex variety of the ideal is disjoint from the multiplicative $$d$$-torus. For such expansive actions it is known that the limit for the growth rate of periodic points exists and is equal to the entropy of the action. In an earlier paper the authors extended this result to ideals whose variety intersects the $$d$$-torus in a finite set. Here we further extend it to the case when the dimension of intersection of the variety with the $$d$$-torus is at most $$d-2$$. The main tool is the construction of homoclinic points which decay rapidly enough to be summable. • ###### Entropy and Growth Rate of Periodic Points of Algebraic $$\mathbb{Z}^d$$-actions (with Klaus Schmidt and Evgeny Verbitskiy) Abstract: Expansive algebraic $$\mathbb{Z}^d$$-actions corresponding to ideals are characterized by the property that the complex variety of the ideal is disjoint from the multiplicative unit torus. For such actions it is known that the limit for the growth rate of periodic points exists and equals the entropy of the action. We extend this result to actions for which the complex variety intersects the multiplicative torus in a finite set. The main technical tool is the use of homoclinic points which decay rapidly enough to be summable. • ###### Non-archimedean Amoebas and Tropical Varieties (with Mikhail Kapranov and Manfred Einsiedler) Abstract: We study the non-archimedean counterpart to the complex amoeba of an algebraic variety, and show that it coincides with a polyhedral set defined by Bieri and Groves using valuations. For hypersurfaces this set is also the tropical variety of the defining polynomial. Using non-archimedean analysis and a recent result of Conrad we prove that the amoeba of an irreducible variety is connected. We introduce the notion of an adelic amoeba for varieties over global fields, and establish a form of the local-global principle for them. This principle is used to explain the calculation of the nonexpansive set for a related dynamical system. • ###### Lehmer’s Problem for Compact Abelian Groups Abstract: We formulate Lehmer’s Problem about the Mahler measure of polynomials for general compact abelian groups, introducing a Lehmer constant for each such group. We show that all nontrivial connected compact groups have the same Lehmer constant, and conjecture the value of the Lehmer constant for finite cyclic groups. We also show that if a group has infinitely many connected components then its Lehmer constant vanishes.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9258160591125488, "perplexity": 371.20084397355316}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585516.51/warc/CC-MAIN-20211022145907-20211022175907-00614.warc.gz"}
https://asmedigitalcollection.asme.org/manufacturingscience/article-abstract/94/2/746/427872/Energy-Transmission-in-Piping-Systems-and-Its?redirectedFrom=fulltext	The piping in a process plant acts as a distribution and radiation system throughout the plant for many significant sources of noise and vibration such as compressors, pumps, valves and other flow discontinuities, and the like. The acoustical and vibrational energy carried by the piping can result in the establishment of undesirable acoustic fields. This paper looks at those factors which are important to the proper description of the energy interaction and propagation in the fluid-filled piping system and discusses their significance in achieving noise control. This content is only available via PDF.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9086471199989319, "perplexity": 1025.9103082217187}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038067400.24/warc/CC-MAIN-20210412113508-20210412143508-00296.warc.gz"}
http://mathhelpforum.com/calculus/2321-improper-integral-question.html	# Thread: Improper integral question 1. ## Improper integral question I got this question and im having trouble with, how do i go about doing it.\ question 7 and also question 10 thanks 2. Hi: As the present time is 2:14 AM, I will get you started with the first problem, and return in a few hours to finish up (hopefully). First, we interpret the improper integral, i.e., int[f(x) dx] ; [a,inf], as the limit of the definite integral on interval [a,n] as n --> inf. In the present case the lower limit of integration is 3, making the interval [3,n]. Now, if we complete the square on denominator x^2+2x+10, the result is (x^2+2x+1) + 9, or (x+1)^2 + 9. So we have: Lim_n->inf [12 int (dx / ((x+1)^2 + 9))]. Yes, I know it's time to learn latex. Letting x+1 = 3tan(ß) gives dx = 3sec^2(ß) * dß. Hence the integral becomes: 123 int [sec^2(ß) dß / (9tan^2(ß) + 9)] = 4 int [sec^2(ß) * dß / (tan^2(ß) + 1)]. Applying the identity tan^2(ß) + 1 = sec^2(ß) leaves 4 int [sec^2(ß) * dß / sec^2(ß) ] = 4 int (dß) = 4ß. Now, recall the substitution x+1 = 3tan(ß). Because the interval on x is [3,n], we can derive an equivalent interval on ß by solving for ß in, 1) 3+1 = 3tan(ß) and 2) n+1 = 3tan(ß). Thus the new interval is [arctan(4/3), arctan((n+1)/3)]. And the integral, i.e., 4ß ; [x=(3,n)] is evaluated as: 4ß ; [arctan(4/3), arctan((n+1)/3)] = 4 * [arctan((n+1)/3) - arctan(4/3)]. I leave the finishing details to you. I'm off to catch up on some sleep. Regards, Rich B.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9746091961860657, "perplexity": 4439.897673670924}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368698196686/warc/CC-MAIN-20130516095636-00046-ip-10-60-113-184.ec2.internal.warc.gz"}
http://math.stackexchange.com/questions/262153/comparison-test-of-sum-frac-sqrtn1-sqrtnn/262159	# comparison test of $\sum\frac{\sqrt{n+1}-\sqrt{n}}{n}$ I want to know if the sum $$\sum_{n=1}^\infty\frac{\sqrt{n+1}-\sqrt{n}}{n}$$ converges or not. I've tried the ratio and root test but they don't fit. So wolframalpha says that the sum converges by the comparison test. So I've tried to find a convergent majorizing sum (I've tried e.g. $\sum\frac1{n^\alpha}$ with $\alpha>1$ etc.) but I can't find one. does anybody know one? - $$\frac{\sqrt{n+1}-\sqrt n}{n}=\frac{1}{n(\sqrt{n+1}+\sqrt n)}<n^{-\frac32}$$ Note that between the first and second steps there's an implicit 'clearing of the numerator' by multiplying by $\dfrac{\sqrt{n+1}+\sqrt{n}}{\sqrt{n+1}+\sqrt{n}}$ and then using $\left(\sqrt{n+1}+\sqrt{n}\right)\cdot\left(\sqrt{n+1}-\sqrt{n}\right) = \left(\sqrt{n+1}\right)^2-\left(\sqrt{n}\right)^2=n+1-n=1$. – Steven Stadnicki Dec 19 '12 at 17:22	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9787834286689758, "perplexity": 170.72322044487774}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-26/segments/1466783396459.32/warc/CC-MAIN-20160624154956-00122-ip-10-164-35-72.ec2.internal.warc.gz"}
https://www.earthdoc.org/content/papers/10.3997/2214-4609.201402168	1887 ### Abstract In standard KPSDM implementations of Kirchhoff prestack depth migration (KPSDM) the wavefield is smeared along two-way-traveltime isochrones. In the case of sparse sampling or limited aperture the image is sometimes affected by significant migration noise. Some modifications have been proposed which aim at reducing these artefacts by constructing a specular path of wave propagation derived from the slowness of coherent phases in the seismogram section and the heuristic restriction of the imaging operator to that wave path. Here we propose another physically frequency-dependent approach by using the concept of Fresnel-Volumes. Firstly the emergence angle at the receiver is determined by a local slowness analysis. Using the emergence angle as the starting direction a ray is propagated into the subsurface and the back-propagation of the wavefield is restricted to the vicinity of this ray according to its approximated Fresnel-Volume. We describe the procedure and show applications to a synthetic model as well as a real data set over a salt pillow in North Germany. Compared with standard KPSDM the image quality of our Fresnel-Volume-Migration is significantly enhanced due to the restriction of the migration operators to the region near the actual reflection point. /content/papers/10.3997/2214-4609.201402168 2006-06-12 2021-02-26	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8448054790496826, "perplexity": 1297.93453459373}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178357984.22/warc/CC-MAIN-20210226205107-20210226235107-00109.warc.gz"}
https://www.taylorfrancis.com/chapters/edit/10.1201/b10712-11/direct-regular-chaotic-tunneling-rates-using-fictitious-integrable-system-approach-arnd-ba%C2%A8cker-roland-ketzmerick-steffen-lo%C2%A8ck?context=ubx&refId=6bacf7da-b3f3-4ece-8373-3a0be8be813f	ABSTRACT Tunneling is one of the most fundamental manifestations of quantum mechanics. For 1D systems, the theoretical description is well established by the Wentzel-Kramers-Brillouin (WKB) method and related approaches [1,2]. However, for higher dimensional systems no such simple description exists. In these systems, typically regular and chaotic motion coexists and in the two-dimensional case regular tori are absolute barriers to the motion. Quantum mechanically, the eigenfunctions either concentrate within the regular islands or in the chaotic sea, as expected from the semiclassical eigenfunction hypothesis [3-5]. These eigenfunctions are coupled by so-called dynamical tunneling [6] through the dynamically generated barriers in phase space. In particular, this leads to a substantial enhancement of tunneling rates between phase-space regions of regular motion due to the presence of chaotic motion, which was termed chaos-assisted tunneling [7-9]. Such dynamical tunneling processes are ubiquitous in molecular physics and were realized with microwave cavities [10] and cold atoms in periodically modulated optical lattices [11,12].	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8510903716087341, "perplexity": 1088.0010388091207}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656104669950.91/warc/CC-MAIN-20220706090857-20220706120857-00509.warc.gz"}
http://www.global-sci.org/intro/article_detail/cicp/7480.html	Volume 10, Issue 5 Numerical Methods for Balance Laws with Space Dependent Flux: Application to Radiotherapy Dose Calculation Commun. Comput. Phys., 10 (2011), pp. 1184-1210. Published online: 2011-10 Preview Full PDF 228 1094 Export citation Cited by • Abstract The present work is concerned with the derivation of numerical methods to approximate the radiation dose in external beam radiotherapy. To address this issue, we consider a moment approximation of radiative transfer, closed by an entropy minimization principle. The model under consideration is governed by a system of hyperbolic equations in conservation form supplemented by source terms. The main difficulty coming from the numerical approximation of this system is an explicit space dependence in the flux function. Indeed, this dependence will be seen to be stiff and specific numerical strategies must be derived in order to obtain the needed accuracy. A first approach is developed considering the 1D case, where a judicious change of variables allows to eliminate the space dependence in the flux function. This is not possible in multi-D. We therefore reinterpret the 1D scheme as a scheme on two meshes, and generalize this to 2D by alternating transformations between separate meshes. We call this procedure projection method. Several numerical experiments, coming from medical physics, illustrate the potential applicability of the developed method. • Keywords • BibTex • RIS • TXT @Article{CiCP-10-1184, author = {}, title = {Numerical Methods for Balance Laws with Space Dependent Flux: Application to Radiotherapy Dose Calculation}, journal = {Communications in Computational Physics}, year = {2011}, volume = {10}, number = {5}, pages = {1184--1210}, abstract = { The present work is concerned with the derivation of numerical methods to approximate the radiation dose in external beam radiotherapy. To address this issue, we consider a moment approximation of radiative transfer, closed by an entropy minimization principle. The model under consideration is governed by a system of hyperbolic equations in conservation form supplemented by source terms. The main difficulty coming from the numerical approximation of this system is an explicit space dependence in the flux function. Indeed, this dependence will be seen to be stiff and specific numerical strategies must be derived in order to obtain the needed accuracy. A first approach is developed considering the 1D case, where a judicious change of variables allows to eliminate the space dependence in the flux function. This is not possible in multi-D. We therefore reinterpret the 1D scheme as a scheme on two meshes, and generalize this to 2D by alternating transformations between separate meshes. We call this procedure projection method. Several numerical experiments, coming from medical physics, illustrate the potential applicability of the developed method. }, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.020810.171210a}, url = {http://global-sci.org/intro/article_detail/cicp/7480.html} } TY - JOUR T1 - Numerical Methods for Balance Laws with Space Dependent Flux: Application to Radiotherapy Dose Calculation JO - Communications in Computational Physics VL - 5 SP - 1184 EP - 1210 PY - 2011 DA - 2011/10 SN - 10 DO - http://dor.org/10.4208/cicp.020810.171210a UR - https://global-sci.org/intro/article_detail/cicp/7480.html KW - AB - The present work is concerned with the derivation of numerical methods to approximate the radiation dose in external beam radiotherapy. To address this issue, we consider a moment approximation of radiative transfer, closed by an entropy minimization principle. The model under consideration is governed by a system of hyperbolic equations in conservation form supplemented by source terms. The main difficulty coming from the numerical approximation of this system is an explicit space dependence in the flux function. Indeed, this dependence will be seen to be stiff and specific numerical strategies must be derived in order to obtain the needed accuracy. A first approach is developed considering the 1D case, where a judicious change of variables allows to eliminate the space dependence in the flux function. This is not possible in multi-D. We therefore reinterpret the 1D scheme as a scheme on two meshes, and generalize this to 2D by alternating transformations between separate meshes. We call this procedure projection method. Several numerical experiments, coming from medical physics, illustrate the potential applicability of the developed method. Christophe Berthon, Martin Frank, Céline Sarazin & Rodolphe Turpault. (2020). Numerical Methods for Balance Laws with Space Dependent Flux: Application to Radiotherapy Dose Calculation. Communications in Computational Physics. 10 (5). 1184-1210. doi:10.4208/cicp.020810.171210a Copy to clipboard The citation has been copied to your clipboard	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9300851225852966, "perplexity": 851.5790293781017}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600402093104.90/warc/CC-MAIN-20200929221433-20200930011433-00152.warc.gz"}
https://arxiv.org/abs/hep-th/0205268	hep-th (what is this?) # Title: Spin Networks for Non-Compact Groups Authors: Laurent Freidel (PI, ENS-Lyon), Etera R. Livine (CPT Marseille) Abstract: Spin networks are natural generalization of Wilson loops functionals. They have been extensively studied in the case where the gauge group is compact and it has been shown that they naturally form a basis of gauge invariant observables. Physically the restriction to compact gauge group is enough for the study of Yang-mills theories, however it is well known that non-compact groups naturally arise as internal gauge groups for Lorentzian gravity models. In this context a proper construction of gauge invariant observables is needed. The purpose of this work is to define the notion of spin network states for non-compact groups. We first built, by a careful gauge fixing procedure, a natural measure and a Hilbert space structure on the space of gauge invariant graph connection. Spin networks are then defined as generalized eigenvectors of a complete set of hermitic commuting operators. We show how the delicate issue of taking the quotient of a space by non compact groups can be address in term of algebraic geometry. We finally construct the full Hilbert space containing all spin network states. Having in mind application to gravity we illustrate our results for the groups SL(2,R), SL(2,C). Comments: 43pages, many figures, some comments added Subjects: High Energy Physics - Theory (hep-th); General Relativity and Quantum Cosmology (gr-qc) Journal reference: J.Math.Phys. 44 (2003) 1322-1356 DOI: 10.1063/1.1521522 Cite as: arXiv:hep-th/0205268 (or arXiv:hep-th/0205268v2 for this version) ## Submission history From: Laurent Freidel [view email] [v1] Mon, 27 May 2002 16:02:51 GMT (96kb) [v2] Mon, 24 Jun 2002 10:06:46 GMT (97kb)	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8649933934211731, "perplexity": 1115.7942141199685}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-26/segments/1498128320685.63/warc/CC-MAIN-20170626064746-20170626084746-00414.warc.gz"}
https://kyushu-u.pure.elsevier.com/en/publications/local-stability-analysis-of-the-azimuthal-magnetorotational-insta	# Local stability analysis of the azimuthal magnetorotational instability of ideal MHD flows Research output: Contribution to journalArticlepeer-review 2 Citations (Scopus) ## Abstract A local stability analysis is carried out of axisymmetric rotating flows of a perfectly conducting fluid subjected to an external azimuthal magnetic field Bθ to non-axisymmetric as well as axisymmetric perturbations. We use the Hain-Lüst equation, capable of dealing with perturbations over a wide range of the axial wavenumber k, which supplements the previous results of \|k\| → ∞ [G. I. Ogilvie and J. E. Pringle, Mon. Not. R. Astron. Soc. 279, 152 (1996)]. When the magnetic field is sufficiently weak, instability occurs for Ro = rΩ′/(2Ω) < Roc with Roc close to zero, with the maximum growth rate given by the Oort A-value \|Ro\|, where Ω(r) is the angular velocity of the rotating flow as a function only of r, the distance from the axis of symmetry, and the prime designates the derivative in r. As the magnetic field is increased the flow becomes unstable to waves of long axial wavelength for a finite range of Ro around 0 when Rb = r2(Bθ/r)′/(2Bθ) < -1/4 with growth rate proportional to \|Bθ\|, giving unstable rotating flows under the current-free background magnetic field. Original language English 113J01 Progress of Theoretical and Experimental Physics 2014 11 https://doi.org/10.1093/ptep/ptu139 Published - 2014 ## All Science Journal Classification (ASJC) codes • Physics and Astronomy(all) ## Fingerprint Dive into the research topics of 'Local stability analysis of the azimuthal magnetorotational instability of ideal MHD flows'. Together they form a unique fingerprint.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8074809312820435, "perplexity": 1795.1515619042523}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499890.39/warc/CC-MAIN-20230131190543-20230131220543-00179.warc.gz"}
https://www.physicsforums.com/threads/singular-external-potential-in-a-field-theory.348212/	Singular external potential in a field theory? 1. Oct 23, 2009 evilcman I have a theoretical question. When doing canonical quantization, and writing the equation for he time evolution of operator in Heisenberg picture, we make use of the statment that the external potential commutes with the field variables $$[V(\phi), \phi] = 0$$ This is obviously true if the external field has a Taylor series expansion, but, I am wondering about two things. * In realistic cases is it possible that the external potential is singular? One thing I can think of is that if we use field theory to describe charge carriers in a metal/semiconductor then an external potential would be the potential of the lattice, which could be singular where the lattice points are. But right now this is just a guess. Is this a good example? Are there others? * If the potential is singular and does not have a Taylor series expansion(which converges everywhere) then is the commutator still 0? Why? 2. Oct 23, 2009 Bob_for_short You probably meant V(x)? Choosing an interaction external potential is dictated with its source. If you forget it, you can easily invent a non physical function and obtain severe physical and mathematical difficulties. In other words, wrong modelling.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9505550265312195, "perplexity": 542.1747098797493}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676593302.74/warc/CC-MAIN-20180722135607-20180722155607-00166.warc.gz"}
https://www.physicsforums.com/threads/free-falling-objects.263100/	# Free falling objects 1. Oct 9, 2008 ### steph35 1. The problem statement, all variables and given/known data A girl throws her set of keys to her friend upward who is in a window 3.90m above. the keys are caught later at 1.30s by the friends hand. 2. Relevant equations a. With what initial velocity were the keys thrown? b. What was the velocity of the keys just before they were caught? 3. The attempt at a solution a. i tried finding change in x/change in t but got the wrong answer b. i tried using a kinematic equation and solving for Vfinal but got the wrong answer... 2. Oct 9, 2008 ### LowlyPion Consider using the equation: $$x = x_0 + v_0 t + (1/2) a t^2$$	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8852827548980713, "perplexity": 2627.7716394559866}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-09/segments/1487501171629.84/warc/CC-MAIN-20170219104611-00320-ip-10-171-10-108.ec2.internal.warc.gz"}
http://bora.uib.no/handle/1956/18033?show=full	dc.contributor.author Wahl, Jens Christian dc.date.accessioned 2018-08-09T12:54:54Z dc.date.available 2018-08-09T12:54:54Z dc.date.issued 2018-06-22 dc.date.submitted 2018-06-21T22:00:12Z dc.identifier.uri http://hdl.handle.net/1956/18033 dc.description.abstract Volatility is a crucial aspect of risk management and important to accurately quantify. A broad range of models and methods tackle this problem, but there is no consensus to exactly which method or model that solves this problem best. We use maximum likelihood and Hamiltonian Monte Carlo to estimate parameters in multivariate factor stochastic volatility models and compare the two alternative methods with the new interweaving strategy proposed in Kastner et al. (2017). Through simulation studies, we show that convergence of the likelihood is unstable and very data dependent. We investigate possible restrictions on our parameters by calculating the characteristic function of our model. We find that restricting the loading matrix (in two dimensions) makes convergence more stable. Furthermore, we introduce the “Nested Laplace Approximation” (where we integrate over the latent variables in a sequential way) and compare to the classical Laplace approximation on two state space models. We also compare the methods on exchange data from 2005-2015. All methods give similar results, but Hamiltonian Monte Carlo is sensitive to the choice of priors. eng dc.language.iso eng eng dc.publisher The University of Bergen eng dc.subject Stochastic volatility eng dc.subject Maximum Likelihood eng dc.subject Markov Chain Monte Carlo eng dc.subject Hamiltonian Monte Carlo eng dc.title Parameter Estimation of Multivariate Factor Stochastic Volatility Models eng dc.type Master thesis eng dc.type.degree Masteroppgave i aktuarfag eng dc.type.course AKTUA399 eng dc.subject.archivecode Mastergrad eng dc.subject.nus 753299 eng dc.type.program MAMN-AKTUA eng dc.date.updated 2018-06-21T22:00:12Z dc.rights.holder Copyright the Author. All rights reserved eng bora.peerreviewed Not peer reviewed eng fs.grade pass fs.unitcode 12-11-0	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8766267895698547, "perplexity": 3853.685217731924}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-43/segments/1539583515352.63/warc/CC-MAIN-20181022155502-20181022181002-00461.warc.gz"}
https://math.stackexchange.com/questions/4040375/exercise-5-s-the-elements-of-integration-and-lebesgue-measure	# Exercise 5.S. the elements of integration and lebesgue measure I'm having problems with this exercise. I've tried to apply the DOMINATED CONVERGENCE THEOREM but I couldn't. Could someone gives me any hint? Suppose the function $$x\rightarrow f(x,t)$$ is $$X$$-measurable for each $$t\in\mathbb{R}$$, and the function $$t\rightarrow f(x,t)$$ is continuous on $$\mathbb{R}$$ for each $$x\in X$$. In addition, suppose that there are integrable functions $$g$$, $$h$$ on $$X$$ such that $$\|f(x,t)\|\leqslant g(x)$$ and such that the improper Riemann integral $$\int_{-\infty}^{+\infty}\|f(x,t)\|dt\leqslant h(x).$$ Show that $$\int_{-\infty}^{+\infty}\left[\int f(x,t)d\mu(x)\right]dt=\int\left[\int_{-\infty}^{+\infty}f(x,t)dt\right]d\mu(x),$$ where the integrals with respect to $$t$$ are improper Riemann integrals. • Have you studied Fubini's or Tonelli's theorems? – P. J. Feb 26 at 5:40 • Can you show how you tried to apply the DCT? Otherwise this type of problem statement question typically gets closed. This point of this problem which appears before any reference to Fubini's theorem is to prove that the Lebesgue and improper Riemann integrals iterate from first principles under these conditions. Improve the question and I can point you in the right direction. – RRL Feb 26 at 21:16	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 13, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9683026671409607, "perplexity": 135.1346574757817}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038075074.29/warc/CC-MAIN-20210413213655-20210414003655-00100.warc.gz"}
https://matheuscmss.wordpress.com/2013/06/11/finiteness-of-algebraically-primitive-closed-sl2r-orbits-in-moduli-spaces/	Posted by: matheuscmss \| June 11, 2013 ## Finiteness of algebraically primitive closed SL(2,R)-orbits in moduli spaces Today I gave a talk at the Second Palis-Balzan conference on Dynamical Systems (held at Institut Henri Poincaré, Paris). In fact, I was not supposed to talk in this conference: as I’m serving as a local organizer (together with Sylvain Crovisier), I was planning to give to others the opportunity to speak. However, Jacob Palis insisted that everyone must talk (the local organizers included), and, since he is the main organizer of this conference, I could not refuse his invitation. Anyhow, my talk concerned a joint work with Alex Wright (currently a PhD student at U. of Chicago under the supervision of Alex Eskin) about the finiteness of algebraically primitive closed ${SL(2,\mathbb{R})}$-orbits on moduli spaces (of Abelian differentials). Below the fold, I will transcript my lecture notes for this talk. 1. Preliminaries and statement of main result The data of a Riemann surface ${M}$ of genus ${g\geq 1}$ and a (non-trivial) Abelian differential (i.e., holomorphic ${1}$-form) ${\omega}$ on ${M}$ determines a structure of translation surface on ${M}$, that is, by local integration of ${\omega}$ outside the (finite) set ${\Sigma}$ of its zeroes, we obtain an atlas on ${M-\Sigma}$ such that all changes of coordinates are given by translation on ${\mathbb{C}\simeq\mathbb{R}^2}$. In particular, after cutting ${M}$ along the edges of a triangulation ${\mathcal{T}}$ such that the set ${\Sigma}$ of zeroes of ${\omega}$ is exactly the set of vertices of ${\mathcal{T}}$, we can think of ${(M,\omega)}$ as a finite collection of triangles ${T_l}$ on the plane ${\mathbb{R}^2}$ whose sides are glued by translations. The set of translation structures on a surface of genus ${g\geq 1}$ is naturally stratified by the subsets obtained by fixing the multiplicities of the zeroes of ${\Sigma}$. Note that, for each ${g\geq 1}$, the number of strata is precisely the number of partitions of ${2g-2}$: indeed, by Riemann-Hurwitz theorem, the sum ${\sum_{s=1}^{\sigma} k_{s}}$ of the orders ${k_{s}}$ of the zeroes of a non-trivial Abelian differential ${\omega}$ is always ${2g-2}$. In this language, given a partition ${(k_1,\dots, k_{\sigma})}$ of ${2g-2}$, the corresponding stratum ${H(k_1,\dots,k_\sigma)}$ of the moduli space of unit area Abelian differentials / translation structures is the set of pairs ${(M,\omega)}$ of Riemann surfaces and Abelian differentials with unit area (that is, the total area of the triangles ${T_l}$ mentioned above is one) and zeroes of orders ${k_s}$ modulo biholomorphisms. In terms of translation structures, ${H(k_1,\dots, k_s)}$ is the set of translation structures with conical singularities at vertices of the triangles ${T_l}$ with a total angle of ${2\pi(k_s+1)}$ and the total area of the triangles ${T_l}$ is one modulo cutting and pasting by translations. For more informations on translation structures and their moduli spaces, the reader might wish to consult these posts here. From the description of ${H(k_1,\dots, k_{\sigma})}$ in terms of translation structures (and triangles glued by translations), it is clear that ${H(k_1,\dots, k_{\sigma})}$ has a natural ${SL(2,\mathbb{R})}$-action. This action of ${SL(2,\mathbb{R})}$ on ${H(k_1,\dots, k_{\sigma})}$ is useful in the investigation of deviations of ergodic averages of interval exchange transformations, translation flows and billiards in rational polygons: for instance, it was recently applied by Delecroix-Hubert-Lelièvre to confirm a conjecture of the physicists Hardy-Weber on the rates of diffusions of typical trajectories in typical realizations of Ehrenfest wind-tree model of Lorentz gases. In particular, a lot of attention was given to the question of understanding the closure of ${SL(2,\mathbb{R})}$-orbits and/or ergodic ${SL(2,\mathbb{R})}$-invariant probability measures. Here, it was widely believed among experts that a Ratner-like result should be true for the ${SL(2,\mathbb{R})}$-action on ${H(k_1,\dots, k_{\sigma})}$ because, philosophically speaking, ${H(k_1,\dots,k_{\sigma})}$ is “very close to a homogenous space” (despite its non-homogeneity). In this direction, Eskin-Mirzakhani showed the following profound result: Theorem 1 (Eskin-Mirzakhani) Any ergodic ${SL(2,\mathbb{R})}$-invariant probability measure ${\mu}$ on any stratum ${H(k_1,\dots,k_{\sigma})}$ is affine, i.e., it is a density (Lebesgue) measure on certain affine subspaces (in period coordinates) associated to its support. Of course, this theorem gives hope for a general classification result of all ergodic ${SL(2,\mathbb{R})}$-invariant probability measures on all strata ${H(k_1,\dots,k_{\sigma})}$. On the other hand, a complete classification of such measures is only available in genus ${2}$ (i.e., for the strata ${H(2)}$ and ${H(1,1)}$) thanks to the works of K. Calta and C. McMullen in 2004. However, as it was pointed out by C. McMullen himself, it is an open problem to generalize his methods to genus ${g\geq 3}$. Nevertheless, some partial progress was made recently for certain types of ergodic ${SL(2,\mathbb{R})}$-invariant probability measures such as the ones supported in a (single) closed ${SL(2,\mathbb{R})}$-orbits — also known as Teichmüller curves. More concretely, M. Möller and his collaborators considered algebraically primitive closed ${SL(2,\mathbb{R})}$-orbits of genus ${g\geq 3}$, that is, Teichmüller curves such that the field generated by the traces of the stabilizers in ${SL(2,\mathbb{R})}$ of any of its points has degree ${g}$ over ${\mathbb{Q}}$. Here, they showed that there are only finitely many algebraically primitive Teichmüller curves in the following connected components of stratum: • the hyperelliptic component ${H(g-1,g-1)^{hyp}}$ consisting of translation surfaces in ${H(g-1,g-1)}$ with an hyperelliptic involution exchanging the two zeroes (cf. this paper of M. Möller in 2008); • ${H(3,1)}$ (cf. this paper of M. Bainbridge and M. Möller in 2011); • the hyperelliptic component ${H(4)^{hyp}}$ of ${H(4)}$ (this is part of a work in progress by Bainbridge, Habegger and Möller). Remark 1 In genus 2, it follows from the works of Calta and McMullen that there are infinitely many algebraically primitive Teichmüller curves in ${H(2)}$ and only one algebraically primitive Teichmüller curve in ${H(1,1)}$. This scenario motivates the following question: Given a connected component ${\mathcal{C}}$ of a stratum ${H(k_1,\dots,k_{\sigma})}$ in genus ${g\geq 3}$, are there only finitely many algebraically primitive Teichmüller curves inside ${\mathcal{C}}$? Of course, a positive answer to it hints that, a priori, it could be a good idea to start this classification by listing ${SL(2,\mathbb{R})}$-invariant probability measures supported on algebraically primitive Teichmüller curves (as there would be only finitely many of them per stratum). In this post, we want to discuss the following modest contribution to this question: Theorem 2 (C. M. and A. Wright) Let ${\mathcal{C}}$ be a connected component of a (minimal) stratum ${H(2g-2)}$ in prime genus ${g\geq 3}$. Then, there are only finitely many algebraically primitive Teichmüller curves in ${\mathcal{C}}$. Before giving the proof of Theorem 2 above, let us make three quick comments. First, the arguments we use to prove our finiteness result are dynamical in nature (based on the analysis of the so-called Kontsevich-Zorich cocycle), while the methods of Möller and his collaborators were more algebro-geometrical (based on the behavior of algebraically primitive Teichmüller curves near the boundary of Deligne-Mumford’s compactification of moduli spaces). Interestingly enough, these two methods seem to be “complementary” in the sense that our dynamical method is particularly adapted to the study of algebraically primitive Teichmüller curves in minimal strata ${H(2g-2)}$, while the algebro-geometrical method of Möller and his collaborators is better adapted to non-minimal strata (and this explains why the result of Bainbridge-Habegger-Möller mentioned above was obtained only very recently compared to the other results). Secondly, A. Wright and I also have partial results for other (non-minimal) strata, e.g., we can show that algebraically primitive Teichmüller curves can’t form a dense subset of a given connected component of a stratum in genus ${g\geq 3}$, but we will restrict our discussion to Theorem 2 because its proof already contain all essential ideas of our forthcoming paper. Finally, it is likely that our methods might give finiteness of algebraically primitive Teichmüller curves in more strata (and also for some non-algebraically Teichmüller curves) as the knowledge of ${SL(2,\mathbb{R})}$-invariant closed subsets of strata is improved. 2. Proof of Theorem 2 The basic idea is very simple. Let us fix ${g\geq 3}$ prime and let us suppose by contradiction that there are infinitely many algebraically primitive Teichmüller curves ${C_n}$ in some connected component ${\mathcal{C}}$ of the minimal stratum ${H(2g-2)}$. A recent equidistribution result of A. Eskin, M. Mirzakhani and A. Mohammadi implies that ${C_n}$ will equidistribute inside the support ${\mathcal{M}}$ of an ergodic ${SL(2,\mathbb{R})}$-invariant probability measure. Actually, for our current purposes, we do not the full strength of this equidistribution result: it is sufficient to know that ${C_n}$ eventually becomes dense inside ${\mathcal{M}}$ in the sense that there exists an integer ${n_0\in \mathbb{N}}$ such that ${C_n\subset\mathcal{M}}$ for all ${n\geq n_0}$ and the subset ${\bigcup\limits_{n\geq n_0} C_n}$ is dense in ${\mathcal{M}}$. Now, we claim that ${\mathcal{M}}$ is equal to the whole connected component ${\mathcal{C}}$ of ${H(2g-2)}$, that is, the sequence of algebraically primitive Teichmüller curves ${C_n}$ becomes dense in ${\mathcal{C}}$. In fact, as we mentioned earlier, by the results of A. Eskin and M. Mirzakhani (cf. Theorem 1 above), ${\mathcal{M}}$ is represented by affine subspaces in period coordinates. As it was shown by A. Wright in this paper here, these affine subspaces representing ${\mathcal{M}}$ are defined over some field ${k(\mathcal{M})}$ of degree ${1\leq deg(\mathcal{M})\leq g}$ over ${\mathbb{Q}}$, and, moreover, since ${\mathcal{M}\subset H(2g-2)}$, one also has the inequality $\displaystyle deg(\mathcal{M})\cdot (\textrm{dim}(\mathcal{M})+1)\leq 4g \ \ \ \ \ (1)$ On the other hand, it is possible to check that the affine subspaces representing a Teichmüller curve (closed ${SL(2,\mathbb{R})}$-orbit) ${D}$ are defined over a field ${k(D)}$ whose degree over ${\mathbb{Q}}$ coincide with the degree of the field generated by the traces of the stabilizers of ${D}$ in ${SL(2,\mathbb{R})}$. In particular, it follows (from the definition of algebraic primitivity) that the affine subspaces representing the algebraically primitive Teichmüller curves ${C_n}$ are defined over some fields whose degrees over ${\mathbb{Q}}$ are ${g}$. Because ${C_n\subset\mathcal{M}}$ for all ${n\geq n_0}$, we deduce that the degree ${deg(\mathcal{M})}$ of ${k(\mathcal{M})}$ divides ${g}$. Since ${g}$ is prime (by hypothesis), we have that ${deg(\mathcal{M})=[k(\mathcal{M}):\mathbb{Q}]=1}$ or ${g}$. We affirm that ${k(\mathcal{M})=\mathbb{Q}}$: indeed, if this is not true, then ${deg(\mathcal{M})=g}$; by inserting this into the inequality (1), we would have ${dim(\mathcal{M})\leq 3}$; since ${\mathcal{M}}$ consists of entire ${SL(2,\mathbb{R})}$-orbits (and ${SL(2,\mathbb{R})}$ has dimension ${3}$), we get that ${dim(\mathcal{M})=3}$; however, this is impossible because ${\mathcal{M}}$ contains infinitely many closed ${SL(2,\mathbb{R})}$-orbits, namely, ${C_n}$ for ${n\geq n_0}$. Once we know that ${k(\mathcal{M})=\mathbb{Q}}$, i.e., ${deg(\mathcal{M})=1}$, we affirm that ${dim(\mathcal{M})=4g-1}$, that is, the inequality (1) is an equality. In fact, let us consider the subset ${\mathbb{R}\cdot\mathcal{M}}$ consisting of Abelian differentials belonging to ${\mathcal{M}}$ after normalization of its total area and let us denote by ${T_x(\mathbb{R}\cdot\mathcal{M})}$ at some point ${x\in C_{n_0}\subset\mathcal{M}}$. By definition, ${T_x(\mathbb{R}\cdot\mathcal{M})}$ contains the ${4}$-dimensional subspace ${T_x(\mathbb{R}\cdot C_{n_0})}$. Since ${T_x(\mathbb{R}\cdot\mathcal{M})}$ is defined over ${k(\mathcal{M})=\mathbb{Q}}$ and ${T_x(\mathbb{R}\cdot C_{n_0})}$ is defined over a field of degree ${g}$ over ${\mathbb{Q}}$, we see that ${T_x(\mathbb{R}\cdot\mathcal{M})}$ must contain all ${g}$ Galois conjugates ${T_x(\mathbb{R}\cdot C_{n_0})}$. Using this information (and some extra arguments), one can show that the linear subspace ${T_x(\mathbb{R}\cdot \mathcal{M})}$ has dimension ${\geq 4g}$, and, a fortiori, ${dim(\mathcal{M})\geq 4g-1}$. By combining this with inequality (1), we deduce that ${dim(\mathcal{M})=4g-1}$. At this point, we can complete the proof of our claim that ${\mathcal{M}=\mathcal{C}\subset H(2g-2)}$ by simply counting dimensions: using the period coordinates on ${H(2g-2)}$, we can identify the tangent space of any point ${x=(M,\omega)\in \mathbb{R}\cdot H(2g-2)}$ with ${H^1(M,\mathbb{R})\oplus i H^1(M,\mathbb{R})}$, a vector space of real dimension ${4g}$; hence ${\mathbb{R}\cdot H(2g-2)}$ has dimension ${4g}$ and, a fortiori, ${dim(\mathcal{C})=dim(H(2g-2))=4g-1}$. In summary, we just proved the following statement (first proved in this paper of A. Wright): Proposition 3 (A. Wright) If there are infinitely many algebraically primitive Teichmüller curves ${C_n}$ in some connected component ${\mathcal{C}}$ of a minimal stratum ${H(2g-2)}$ with ${g\geq 2}$ prime, then they must form a dense subset of ${\mathcal{C}}$, that is, $\displaystyle \overline{\bigcup\limits_{n\in\mathbb{N}}C_n}=\mathcal{C}$ Our plan is to contradict this denseness property of ${C_n}$ along the following lines. Let us consider the so-called Kontsevich-Zorich (KZ) cocycle over ${\mathcal{C}\subset H(2g-2)}$, that is, we use the ${SL(2,\mathbb{R})}$-action to move around ${\mathcal{C}\subset H(2g-2)}$, and every time that we cut and paste a translation structure ${g\cdot (M,\omega)}$ obtained from ${(M,\omega)\in\mathcal{C}}$ by applying ${g\in SL(2,\mathbb{R})}$ to get a shape “close” to ${(M,\omega)}$, we keep track of how the homology cycles on ${(M,\omega)}$ were changed. In practice, by selecting an appropriate symplectic basis of ${H_1(M,\mathbb{R})}$ (with respect to the usual intersection form on homology), this amounts to “attach” a symplectic matrix ${A_{g,(M,\omega)}\in Sp(2g,\mathbb{Z})}$ to each ${g\in SL(2,\mathbb{R})}$ and ${(M,\omega)\in\mathcal{C}}$. Generally speaking, the Kontsevich-Zorich cocycle preserves a family of 2-dimensional symplectic plane ${P_{(M,\omega)}\subset H_1(M,\mathbb{R})}$ (called tautological planes). Since this cocycle is symplectic, the symplectic orthogonals ${P_{(M,\omega)}^{\dagger}}$ of the planes ${P_{(M,\omega)}}$ are also preserved, so that the Kontsevich-Zorich cocycle along any ${SL(2,\mathbb{R})}$-orbit decomposes into a ${2\times 2}$-block corresponding to its restriction to ${P_{(M,\omega)}}$ and a ${(2g-2)\times(2g-2)}$-block corresponding to its restriction to ${P_{(M,\omega)}^{\dagger}}$. For later use, we will call restricted Kontsevich-Zorich cocycle the restriction of the Kontsevich-Zorich cocycle to ${P_{(M,\omega)}^{\dagger}}$. By definition, the restricted KZ cocycle is a ${Sp(2g-2,\mathbb{R})}$-cocycle over ${SL(2,\mathbb{R})}$-orbits. As it turns out, the restricted KZ cocycle tends to behave poorly or richly depending on the ${SL(2,\mathbb{R})}$-orbit we look at. For example, the restricted KZ cocycle over an algebraically primitive Teichmüller curve ${D}$ in a stratum of genus ${g}$ decomposes into ${g-1}$ blocks consisting of its restrictions to the ${g-1}$ non-trivial Galois conjugates of the tautological planes (because these tautological planes are defined over a field of degree ${g}$ over ${\mathbb{Q}}$ when the Teichmüller curve ${D}$ is algebraically primitive). In particular, the matrices of the restricted KZ cocycle over ${D}$ belong to a “poor” subgroup isomorphic to ${SL(2,\mathbb{R})^{g-1}}$ of the symplectic group ${Sp(2g-2,\mathbb{R})}$. On the other hand, it is expected that the restricted KZ cocycle over “most” ${SL(2,\mathbb{R})}$-orbits (in any connected component of any [not necessarily minimal] stratum) has a “rich” behavior: for instance, the simplicity result of Avila-Viana (as well as some numerical experiments) gives hope that the monoid of matrices of the restricted KZ cocycle over a typical ${SL(2,\mathbb{R})}$-orbit has full Zariski closure ${Sp(2g-2,\mathbb{R})}$. So, if some family of algebraically primitive Teichmüller curves ${C_n}$ becomes dense in some connected component ${\mathcal{C}}$ of some stratum in genus ${g\geq 3}$, we would have to two opposite properties fighting one against the other: on one hand, the restricted KZ cocycle over the dense subset ${\bigcup\limits_{n\in\mathbb{N}} C_n}$ has a very rigid structure, namely, they decompose into ${(g-1)}$ blocks of sizes ${2\times 2}$; on the other hand, the typical ${SL(2,\mathbb{R})}$-orbit has a “rich” restricted KZ cocycle, say Zariski dense in ${Sp(2g-2,\mathbb{R})}$, and thus it can not be have a very rigid structure. Therefore, we will contradict the denseness property of ${C_n}$ in ${\mathcal{C}}$ if we can show that: • (i) some non-trivial piece of information coming from the very rigid structure of the restricted KZ cocycle over ${C_n}$ “pass to the limit” in the sense that the restricted KZ cocycle over all ${SL(2,\mathbb{R})}$-orbits in ${\mathcal{C}}$ would have some (partial) rigidity property; • (ii) show that the restricted KZ cocycle over some ${SL(2,\mathbb{R})}$-orbit in ${\mathcal{C}}$ is rich enough so that no rigidity property can occur (not even partially). Concerning (i), it is tempting to use the decomposition into ${(g-1)}$ blocks of sizes ${2\times 2}$ as the very rigid property that passes through the limit. However, it is not obvious that the limit of a decomposition into ${(g-1)}$ blocks is still a set of ${(g-1)}$ blocks: indeed, even though the Grassmanian of planes is compact (and hence we can extract limits), some of these blocks might collapse. Nevertheless, it possible to check that the restricted KZ cocycle over ${C_n}$ preserves a very special (projective) subset of Grassmanian of (symplectic) planes that A. Wright and I call Hodge-Teichmüller planes or HT planes for short. More precisely, we say that a plane ${P\subset H^1(M,\mathbb{R})}$ is a HT plane if the images of its complexification ${P_{\mathbb{C}}:=P\otimes\mathbb{C}\subset H^1(M,\mathbb{C})}$ under all matrices ${A_{g,(M,\omega)}}$ of the KZ cocycle intersect non-trivially the ${(1,0)}$ and ${(0,1)}$ subspaces ${H^{1,0}(g(M,\omega))}$ and ${H^{0,1}(g(M,\omega))}$ of the Hodge filtration of ${H^1(g(M,\omega),\mathbb{C})}$ (into holomorphic and anti-holomorphic forms). Note that HT planes are really very special: since the usual action of ${SL(2,\mathbb{R})}$ on ${\mathbb{R}^2\equiv\mathbb{C}}$ is not holomorphic, the complex structure of ${(M,\omega)}$ changes when we apply the elements ${g\in SL(2,\mathbb{R})}$, so that the subspaces ${H^{1,0}(g(M,\omega))}$ and ${H^{0,1}(g(M,\omega))}$ move around ${H^1(M,\mathbb{C})}$ and hence it is not easy for the images of a plane ${P}$ to keep intersecting the “moving targets” ${H^{1,0}}$ and ${H^{0,1}}$. As it turns out, the (rigid) property of preservation of a non-trivial set of HT planes passes through the limit: indeed, this follows from the facts that the HT planes ${P}$ were defined in terms of the intersections of ${A_{g,(M,\omega)}P}$ with ${H^{1,0}(g(M,\omega))}$ and ${H^{0,1}(g(M,\omega)}$ and it is known that ${H^{1,0}(M)}$ and ${H^{0,1}(M)}$ vary continuously with ${(M,\omega)}$. Also, in the case of the algebraically primitive Teichmüller curves, work of Martin Moller implies that the Galois conjugates of the tautological planes are HT planes and Hodge orthogonal, that is, they are mutually orthogonal with respect to the Hodge inner product in cohomology. Since the Hodge inner product is also known to depend continuously on ${(M,\omega)}$, we have that the limits of the ${(g-1)}$ HT planes associated to ${C_n}$ lead to ${(g-1)}$ Hodge-orthogonal HT planes. In other words, we showed the following result (that formalizes item (i) above): Proposition 4 Let ${C_n}$, ${n\in\mathbb{N}}$, be a family algebraically primitive Teichmüller curves in some connected component ${\mathcal{C}}$ of some stratum in genus ${g\geq 2}$ such that $\displaystyle \overline{\bigcup\limits_{n\in\mathbb{N}}C_n}=\mathcal{C}.$ Then, the restricted Kontsevich-Zorich cocycle over any ${SL(2,\mathbb{R})}$-orbit in ${\mathcal{C}}$ preserves the non-trivial set of at least ${(g-1)}$ HT planes that are mutually Hodge-orthogonal. From Propositions 3 and 4, we see that the proof of Theorem 2 is reduced to the following proposition: Proposition 5 In any connected component ${\mathcal{C}}$ of any stratum in genus ${g\geq 3}$ one can find ${SL(2,\mathbb{R})}$-orbits such that the restricted KZ cocycle can not have ${(g-1)}$ Hodge-orthogonal HT planes. The proof of Proposition 5 is based on the following observations. It is not hard to see that the set of at least ${g-1}$ HT planes preserved by the restricted KZ cocycle is also preserved by its Zariski closure (in ${Sp(2g-2,\mathbb{R})}$). Moreover, this set of HT planes is a proper compact subset of the Grassmanian of symplectic planes. Therefore, since ${Sp(4,\mathbb{R})}$ has no proper compact invariant subset in the Grassmanian of symplectic planes, we have that any ${SL(2,\mathbb{R})}$-orbit such that the Zariski closure of the restricted KZ cocycle is “at least as rich as ${Sp(4,\mathbb{R})}$” is likely to do not have ${(g-1)}$ HT planes. In other words, the proof of Proposition 5 follows if we construct on each connected component ${\mathcal{C}}$ of any stratum in genus ${g\geq 3}$ some ${SL(2,\mathbb{R})}$-orbit whose restricted KZ cocycle has a “mild rich Zariski closure”. In our first attempts, A. Wright and I tried to construct such ${SL(2,\mathbb{R})}$-orbits by taking adequate ramified covers of certain closed ${SL(2,\mathbb{R})}$-orbits in genus ${3}$. In fact, this idea works to produce nice examples in some connected components of some minimal strata ${H(2g-2)}$ (where ${g}$ runs along some arithmetic progression), but if one wishes to construct examples in all connected components of all strata in genus ${g\geq 3}$, then one runs into some nasty combinatorial problems that we do not know how to solve. For this reason, we decided to change our initial strategy outlined in the previous paragraph into the following one. Firstly, we considered the following two square-tiled surfaces (i.e., translation surfaces obtained from finite coverings of the unit torus ${\mathbb{R}^2/\mathbb{Z}^2}$ that are ramified only at the origin ${0\in\mathbb{R}^2/\mathbb{Z}^2}$): A square-tiled surface in ${M_\ast\in H(4)^{hyp}}$. A square-tiled surface in ${M_{\ast\ast}\in H(4)^{odd}}$. These square-tiled surfaces generate closed ${SL(2,\mathbb{R})}$-orbits in the two connected components ${H(4)^{hyp}}$ and ${H(4)^{odd}}$ of ${H(4)}$. Moreover, by computing some matrices of the restricted KZ cocycle over these square-tiled surfaces (using appropriate Dehn twists along the directions indicated above), we can show that the Zariski closure of the restricted KZ cocycle is ${Sp(4,\mathbb{R})}$ in both cases. In particular, from our discussion so far, this proves Proposition 5 for the cases of the two connected components of ${H(4)}$. Now, the idea to prove Proposition 5 in the general case of a connected component ${\mathcal{C}}$ is to find some translation surface (in ${\mathcal{C}}$) whose restricted KZ cocycle behaves a “little bit” like the corresponding cocycle for ${M_\ast}$ or ${M_{\ast\ast}}$. However, we are not quite able to do this and we proceed in a slightly different way. Formally, we show Proposition 5 by contradiction and induction using the fact that this proposition was already shown to be true for the connected components of ${H(4)}$. More concretely, assume that all translation surfaces ${M}$ in some connected component ${\mathcal{C}}$ of some stratum have ${(g-1)}$ Hodge-orthogonal HT planes. It was shown by Kontsevich-Zorich (in their classification of connected components of strata) that, by carefully collapsing zeroes, and pinching off the handles of ${M}$ like in this picture here and by deleting the torus component, we reach one of the connected components of ${H(4)}$. Furthermore, by a close inspection of Kontsevich-Zorich procedure, we can see that, if the initial translation surface has ${(g-1)}$ Hodge-orthogonal HT planes and if the pinching procedure is really careful, then each time we pinch off a handle, we kill off at most one of the HT planes. In particular, among the ${(g-1)}$ Hodge-orthogonal HT planes of the initial translation surfaces, we know that at least two of them survive by the end of the procedure, and, hence, we obtain some translation surface in some of the connected components of ${H(4)}$ having two Hodge-orthogonal HT planes, a contradiction. ## Responses 1. Update (June 15, 2013): Alex Wright pointed out to me (by email) that some parts of this post were misleading or badly phrased. (Thanks Alex!) In particular, I followed his suggestions and I hope this post is now more clear. 2. Are there any videos from this conference ? • Unfortunately, there were no videos from this conference (contrary to the first Palis-Balzan conference at Rio). On the other hand, as a form of “compensation”, J. Palis asked the invited speakers to supply lecture notes and/or slides of their talks, and this material is available here http://www.impa.br/opencms/pt/eventos/store_old/evento_1305?link=2 (see the links “Presentation” next to the titles of some talks). 3. Hello Matheus, Is there any simpler proof of the denseness of infinite Teichmuller curves in stratum H(2), the stratum of holomorphic 1-form with double zero in genus 2? I’m a first-year PhD student and don’t much about it. I’m reading the paper “Teichmuller curves in genus two: Discriminant and spin” of McMullen and trying to prove it but not yet. Thank you. • Hi Amateur, I believe that you are interested in non-arithmetic Teichmuller curves, right? (Otherwise, arithmetic Teichmuller curves [SL(2,R)-orbits of square-tiled surfaces] always form a dense subset of any connected component of any given stratum) In this case, I’m not aware of a better reference than McMullen’s papers for the construction of Teichmuller curves in genus 2. Best, Matheus 4. I’m sorry. I means infinite union of Teichmuller curves.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 310, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.936622142791748, "perplexity": 283.4325004763872}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886120194.50/warc/CC-MAIN-20170823113414-20170823133414-00717.warc.gz"}
http://cms.math.ca/cjm/msc/11?page=6	location: Publications → journals Search results Search: MSC category 11 ( Number theory ) Expand all Collapse all Results 126 - 150 of 220 126. CJM 2005 (vol 57 pp. 812) Trifković, Mak On the Vanishing of $\mu$-Invariants of Elliptic Curves over $\qq$ Let $E_{/\qq}$ be an elliptic curve with good ordinary reduction at a prime $p>2$. It has a well-defined Iwasawa $\mu$-invariant $\mu(E)_p$ which encodes part of the information about the growth of the Selmer group $\sel E{K_n}$ as $K_n$ ranges over the subfields of the cyclotomic $\zzp$-extension $K_\infty/\qq$. Ralph Greenberg has conjectured that any such $E$ is isogenous to a curve $E'$ with $\mu(E')_p=0$. In this paper we prove Greenberg's conjecture for infinitely many curves $E$ with a rational $p$-torsion point, $p=3$ or $5$, no two of our examples having isomorphic $p$-torsion. The core of our strategy is a partial explicit evaluation of the global duality pairing for finite flat group schemes over rings of integers. Category:11R23 127. CJM 2005 (vol 57 pp. 449) Alkan, Emre On the Sizes of Gaps in the Fourier Expansion of Modular Forms Let $f= \sum_{n=1}^{\infty} a_f(n)q^n$ be a cusp form with integer weight $k \geq 2$ that is not a linear combination of forms with complex multiplication. For $n \geq 1$, let $$i_f(n)=\begin{cases}\max\{ i : a_f(n+j)=0 \text{ for all } 0 \leq j \leq i\}&\text{if a_f(n)=0,}\\ 0&\text{otherwise}.\end{cases}$$ Concerning bounded values of $i_f(n)$ we prove that for $\epsilon >0$ there exists $M = M(\epsilon,f)$ such that $\# \{n \leq x : i_f(n) \leq M\} \geq (1 - \epsilon) x$. Using results of Wu, we show that if $f$ is a weight 2 cusp form for an elliptic curve without complex multiplication, then $i_f(n) \ll_{f, \epsilon} n^{\frac{51}{134} + \epsilon}$. Using a result of David and Pappalardi, we improve the exponent to $\frac{1}{3}$ for almost all newforms associated to elliptic curves without complex multiplication. Inspired by a classical paper of Selberg, we also investigate $i_f(n)$ on the average using well known bounds on the Riemann Zeta function. Category:11F30 128. CJM 2005 (vol 57 pp. 535) Kim, Henry H. On Local $L$-Functions and Normalized Intertwining Operators In this paper we make explicit all $L$-functions in the Langlands--Shahidi method which appear as normalizing factors of global intertwining operators in the constant term of the Eisenstein series. We prove, in many cases, the conjecture of Shahidi regarding the holomorphy of the local $L$-functions. We also prove that the normalized local intertwining operators are holomorphic and non-vaninishing for $\re(s)\geq 1/2$ in many cases. These local results are essential in global applications such as Langlands functoriality, residual spectrum and determining poles of automorphic $L$-functions. Categories:11F70, 22E55 129. CJM 2005 (vol 57 pp. 494) Friedlander, John B.; Iwaniec, Henryk Summation Formulae for Coefficients of $L$-functions With applications in mind we establish a summation formula for the coefficients of a general Dirichlet series satisfying a suitable functional equation. Among a number of consequences we derive a generalization of an elegant divisor sum bound due to F.~V. Atkinson. Categories:11M06, 11M41 130. CJM 2005 (vol 57 pp. 616) Muić, Goran Reducibility of Generalized Principal Series In this paper we describe reducibility of non-unitary generalized principal series for classical $p$-adic groups in terms of the classification of discrete series due to M\oe glin and Tadi\'c. Categories:22E35, and, 50, 11F70 131. CJM 2005 (vol 57 pp. 267) Partial Euler Products on the Critical Line The initial version of the Birch and Swinnerton-Dyer conjecture concerned asymptotics for partial Euler products for an elliptic curve $L$-function at $s = 1$. Goldfeld later proved that these asymptotics imply the Riemann hypothesis for the $L$-function and that the constant in the asymptotics has an unexpected factor of $\sqrt{2}$. We extend Goldfeld's theorem to an analysis of partial Euler products for a typical $L$-function along its critical line. The general $\sqrt{2}$ phenomenon is related to second moments, while the asymptotic behavior (over number fields) is proved to be equivalent to a condition that in a precise sense seems much deeper than the Riemann hypothesis. Over function fields, the Euler product asymptotics can sometimes be proved unconditionally. Keywords:Euler product, explicit formula, second momentCategories:11M41, 11S40 132. CJM 2005 (vol 57 pp. 298) Kumchev, Angel V. On the Waring--Goldbach Problem: Exceptional Sets for Sums of Cubes and Higher Powers We investigate exceptional sets in the Waring--Goldbach problem. For example, in the cubic case, we show that all but $O(N^{79/84+\epsilon})$ integers subject to the necessary local conditions can be represented as the sum of five cubes of primes. Furthermore, we develop a new device that leads easily to similar estimates for exceptional sets for sums of fourth and higher powers of primes. Categories:11P32, 11L15, 11L20, 11N36, 11P55 133. CJM 2005 (vol 57 pp. 328) Kuo, Wentang; Murty, M. Ram On a Conjecture of Birch and Swinnerton-Dyer Let $$E/\mathbb{Q}$$ be an elliptic curve defined by the equation $$y^2=x^3 +ax +b$$. For a prime $$p, \linebreak p \nmid\Delta =-16(4a^3+27b^2)\neq 0$$, define $N_p = p+1 -a_p = \|E(\mathbb{F}_p)\|.$ As a precursor to their celebrated conjecture, Birch and Swinnerton-Dyer originally conjectured that for some constant $c$, $\prod_{p \leq x, p \nmid\Delta } \frac{N_p}{p} \sim c (\log x)^r, \quad x \to \infty.$ Let $$\alpha _p$$ and $$\beta _p$$ be the eigenvalues of the Frobenius at $$p$$. Define $\tilde{c}_n = \begin{cases} \frac{\alpha_p^k + \beta_p^k}{k}& n =p^k, p \textrm{ is a prime, k is a natural number, p\nmid \Delta} . \\ 0 & \text{otherwise}. \end{cases}.$ and $$\tilde{C}(x)= \sum_{n\leq x} \tilde{c}_n$$. In this paper, we establish the equivalence between the conjecture and the condition $$\tilde{C}(x)=\mathbf{o}(x)$$. The asymptotic condition is indeed much deeper than what we know so far or what we can know under the analogue of the Riemann hypothesis. In addition, we provide an oscillation theorem and an $$\Omega$$ theorem which relate to the constant $c$ in the conjecture. Categories:11M41, 11M06 134. CJM 2005 (vol 57 pp. 338) Lange, Tanja; Shparlinski, Igor E. Certain Exponential Sums and Random Walks on Elliptic Curves For a given elliptic curve $\E$, we obtain an upper bound on the discrepancy of sets of multiples $z_sG$ where $z_s$ runs through a sequence $\cZ=$$z_1, \dots, z_T$$$ such that $k z_1,\dots, kz_T$ is a permutation of $z_1, \dots, z_T$, both sequences taken modulo $t$, for sufficiently many distinct values of $k$ modulo $t$. We apply this result to studying an analogue of the power generator over an elliptic curve. These results are elliptic curve analogues of those obtained for multiplicative groups of finite fields and residue rings. Categories:11L07, 11T23, 11T71, 14H52, 94A60 135. CJM 2005 (vol 57 pp. 180) Somodi, Marius On the Size of the Wild Set To every pair of algebraic number fields with isomorphic Witt rings one can associate a number, called the {\it minimum number of wild primes}. Earlier investigations have established lower bounds for this number. In this paper an analysis is presented that expresses the minimum number of wild primes in terms of the number of wild dyadic primes. This formula not only gives immediate upper bounds, but can be considered to be an exact formula for the minimum number of wild primes. Categories:11E12, 11E81, 19F15, 11R29 136. CJM 2004 (vol 56 pp. 897) Borwein, Jonathan M.; Borwein, David; Galway, William F. Finding and Excluding $b$-ary Machin-Type Individual Digit Formulae Constants with formulae of the form treated by D.~Bailey, P.~Borwein, and S.~Plouffe (\emph{BBP formulae} to a given base $b$) have interesting computational properties, such as allowing single digits in their base $b$ expansion to be independently computed, and there are hints that they should be \emph{normal} numbers, {\em i.e.,} that their base $b$ digits are randomly distributed. We study a formally limited subset of BBP formulae, which we call \emph{Machin-type BBP formulae}, for which it is relatively easy to determine whether or not a given constant $\kappa$ has a Machin-type BBP formula. In particular, given $b \in \mathbb{N}$, $b>2$, $b$ not a proper power, a $b$-ary Machin-type BBP arctangent formula for $\kappa$ is a formula of the form $\kappa = \sum_{m} a_m \arctan(-b^{-m})$, $a_m \in \mathbb{Q}$, while when $b=2$, we also allow terms of the form $a_m \arctan(1/(1-2^m))$. Of particular interest, we show that $\pi$ has no Machin-type BBP arctangent formula when $b \neq 2$. To the best of our knowledge, when there is no Machin-type BBP formula for a constant then no BBP formula of any form is known for that constant. Keywords:BBP formulae, Machin-type formulae, arctangents,, logarithms, normality, Mersenne primes, Bang's theorem,, Zsigmondy's theorem, primitive prime factors, $p$-adic analysisCategories:11Y99, 11A51, 11Y50, 11K36, 33B10 137. CJM 2004 (vol 56 pp. 673) Cali, Élie DÃ©faut de semi-stabilitÃ© des courbes elliptiques dans le cas non ramifiÃ© Let $\overline {\Q_2}$ be an algebraic closure of $\Q_2$ and $K$ be an unramified finite extension of $\Q_2$ included in $\overline {\Q_2}$. Let $E$ be an elliptic curve defined over $K$ with additive reduction over $K$, and having an integral modular invariant. Let us denote by $K_{nr}$ the maximal unramified extension of $K$ contained in $\overline {\Q_2}$. There exists a smallest finite extension $L$ of $K_{nr}$ over which $E$ has good reduction. We determine in this paper the degree of the extension $L/K_{nr}$. Category:11G07 138. CJM 2004 (vol 56 pp. 612) Pál, Ambrus Solvable Points on Projective Algebraic Curves We examine the problem of finding rational points defined over solvable extensions on algebraic curves defined over general fields. We construct non-singular, geometrically irreducible projective curves without solvable points of genus $g$, when $g$ is at least $40$, over fields of arbitrary characteristic. We prove that every smooth, geometrically irreducible projective curve of genus $0$, $2$, $3$ or $4$ defined over any field has a solvable point. Finally we prove that every genus $1$ curve defined over a local field of characteristic zero with residue field of characteristic $p$ has a divisor of degree prime to $6p$ defined over a solvable extension. Categories:14H25, 11D88 139. CJM 2004 (vol 56 pp. 406) Pál, Ambrus Theta Series, Eisenstein Series and PoincarÃ© Series over Function Fields We construct analogues of theta series, Eisenstein series and Poincar\'e series for function fields of one variable over finite fields, and prove their basic properties. Category:11F12 140. CJM 2004 (vol 56 pp. 356) Murty, M. Ram; Saidak, Filip Non-Abelian Generalizations of the Erd\H os-Kac Theorem Let $a$ be a natural number greater than $1$. Let $f_a(n)$ be the order of $a$ mod $n$. Denote by $\omega(n)$ the number of distinct prime factors of $n$. Assuming a weak form of the generalised Riemann hypothesis, we prove the following conjecture of Erd\"os and Pomerance: The number of $n\leq x$ coprime to $a$ satisfying $$\alpha \leq \frac{\omega(f_a(n)) - (\log \log n)^2/2 }{ (\log \log n)^{3/2}/\sqrt{3}} \leq \beta$$ is asymptotic to $$\left(\frac{ 1 }{ \sqrt{2\pi}} \int_{\alpha}^{\beta} e^{-t^2/2}dt\right) \frac{x\phi(a) }{ a},$$ as $x$ tends to infinity. Keywords:Tur{\' a}n's theorem, Erd{\H o}s-Kac theorem, Chebotarev density theorem,, Erd{\H o}s-Pomerance conjectureCategories:11K36, 11K99 141. CJM 2004 (vol 56 pp. 373) Orton, Louisa An Elementary Proof of a Weak Exceptional Zero Conjecture In this paper we extend Darmon's theory of integration on $\uh_p\times \uh$'' to cusp forms $f$ of higher even weight. This enables us to prove a weak exceptional zero conjecture'': that when the $p$-adic $L$-function of $f$ has an exceptional zero at the central point, the $\mathcal{L}$-invariant arising is independent of a twist by certain Dirichlet characters. Categories:11F11, 11F67 142. CJM 2004 (vol 56 pp. 168) Pogge, James Todd On a Certain Residual Spectrum of $\Sp_8$ Let $G=\Sp_{2n}$ be the symplectic group defined over a number field $F$. Let $\mathbb{A}$ be the ring of adeles. A fundamental problem in the theory of automorphic forms is to decompose the right regular representation of $G(\mathbb{A})$ acting on the Hilbert space $L^2\bigl(G(F)\setminus G(\mathbb{A})\bigr)$. Main contributions have been made by Langlands. He described, using his theory of Eisenstein series, an orthogonal decomposition of this space of the form: $L_{\dis}^2 \bigl( G(F)\setminus G(\mathbb{A}) \bigr)=\bigoplus_{(M,\pi)} L_{\dis}^2(G(F) \setminus G(\mathbb{A}) \bigr)_{(M,\pi)}$, where $(M,\pi)$ is a Levi subgroup with a cuspidal automorphic representation $\pi$ taken modulo conjugacy (Here we normalize $\pi$ so that the action of the maximal split torus in the center of $G$ at the archimedean places is trivial.) and $L_{\dis}^2\bigl(G(F)\setminus G(\mathbb{A})\bigr)_{(M,\pi)}$ is a space of residues of Eisenstein series associated to $(M,\pi)$. In this paper, we will completely determine the space $L_{\dis}^2\bigl(G(F)\setminus G(\mathbb{A})\bigr)_{(M,\pi)}$, when $M\simeq\GL_2\times\GL_2$. This is the first result on the residual spectrum for non-maximal, non-Borel parabolic subgroups, other than $\GL_n$. Categories:11F70, 22E55 143. CJM 2004 (vol 56 pp. 55) Harper, Malcolm $\mathbb{Z}[\sqrt{14}]$ is Euclidean We provide the first unconditional proof that the ring $\mathbb{Z} [\sqrt{14}]$ is a Euclidean domain. The proof is generalized to other real quadratic fields and to cyclotomic extensions of $\mathbb{Q}$. It is proved that if $K$ is a real quadratic field (modulo the existence of two special primes of $K$) or if $K$ is a cyclotomic extension of $\mathbb{Q}$ then: \begin{center} \emph{% the ring of integers of $K$ is a Euclidean domain if and only if it is a principal ideal domain.} \end{center} The proof is a modification of the proof of a theorem of Clark and Murty giving a similar result when $K$ is a totally real extension of degree at least three. The main changes are a new Motzkin-type lemma and the addition of the large sieve to the argument. These changes allow application of a powerful theorem due to Bombieri, Friedlander and Iwaniec in order to obtain the result in the real quadratic case. The modification also allows the completion of the classification of cyclotomic extensions in terms of the Euclidean property. Categories:11R04, 11R11 144. CJM 2004 (vol 56 pp. 23) Bennett, Michael A.; Skinner, Chris M. Ternary Diophantine Equations via Galois Representations and Modular Forms In this paper, we develop techniques for solving ternary Diophantine equations of the shape $Ax^n + By^n = Cz^2$, based upon the theory of Galois representations and modular forms. We subsequently utilize these methods to completely solve such equations for various choices of the parameters $A$, $B$ and $C$. We conclude with an application of our results to certain classical polynomial-exponential equations, such as those of Ramanujan--Nagell type. Categories:11D41, 11F11, 11G05 145. CJM 2004 (vol 56 pp. 71) Harper, Malcolm; Murty, M. Ram Euclidean Rings of Algebraic Integers Let $K$ be a finite Galois extension of the field of rational numbers with unit rank greater than~3. We prove that the ring of integers of $K$ is a Euclidean domain if and only if it is a principal ideal domain. This was previously known under the assumption of the generalized Riemann hypothesis for Dedekind zeta functions. We now prove this unconditionally. Categories:11R04, 11R27, 11R32, 11R42, 11N36 146. CJM 2004 (vol 56 pp. 194) Saikia, A. Selmer Groups of Elliptic Curves with Complex Multiplication Suppose $K$ is an imaginary quadratic field and $E$ is an elliptic curve over a number field $F$ with complex multiplication by the ring of integers in $K$. Let $p$ be a rational prime that splits as $\mathfrak{p}_{1}\mathfrak{p}_{2}$ in $K$. Let $E_{p^{n}}$ denote the $p^{n}$-division points on $E$. Assume that $F(E_{p^{n}})$ is abelian over $K$ for all $n\geq 0$. This paper proves that the Pontrjagin dual of the $\mathfrak{p}_{1}^{\infty}$-Selmer group of $E$ over $F(E_{p^{\infty}})$ is a finitely generated free $\Lambda$-module, where $\Lambda$ is the Iwasawa algebra of $\Gal\bigl(F(E_{p^{\infty}})/ F(E_{\mathfrak{p}_{1}^{\infty}\mathfrak{p}_{2}})\bigr)$. It also gives a simple formula for the rank of the Pontrjagin dual as a $\Lambda$-module. Categories:11R23, 11G05 147. CJM 2003 (vol 55 pp. 1191) Granville, Andrew; Soundararajan, K. Decay of Mean Values of Multiplicative Functions For given multiplicative function $f$, with $\|f(n)\| \leq 1$ for all $n$, we are interested in how fast its mean value $(1/x) \sum_{n\leq x} f(n)$ converges. Hal\'asz showed that this depends on the minimum $M$ (over $y\in \mathbb{R}$) of $\sum_{p\leq x} \bigl( 1 - \Re (f(p) p^{-iy}) \bigr) / p$, and subsequent authors gave the upper bound $\ll (1+M) e^{-M}$. For many applications it is necessary to have explicit constants in this and various related bounds, and we provide these via our own variant of the Hal\'asz-Montgomery lemma (in fact the constant we give is best possible up to a factor of 10). We also develop a new type of hybrid bound in terms of the location of the absolute value of $y$ that minimizes the sum above. As one application we give bounds for the least representatives of the cosets of the $k$-th powers mod~$p$. Categories:11N60, 11N56, 10K20, 11N37 148. CJM 2003 (vol 55 pp. 897) Archinard, Natália Hypergeometric Abelian Varieties In this paper, we construct abelian varieties associated to Gauss' and Appell--Lauricella hypergeometric series. Abelian varieties of this kind and the algebraic curves we define to construct them were considered by several authors in settings ranging from monodromy groups (Deligne, Mostow), exceptional sets (Cohen, Wolfart, W\"ustholz), modular embeddings (Cohen, Wolfart) to CM-type (Cohen, Shiga, Wolfart) and modularity (Darmon). Our contribution is to provide a complete, explicit and self-contained geometric construction. Categories:11, 14 149. CJM 2003 (vol 55 pp. 933) Beineke, Jennifer; Bump, Daniel Renormalized Periods on $\GL(3)$ A theory of renormalization of divergent integrals over torus periods on $\GL(3)$ is given, based on a relative truncation. It is shown that the renormalized periods of Eisenstein series have unexpected functional equations. Categories:11F12, 11F55 150. CJM 2003 (vol 55 pp. 711) Broughan, Kevin A. Adic Topologies for the Rational Integers A topology on $\mathbb{Z}$, which gives a nice proof that the set of prime integers is infinite, is characterised and examined. It is found to be homeomorphic to $\mathbb{Q}$, with a compact completion homeomorphic to the Cantor set. It has a natural place in a family of topologies on $\mathbb{Z}$, which includes the $p$-adics, and one in which the set of rational primes $\mathbb{P}$ is dense. Examples from number theory are given, including the primes and squares, Fermat numbers, Fibonacci numbers and $k$-free numbers. Keywords:$p$-adic, metrizable, quasi-valuation, topological ring,, completion, inverse limit, diophantine equation, prime integers,, Fermat numbers, Fibonacci numbersCategories:11B05, 11B25, 11B50, 13J10, 13B35 Page Previous 1 ... 5 6 7 ... 9 Next	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 2, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9870813488960266, "perplexity": 787.7766003948227}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1408500832738.80/warc/CC-MAIN-20140820021352-00153-ip-10-180-136-8.ec2.internal.warc.gz"}
http://www.megacz.com/software/wix/tutorial.html	# Tutorial Under construction, a long, long way from being useful. Right now this is mostly just notes to myself about topics I don't want to forget to cover. ## Commands Just as in TEX, a backslash followed by alphanumeric characters is a command: \command Also like TEX, an argument can be supplied to a command by surrounding it with curly braces and putting it immediately after the command (no whitespace). \command{argument} To supply multiple arguments, use square braces for all but the last argument. \command[arg1][arg2][arg3]{argument} What should round braces mean? \command(arg1){argument} Should we allow whitespace after commands for spacing purposes? \command (arg1) (arg2) {argument} ## Tokens There are several places where WIX interprets two tokens differently if they are separated by whitespace. For example, a set of curly braces after a command is considered an argument to the command if there is no intervening whitespace: \command{argument} Adding whitespace prevents WIX from interpreting the curly-braced text as an argument: \command {not-an-argument} However, this will also cause whitespace to appear between the command and the argument in the output. Sometimes this is not desirable; for example, consider the euro symbol: \euro 3.20 This is rendered as “€ 3.20” (note the space). If we want to typeset the “€” immediately adjacent to the “3” in order to get “€3.20” we cannot include whitespace there. In this situation – and any other situation where you want to separate two tokens but not introduce space in the output – the solution is to use a pair of empty braces: \euro{}3.20 You can think of the empty brace pair as being “a whitespace of width zero”. Another alternative is to put curly braces around the command {\euro}3.20 In WIX, you can add a link to any atom by putting the characters -> and a url after it, like this: Note that you can leave whitespace after the arrow – but not before it. This makes the paragraphs in your source code fill more easily (for example, with emacs' fill-paragraph command). WIX is pretty smart about figuring out where your url ends and the text begins again – it will try to build the longest valid url it can, but it will also try to avoid getting punctuation “stuck” on the end of an url. For example, if you type this: I like apples->http://wikipedia.org/apples, oranges, and pears. The url http://wikipedia.org/apples, (note the comma) is technically a valid url. However, most of the time when you type something like this, you probably didn't intend for the comma to be part of the url. So, WIX has special grammar rules that tell it to try to avoid including trailing punctuation in urls. If you really mean for an url to have trailing punctuation (some do), just put curly braces around the url: I like apples->{http://wikipedia.org/apples,} oranges, and pears. You can also use curly braces to have a link span multiple words: I like {apples, oranges, and pears}->{http://wikipedia.org/fruit} Also, wherever an url is allowed to appear, you can include a bare email address (ie without the usual mailto: prefix):	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8721863627433777, "perplexity": 1086.2362361084074}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368709906749/warc/CC-MAIN-20130516131146-00037-ip-10-60-113-184.ec2.internal.warc.gz"}
https://www.physicsforums.com/threads/is-1-a-prime-number.13647/	# Is 1 a prime number? 1. Feb 1, 2004 I've heard ppl say it is and it isn't, so is it a prime number? :P 2. Feb 1, 2004 ### chroot Staff Emeritus 3. Feb 1, 2004 yeah, I was kind of leaning towards that it's not a prime number, thanks for confirming :) 4. Feb 2, 2004 ### mathman Excluding 1 from the definition of prime number makes it easier to state many theorems in number theory. For example, every integer can be factored uniquely into a product of (unique) powers of primes. If 1 was called a prime, its exponent could be anything.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8393218517303467, "perplexity": 1196.2901547281178}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-09/segments/1487501171781.5/warc/CC-MAIN-20170219104611-00116-ip-10-171-10-108.ec2.internal.warc.gz"}
https://groups.google.com/g/sci.physics/c/cBaGg1AZc-k/m/v3hvP0ZRcncJ	# How was Dark Matter calculated? 5 views ### Jeffrey H Oct 30, 1999, 3:00:00 AM10/30/99 to Does anyone know what formula for gravity they used when they simulated the milky way on a supercomputer and discovered a need for additional mass ("dark matter")? I wonder if they used Newtonian gravity or Einsteinian? ### Nathan Urban Oct 30, 1999, 3:00:00 AM10/30/99 to Newtonian. You can show that Einsteinian corrections are negligible. [Followups to sci.astro.] ### Louis S Pogoda Jr Oct 30, 1999, 3:00:00 AM10/30/99 to I think you're assuming the problem is harder to see than it is. Matter was first found to be "missing" back in the 30's by Dutch astronomer van Oort. He looked at stars in the solar neighborhood whose motions were known. He found that, if you assume these nearby stars are part of the Milky Way and not just "passing through", they were moving too fast to be gravitationally bound by the visible mass of the galaxy. Based on the velocities of nearby stars, there must be about three times more mass in the galaxy than is visible. Subsequent studies seem to show that the larger the scale at which you consider things, the more mass that seems to be missing. Jeffrey H wrote in message <[email protected]>... ### Aubrey Oct 30, 1999, 3:00:00 AM10/30/99 to That's very interesting. Been wondering about that myself. Now I'm wondering why a large black hole at the center of our galaxy couldn't explain this excess speed instead of dark matter. Do you know? Louis S Pogoda Jr wrote in message ... ### Mark Folsom Oct 30, 1999, 3:00:00 AM10/30/99 to I think that the orbital velocity profile, speed versus radius, of stars in the galaxy isn't consistent with a large central mass. Rather, it appears that the excess mass is distributed over a large volume. Mark Folsom Aubrey <[email protected]> wrote in message news:#KvGdUwI\$GA.307@cpmsnbbsa05... ### Louis S Pogoda Jr Oct 31, 1999, 2:00:00 AM10/31/99 to Ah, wouldn't this "large black hole" BE dark? The problem is that there's apparently more gravitational attraction than that due to visible (which more or less means "luminous") matter - more than can be accounted for by the stars and gas clouds we see. Notice that for the solar neighborhood, there apparently needs to be three times more matter than we can see. Last I heard, large black holes at the center of galaxies, ours included, where in the millions of solar masses. That's nowhere near the 100 billion stars in the Milky Way, let alone three times as great. My original point was that the existence of some sort of dark matter can be, and was, inferred before the existence of supercomputers. It isn't necessary to model the galaxy in anything like realistic detail to come to that conclusion. Aubrey wrote in message <#KvGdUwI\$GA.307@cpmsnbbsa05>... ### z@z Nov 1, 1999, 3:00:00 AM11/1/99 to Does anyone know what formula for gravity they used when they simulated the milky way on a supercomputer and discovered a need for additional mass ("dark matter")? > Louis S Pogoda Jr replied: I think you're assuming the problem is harder to see than it is. Matter was first found to be "missing" back in the 30's by Dutch astronomer van Oort. He looked at stars in the solar neighborhood whose motions were known. He found that, if you assume these nearby stars are part of the Milky Way and not just "passing through", they were moving too fast to be gravitationally bound by the visible mass of the galaxy. Based on the velocities of nearby stars, there must be about three times more mass in the galaxy than is visible. Subsequent studies seem to show that the larger the scale at which you consider things, the more mass that seems to be missing. > > Aubrey wrote: That's very interesting. Been wondering about that myself. Now I'm wondering why a large black hole at the center of our galaxy couldn't explain this excess speed instead of dark matter. Do you know? > > > Mark Folsom replied: I think that the orbital velocity profile, speed versus radius, of stars in the galaxy isn't consistent with a large central mass. Rather, it appears that the excess mass is distributed over a large volume. I wonder if they used Newtonian gravity or Einsteinian? Newtonian. You can show that Einsteinian corrections are negligible. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - There is a very simple solution to at least a part of this dark-matter problem: the inersis hypothesis, i.e. the assumption that inertial motions do not follow straight lines but depend on the motions of the surrounding objects. The inersis concept can be considered a quantitative version of Mach's principle, based on 'weighted averages' of the velocities of the surrounding material objects (particles). Inersis can also be considered a relational ether which is dragged by all particles proportional to their mass and inversely proportional to their distance (i.e. proportional to lost gravitational potential). According to this concept, the higher velocities of stars in galaxies are caused by the same effect as Mercury's non-classical perihelion shift. Whereas the deviations from classical mechanics are still very small in planetary systems, they become the higher, the larger the scale at which one considers things. The inersis concept as first developed in 1987 is descibed in: http://members.lol.li/twostone/f31.html (in German) Assume an isolated group of six identical galaxies forming an equilateral hexagon. Further assume a little satellite galaxy s whose distance from galaxy 1 is the same as the distance between two neighbouring galaxies. 5 6 4 . 1 s 3 2 In the short term the shape of the group can remain stable if the six galaxies orbit at a given velocity V their common center. If we assume that inersis of s is affected only by these six galaxies, we get at the current location of s a tangential inersis velocity of 27% of the rotation speed of the group. This velocity is added to the velocity which is needed for s to compensate gravitational attraction of the group. (See reference above). In http://members.lol.li/twostone/a4.html (also in German) a quantitatively more accurate version of the explanation of the perihelion shifts of the planets can be found. Mercury's shift results primarily from the sun's rotation. The small shift of Mars however results primarily from Jupiter. It should be very easy to introduce the inersis concept into existing simulations of galaxies. I'm sure that much better agreement with empirical data will be the result. Maybe also the problem of the spiral forms of galaxies can be solved in a simpler and more transparent way. Wolfgang Gottfried G. http://members.lol.li/twostone/E/physics1.html	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9002566337585449, "perplexity": 1929.925564244138}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964363465.47/warc/CC-MAIN-20211208083545-20211208113545-00407.warc.gz"}
https://www.physicsforums.com/threads/astro-simulation-program-problem-program-bugs.158700/	# Astro simulation program problem - program bugs? 1. Mar 1, 2007 ### concious Hello all, for the past few weeks my friend and me have been working on a project involving simulation of orbital objects interaction. For example, one could see how everything looks if he/she entered all the details about the objects (that is: mass, position and velocity vector). The simulation is based on timesteps, that is, the system calculates new positions of objects after a timestep (default is one day). One can see how everything looks by "timestepping". The problem is, the calculations are bogus! Moreover, the formulas used MUST be right and this puts us into confusion: the problem lies not in the program but in the principle used. We pick each two objects, calculate the distance between them, the gravitational force between the objects, the average of speed gained ant the latter is added to objects' speed (obj.speed.x += accel.x * time_step / 2). After all that we move all the objects (like obj.place.x += obj.speed.x * time_step). Everything is calculated in the SI. We tested the system with the Earth and the Sun and it results in Earth going out of the orbit! So something's certainly wrong... We would appreciate any help. If there is any interest, we can send you the sources (C++, GPL license) which compile great on Win32 and Linux (OpenGL is used for displaying 3D graphics). Last edited: Mar 1, 2007 2. Mar 1, 2007 ### whatta aren't you supposed to sum all the forces and then divide by object mass and multiply that by delta_t and add that to speed? 3. Mar 1, 2007 ### gagiz What difference does that make if we sum the forces or the delta_speed? :? 4. Mar 1, 2007 ### whatta I thought you were saying you averaging them 5. Mar 1, 2007 ### gagiz Yes, that was a mistake, there should be accel * time / 2... Though it doesn't help :/. Last edited: Mar 2, 2007	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8819878697395325, "perplexity": 2550.894253832426}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988721008.78/warc/CC-MAIN-20161020183841-00280-ip-10-171-6-4.ec2.internal.warc.gz"}
https://dewesoft.pro/online/course/power-quality	EN English Italian German Languages # Power quality The different power quality parameters describe the deviation of the voltage from its ideal sinusoidal waveform at a certain frequency. These deviations can lead to disturbances, outages or damages of electrical equipment connected to the grid. ## Power quality The different power quality parameters describe the deviation of the voltage from its ideal sinusoidal waveform at a certain frequency. These deviations can lead to disturbances, outages or damages of electrical equipment connected to the grid. It is essential to permanently track these parameters: starting during the development phase (of the electrical equipment) until live operation: e.g. continuous monitoring of a couple of points in the electrical grid in order to prevent and correct quality disturbances The Dewesoft Power Analyser is able to measure all of these parameters according to IEC 61000-4-30 Class A. In comparison to other power quality analysers it’s possible to do more detailed analysis (e.g. raw data storing, behavior at faults, calculation of additional parameters etc.). The purpose of this chapter is to cover all power quality parameters which can be calculated in Dewesoft X. ## Harmonics Harmonics are integer multiples of the fundamental frequency (e.g. 50 Hz) and cause a distortion in voltage and current of the original waveform. Harmonic voltages and currents caused by non-sinusoidal loads can affect operation and lifetime of electrical equipment and devices. Harmonic frequencies in motors and generators can increase heating (iron & copper losses), can affect torque (pulsating or reduced torque), can create mechanical oscillations and higher audible noise, causes ageing of shaft, insulation and mechanical parts and reduce the efficiency. Current harmonics in transformers increase copper and stray flux losses. Voltage harmonics increase iron losses. The losses are directly proportional to the frequency and, therefore, higher frequency harmonic components are more important than lower frequency components. Harmonics can also cause vibrations and higher noise. The effects to other electrical equipment and devices are very similar and are mainly reduced efficiency and lifetime, increased heating, malfunction or even unpredictable behaviour. Dewesoft allows measuring harmonics for voltage, current and additional active and reactive power up to the 3000th order. All calculations are implemented according to IEC 61000-4-7 and can be selected in the power module according to the next picture: ## Harmonics Up to 500 harmonics can be calculated, in addition there is the option to choose all harmonics or just even or odd ones. If there are current channels used in the power module it is also possible to calculate phase angles, P, Q and the impedance. ## Number of sidebands The basic idea of sidebands is that a certain frequency range is considered as one harmonic. Example: 1 full sideband (equals +/-5Hz) at a frequency of 50 Hz means that a frequency range from 45-55 Hz is considered to be the first harmonic (it's the same for all other harmonics). If you select 2 sidebands the first harmonic will reach from 40 to 60 Hz. ## Number of halfbands The IEC 61000-4-7 (page 22) requires for the grouping of the harmonics sidebands where only the square root of the quadratic half of them should be added. This is required for the lowest and highest line and is defined as halfbands in Dewesoft. Example I: 1 sideband and 1 halfband at a frequency of 50 Hz mean that a frequency range from 45-55 Hz and the square root of the quadratic half of the 40 Hz to 60 Hz lines are considered to be the first harmonic. Example II: 2 sidebands and 1 halfband at a frequency of 50 Hz mean that the lines from 40 Hz to 60 Hz have the full amplitude, while the lines at 35 Hz and 65 Hz are only considered with the square root of the quadratic half. ## Interharmonics The interharmonics cover all lines not covered by the Harmonics (see page 26 of IEC 61000-4-7). Example: 1 sideband and 1 halfband at a frequency of 50 Hz, the first interharmonic is the area between 0 Hz and 45Hz. ## Group FFT lines The higher frequency parts can be grouped in 200 Hz bands up to 150 kHz. Attention: Please be aware that according to page 29 of IEC 61000-4-7 these groups start at -95 Hz to +100 Hz around the middle frequency. Example: For 2.1 kHz, lines from 2005 Hz to 2200 Hz are considered to be one group. ## Full FFT This option calculates the full FFT which can be exported to the database and displayed via 2D graph. ## Harmonics smoothing filter This option enables the low pass filter which is required in IEC 61000-4-7 according to page 23. ## Background harmonics With this option it is possible to subtract an existing and known harmonic pattern (magnitude and phase) from measured values. This is a typical application for the commissioning a powerful power converter in order to know the noise of this converter. You can use this function for the voltage and/or the current, as you can see in the following picture. All you need to do is to enter the magnitude und phase angle of the harmonic pattern into the following input mask. ## Example The following screens show a certain harmonic pattern measured and then subtracted in Dewesoft X using the background harmonics function. ## Measurement with Dewesoft X Voltage, current, active and reactive power, phase angle, impedance, interharmonics and higher frequencies can be displayes in Dewesoft X whether as numeric display, recorder or 2D graph. There are two possibilities for displaying harmonics. You can choose between: • Harmonic FFT • 2D graph With the Harmonic FFT you can display the harmonics of the voltage, the current, power or reactive power. In the picture below you can see the voltage harmonics of a three-phase system. With the 2D graph you can display voltages and currents of different phases in one graph. In addition, you have a wide range of display options. In the following figure you can see the harmonics of the phase voltage L1. On the picture below you can see the display options of the 2D graph. There you can choose the graph type (line or histogram) and linear or logarithmic display. Persistence means that when the harmonics change during a measurement, the changes are displayed blurred, for example like in the following figure: Higher frequencies Example: higher frequencies from 2 Hz up to 20 kHz shown in a 2D graph as histogram (application: HVDC converter station). Interharmonics Example: interharmonics shown in 2D-Graph as histogram. Peak at 900 Hz which is the switching frequency of a HVDC converter operated in the public grid. ## Dewesoft X calculations Dewesoft X calculations for each harmonic/whole waveform: Dewesoft X calculations for a 3 phase system: ## THD - total harmonic distortion The total harmonic distortion (THD) for voltage and current can be calculated up to the 3000th order. In general it is defined as sum of all harmonics to the fundamental. The most important producers of harmonics are loads which are controlled by converters (diodes, thyristors, transistors). In the following two pictures you find a typical comparison of lamps and the produced harmonics and a representative illustration of different converter technologies and the according THD value. ## Fundamental symmetrical components Normally the electric power system operates in a balanced three-phase sinusoidal steady-state mode. Disturbances, for example a fault or short circuit, lead to a unbalanced condition. By using the method of the symmetrical components it is possible to transform any unbalanced threephase system into 3 separated sets of balanced three-phase components, the positive, negative and zero sequence. The main advantage of the symmetrical components is that it makes life (and calculation) much easier. In case of a fault or short circuit the unbalanced system can be easily transformed into symmetrical components, wherewith the calculations can be done straight forward. In the end the results are transformed back into the „real-life“ phase voltages and currents. In general a 3-phase system can be displayed and described as following: A balanced 3-phase system may look like this: same RMS-value for all line voltages and currents, and a 120° phase shift between each of them. In order to explain the basic idea of the symmetrical components, the first step is to define the operator „a“ as unit vector with an phase angle of 120° or 2pi/3. So the voltages can be described in different ways now: ## Calculation of zero-sequence system In an symmetrical system the following equation is valid: In a real system this sum is not equal to zero. A voltage difference occurs: This voltage difference divided through 3 represents the so called zero-sequence system: The zero-sequence systems for the three phases (u10, u20, u30) have the same aplitude and phase. Therefore, only value for the zero-sequence system "U_0" will be shown. The calculation of the zero-sequence current is analogue to this procedure. HINT: If you multiply the currents for the zero-sequences system with 3 (=I_0 x 3) you will get the current of the neutral line U_N. ## Calculation of positive-sequence system The positive sequence system has the same rotating direction as the original system (right). This means it will have the same rotating direction of an electrical machine connected to the grid. As the values of the positive-sequence system for all three phases have the same amplitude (now they are symmetrical) and an phase shift of exactly 120°, it's enough to show one value. The value for the positive-sequence system in Dewesoft X is called „U_1“. ## Calculation of negative-sequence system The negative sequence system has the opposite rotating direction as the original system (left). This means it will rotate in opposite direction of an electrical machine connected to the grid. As the values of the negative-sequence system for all three phases have the same amplitude (now they are symmetrical) and an phase shift of exactly 120°, it's enough to show one value. The value for the negative-sequence system in Dewesoft X is called „U_2“. ## Matrix of zero, positive and negative-sequence system According to the following equations the phase voltages and currents are transformed into the symmetrical components. The result are three balanced 3-phase systems, the positive (U 1, I1), negative (U 2, I2) and zero sequence (U0, I0). NOTE: The basic values of symmetrical components (U_0, U_1, U_2, I_0, I_1, I_2) are calculated for each harmonic and added up geometrically. As you can see in the next pictures, a unbalanced system can be composed by using the positive, negative and zero symmetrical components. The following picture shows an screenshot of Dewesoft X showing the real system: The following picture shows an screenshot of Dewesoft X showing the three systems (positive, negative and zero) of the symmetrical components: This screen is provided by Kurt Stranner (KNG Netz GmbH). Out of the parameters of the symmetrical components (positive-, negative- zero- sequence) the original system can easily be rebuild: The following variables are calculated in Dewesoft X and shows the components of the zero- and negative-sequence system compared to the positive-sequence system (for total and fundamental harmonic). ## Extended positive sequence parameters (according to IEC 614000) The following calculations are based on Annex C of IEC 61400-21. Based on the measured phase voltages and currents, the fundamental's Fourier coefficients are calculated over one fundamental cycle T as first step. It is important to mention that the index a stands for the line voltage L1. The coefficients for L2 (ub) and L 3 (uc) as well as the coeffiecients for the currents (ia, ib, ic) are calculated exactly the same. Furthermore f 1 is the frequency of the fundamental. The RMS value of the fundamental line voltage is: ## Flicker Flicker is a term for the fluctuations (repeated variations) of voltage. Flashing light bulbs are indicators for a high flicker exposure. Flicker is especially present at grids with a low short-circuit resistance and is caused by the frequent connection and disconnection (e.g. heat pumps, rolling mills, etc.) of loads which affects the voltage. A high level of flicker is perceived as psychologically irritating and can be harmful to people. The Dewesoft power analyser allows to measure all flicker parameters according to IEC 61000-4-15. The flicker emission calculation is implemented according to IEC 61400-21 and allows the evaluation of flicker emission in to the grid caused by wind power plants or other generation units. The flickermeter architecture is described by the block diagram of the next figure. It is divided into two parts, simulation of the response of the lamp-eye-brain chain und the on-line statistical analysis of the flicker signal leading to the known parameters. ## Block 1 This block contains a voltage adapting circuit that scales the input mains frequency voltage to an internal reference level. This method permits flicker measurements to be made, independently of the actual input carrier voltage level and may be expressed as a per cent ratio. ## Block 2 The purpose of the second block is to recover the voltage fluctuation by squaring the input voltage scaled to the reference leve, thus simulating the behaviour of a lamp. ## Block 3 This block is composed of a cascade of two filters, which can precede or follow the selective filter circuit. The first low-pass filter eliminates the double mains frequency ripple components of the demodulator ouptut. The high pass filter can be used to eliminate any DC voltage component. The second filter is a weighting filter block that simulates the frequency response of the human visual system to sinusoidal voltage fluctuations of a coiled filament gas-filled lamp (60W/230V and/or 60W/120V). ## Block 4 Is composed of a squaring multiplier and a first order low-pass filter. The human flicker perception, by the eye and brain combination, to voltage fluctuations applied to the reference lamp, is simulated by the combined non-linear response of blocks 2,3 and 4. ## Block 5 The last block of the whole chain performs an on-line analysis of the flicker level, thus allowing direct calculation of significant evaluation parameters. In the next figure you can see an example of a rectangular voltage flicker. ## Measurement with Dewesoft X - flicker With Dewesoft X you can calculate the Pst and Plt value according to the IEC norm with a calculation time of 10 minutes respectively 120 minutes, though it is also possible to adapt the calculation time to your needs, set a calculation overlap and filter. ## Flicker emission The flicker emission (also called current flicker) calculates the proportion of the flicker, which is added to the grid by a producer or a consumer. In addition the internal voltage drop is calculated by the grid impedance of the current flow. The voltage drop is added to an idealized voltage source vectorial (U = Usim + RI + L di/dt). Using the flicker algorithm and the new voltage the current flicker values are calculated. Enable "Flicker" and "Current Flicker" and input the grid parameters. You can type in the short circuit power or the impedance of the grid. The phase is the impedance angle of the grid. You can also type in a number of different phase angles (e.g. 30;50;70;85). NOTE: In the power module the nominal voltage must be set. This value is also the value for the idealized voltage source! Now let's take a look at a flicker measurement with the settings from above. The following parameters are calculated with Dewesoft X: ## Rapid voltage changes The rapid voltage changes are parameters which are added as a supplement to the flicker standard. Rapid voltage changes describe all voltage changes which are changing the voltage for more than 3% at a certain time interval. These voltage changes can afterwards be analysed with different parameters (depth of voltage change, duration, steady state deviation, etc.). The rapid voltage changes (RVC) are a special calculation in Dewesoft X which allows to calculate the maximal voltage drop (d max), the stationary deviation after the voltage drop (dc) and the time where the voltage drop is below 3,3% of U n. All values are calculated according to the IEC 61000-4-15. Analysis can be done for example for IEC 61000-3-3 and IEC 61000-3-11. The following picture shows the calculated parameters (IEC 61000-4-15 page 35). ## Measurements with Dewesoft X Let's take a look at the rapid voltage changes with Dewesoft X: Steady state duration is defined in seconds. Hysteresis is the condition for the stationary deviation (du_dc), see IEC 61000-4-15 page 8. EXAMPLE: If you define a hysteresis of a 0,2% and a steady state duration of 1s, the stationary condition is reached if the voltage doesn't leave +- 0,2% for 1 second. NOTE: The rapid voltage changes values (du_max, du_duration, du_dc) are calculated out of the defined settings for period values (number of periods and overlap). Take care of right settings for analysis according to related standards (½ period values for RVC determination according to IEC61000-4-15) ## Channel list in Dewesoft X Dewesoft has created a special FFT algorithm (software PLL) to determine the periodic time (frequency). The algorithm determines the periodic time of the signal via a special FFT algorithm at a sampling window of multiple periods (typically 10 periods, definable in power module). The calculated frequency is highly accurate (mHz) and works for every application (motor, inverter, grid, …). How Dewesoft X calculates the power of an AC system While other power analysers calculate the power in the time domain, in Dewesoft X it is calculated in the frequency domain. With the before determined period time, an FFT analysis for voltage and current is done for a definable number of periods (typically 10 with electrical applications) and a definable sampling rate. Out of this FFT analysis, we get an amplitude for the voltage, current and cos phi for each harmonic. One major benefit of this FFT transformation is that we can now correct the behaviour of amplifiers, current or voltage transducers in amplitude and phase for the whole frequency range (using the Sensor XML). This way of power analysis has the highest possible accuracy. Another benefit is that harmonic analysis and other power quality analysis can be done completely synchronized to the fundamental frequency. With the FFT corrected values, the RMS voltages and currents are calculated out of the RMS values of each harmonic. The power values for each harmonic and the total values are calculated with the following formulas: ## Applications Power grid • fault and transient recording • power quality analysis (IEEE 1159, EN50160) Transformer • efficiency analysis (IEC 60076-1) • no-load and short circuit testing • vibration, noise Wind, solar and CHP • power performance (IEC 61400-12) • power quality (IEC 61400-21 / FGW-TR3) • active and reactive power (FGW-TR3) • behaviour at faults (FGW-TR3) Nuclear power plant • turbine and generator • testing rod drop • castor testing Turbine and generator • modal analysis • order tracking • balancing • rotational vibration • efficiency measurement Smart grid and energy management • power system testing • demand side management Aircraft • power system testing • fault and transient recording • hybrid testing • harmonics analysis Marine • power system testing • fault and transient recording • hybrid testing Railway • power system testing (AC and DC rails) • power quality analysis • fault and transient recording • short-circuit analysis • pantograph and current shoe testing E-mobility • electric two wheeler • electric vehicle • hybrid vehicle (series and parallel) • hydrogen vehicle Equipment testing • fans and pumps testing • circuit breaker testing • filter analysis • harmonics analysis according to IEC 61000-3-2/-12 • voltage changes according to IEC 61000-3-3/-11 • CE conformity of electrical devices (harmonics, flicker) and a lot more	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8117043972015381, "perplexity": 1914.0064235968837}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-09/segments/1550247494125.62/warc/CC-MAIN-20190220003821-20190220025821-00312.warc.gz"}
https://aas.org/archives/BAAS/v29n2/aas190/abs/S050001.html	Session 50 - Invited Talk. Invited session, Thursday, June 12 North Main Hall A, ## [50.01] ISO's View on the Interstellar Medium and Star Formation E. F. van Dishoeck (Leiden Obs.) The Infrared Space Observatory (ISO), launched November 17 1995, provides the first opportunity for spectroscopic observations over the complete 2.4-200 \mum wavelength range above the atmosphere. Its scientific payload consists of 4 instruments: a short wavelength spectrometer (SWS), a long wavelength spectrometer (LWS), a camera (ISOCAM) and an imaging photopolarimeter (ISOPHOT). The two spectrometers provide resolving powers ranging from 200 to 30,000. The wavelength range covered by ISO is very rich in atomic and molecular lines, so that a wide variety of astrophysical objects (ionized and neutral interstellar clouds, star-forming regions, stars, galaxies, planetary atmospheres, comets) are being observed. In this talk, an overview will be presented of the most exciting new results on the interstellar medium and star formation obtained with the SWS and LWS. These include direct observations of the rotational transitions of the dominant interstellar molecule, H_2, which are readily detected in warm molecular gas found in photon-dominated regions, shocks, and the envelopes and circumstellar disks around young stellar objects. The data provide important constraints on the physical state of these regions, the heating and cooling mechanisms, and the fraction of warm gas at \geq100 K. Other topics include the surprising detection of new solid--state features due to crystalline silicates and PAH's in the mid-infrared spectra of old and young stars and comets, the observation of high ionization lines in planetary nebulae, the identification of new molecules in interstellar ices, and the wealth of data on gas-phase H_2O in a variety of objects. The observations will be discussed in the context of the physical and chemical evolution of low- and high-mass young stellar objects.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8158872127532959, "perplexity": 3714.0096768195904}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-22/segments/1464049272349.32/warc/CC-MAIN-20160524002112-00164-ip-10-185-217-139.ec2.internal.warc.gz"}
https://www.gradesaver.com/textbooks/math/algebra/elementary-and-intermediate-algebra-concepts-and-applications-6th-edition/chapter-2-equations-inequalities-and-problem-solving-2-6-solving-inequalities-2-6-exercise-set-page-135/36	## Elementary and Intermediate Algebra: Concepts & Applications (6th Edition) $\text{Set Builder Notation: } \{x\|x\ge-6\} \\\text{Interval Notation: } [-6,\infty)$ The given inequality, $x\ge-6 ,$ can be written in the following formats: \begin{array}{l}\require{cancel} \text{Set Builder Notation: } \{x\|x\ge-6\} \\\text{Interval Notation: } [-6,\infty) .\end{array} Note that in the interval notation, use parentheses for $\lt$ or $\gt.$ Use a bracket for $\le$ or for $\ge.$ In the graph, a hollowed dot is used for $\lt$ or $\gt.$ A solid dot is used for $\le$ or $\ge.$	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9614199995994568, "perplexity": 1085.2322040047623}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439738699.68/warc/CC-MAIN-20200810205824-20200810235824-00309.warc.gz"}
https://www.physicsforums.com/threads/integration-partial-fractions.103180/	# Integration: Partial Fractions 1. Dec 7, 2005 ### Alw How does this work? All i really understood from class was that you would factor the integrand and then somehow A and B were involved, and you would use systems of equations to find A and B. What's the middle ground? Thanks in advance! 2. Dec 7, 2005 ### Jameson Alright, so let's say you have to integrate the following expression. (1)$$\int \frac{1}{x^2-1}dx$$ It should first be noticed that this doesn't follow any of the standard integration rules, like the natural log one for example and that another method should be employed. So to break this up into partial fractions you should factor the denominator and split the expression into two fractions like so. (2)$$\frac{1}{x^2-1}= \frac{A}{x-1}+\frac{B}{x+1}$$ Now multiply through by (x-1)(x+1) to get this into something more workable. Doing this you'll get: (3)$$1=A(x+1)+B(x-1)$$ You can use this expression and your previous one to do a system of equations, but another method is much simpler. Let x=-1 so that the A term will be zero and you can solve for B. Now let x=1 so that the B term will be zero and you can solve for A. I'll let you finish this one, but I hope the concept explanation helps! Jameson	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9651172757148743, "perplexity": 417.84497796902343}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547583659063.33/warc/CC-MAIN-20190117184304-20190117210304-00200.warc.gz"}
http://eprints.iisc.ernet.in/2459/	Home \| About \| Browse \| Latest Additions \| Advanced Search \| Contact \| Help Mixed alkali effect in borate glasses-electron paramagnetic resonance and optical absorption studies in $Cu^{2+}$ doped $xNa_2O-(30 - x)K_2O-70B_2O_3$ glasses Chakradhar, Sreekanth RP and Ramesh, KP and Rao, JL and Ramakrishna, J (2003) Mixed alkali effect in borate glasses-electron paramagnetic resonance and optical absorption studies in $Cu^{2+}$ doped $xNa_2O-(30 - x)K_2O-70B_2O_3$ glasses. In: Journal of Physics: Condensed Matter, 15 (9). pp. 1469-1486. Preview PDF 2003Cu.pdf The mixed alkali borate glasses $xNa_2O-(30 - x)K_2O-70B_2O_3$ (5 \leq x \leq 25, doped with 0.5 mol% of CuO, have been investigated, using electron paramagnetic resonance (EPR) and optical absorption techniques, as a function of mixed alkali content, to look for the 'mixed alkali effect' (MAE) on the spectral properties of the glasses. The EPR spectra of all the investigated samples exhibit resonance signals which are characteristic of the $Cu^{2+}$ ions in octahedral sites with tetragonal distortion. From the observed EPR spectra, the spin-Hamiltonian parameters have been determined. It is observed that the spin-Hamiltonian parameter $g_{\\|}$ goes through a minimum around x = 10-15 whereas $A_{\\|}$ goes through a maximum around x = 15 showing the MAE. The number of spins participating in resonance $(N_2)$ and the calculated paramagnetic susceptibilities (\chi) exhibit a shallow minimum around x = 20 showing the MAE in these glasses. The optical absorption spectrum of the x = 5 glass exhibits two bands: a strong band centred at 14 240 $cm^{-1}$ corresponding to the transition $(^2B_1g \rightarrow ^2B_2g)$ and a weak band on the higher energy side at 22 115 $cm^{-1}$ corresponding to the transition $(^2B_1g \rightarrow ^2E_g)$. With x > 5, the higher energy band disappears and the lower energy band shifts slightly to the lower energy side. By correlating the EPR and optical absorption data, the molecular orbital coefficients ${\alpha}^2$ and ${\beta}^2_1$ are evaluated for the different glasses investigated. The values indicate that the in-plane \sigma bonding is moderately covalent while the in-plane \pi bonding is significantly ionic in nature; these exhibit a minimum with x = 15, showing the MAE. The theoretical values of optical basicity of the glasses have also been evaluated. From optical absorption edges, the optical bandgap energies have been calculated and are found to lie in the range 3.00-3.40 eV. The physical properties of the glasses studied have also been evaluated with respect to the composition.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8700911402702332, "perplexity": 1958.8526757672585}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891812841.74/warc/CC-MAIN-20180219211247-20180219231247-00135.warc.gz"}
https://www.physicsforums.com/threads/momentum-enegy-and-collisions.296406/	# Momentum, enegy and collisions 1. Mar 1, 2009 ### hellothere123 A proton (mass 1 u) moving at 7.80 x10^6m/s collides elastically and head-on with a second particle moving in the opposite direction at 2.40x10^6 m/s. After the collision, the proton is moving opposite to its initial direction at 6.60x10^6m/s. Find the mass and final velocity of the second particle. [Take the proton's initial velocity to be in the positive direction.] I tried using the conservation of momentum and kinetic energy to do this. I get this big mess that i cannot solve for.. Please show me how i would do this. i would like to learn to do these problems. thanks. 2. Mar 1, 2009 ### LowlyPion Re: momentum You should have 2 equations with 2 unknowns from each of the conservations. Make it simpler by dropping the powers of 10 and add it back as a factor at the end. Try writing out your equations here. 3. Mar 2, 2009 ### hellothere123 Re: momentum so if we let m1 be the proton. we have.. (7.8)-(m2)(2.4)=(-6.6)+m2v2 (7.8^2)+m2(2.4^2)=(6.6^2)+m2(v2)^2 m2 = [(7.8^2)-(6.6^2)]/[(v2)^2-(2.4^2)] and then when i plug it back in, i get a quadratic. that gives me a mass considerably larger than the other one. 4. Mar 2, 2009 ### The Bob Re: momentum What do you mean considerably? If your maths is right then what stops the second particle being very large? The Bob 5. Mar 2, 2009 ### hellothere123 Re: momentum my math is probably bad, that is why i was hoping someone could show me the math so i can see where i went wrong 6. Mar 2, 2009 ### LowlyPion Re: momentum OK. So you can say that m(v + 2.4) = 14.4 from the first equation. And ... m(V2 - 2.42) = 7.82 - 6.62 Note that this factors easily into m(V + 2.4)(V - 2.4) = 7.82 - 6.62 But from the first equation you know m(v + 2.4) = 14.4 So ... 14.4*(V - 2.4) = 7.82 - 6.62 Much easier than a quadratic to solve. EDIT: Sorry the 7.82 and 6.62 terms got translated incorrectly. I fixed them now. Last edited: Mar 2, 2009 7. Mar 2, 2009 ### hellothere123 Re: momentum that is much easier and such.. and after working it out.. i did not get the right answer. the answer was 3.6 doing that gave me 1.2 8. Mar 2, 2009 ### LowlyPion Re: momentum Sorry. I apparently switched two terms inadvertently in typing it out. I just fixed it.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9395736455917358, "perplexity": 1704.0958115643868}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-09/segments/1550247500089.84/warc/CC-MAIN-20190221051342-20190221073342-00369.warc.gz"}
https://www.physicsforums.com/threads/shm-question-involving-a-tunnel-through-earth.534272/	# SHM question involving a tunnel through earth 1. Sep 27, 2011 ### craig.16 1. The problem statement, all variables and given/known data A tunnel is bored through the centre of the Earth from Liverpool to New Zealand and a travel pod is dropped into it. The gravitational force on the pod is proportional to its distance from the centre of the Earth and so it undergoes simple harmonic motion described by z=R0cos$\omega$t where z=position relative to the centre of the Earth $\omega$=1.2*10-3rads-1 1) What is the velocity of the pod when it reaches the centre of the Earth? 2) How long does the pod take to travel to the opposite side of the Earth? 2. Relevant equations The one displayed above 3. The attempt at a solution For question 1) since its at the centre of the Earth z=0 giving: 0=R0cos$\omega$t cos$\omega$t=0 $\omega$t=1 t=1/$\omega$ and thats where I hit a dead end. I know my approach is completely wrong since the end result for velocity is faster than the speed of light however I don't know exactly how to remedy this situation. Any hints will be greatly appreciated as this is frustrating me how I can't answer a basic SHM question like this. 1. The problem statement, all variables and given/known data 2. Relevant equations 3. The attempt at a solution 2. Sep 27, 2011 ### SammyS Staff Emeritus I don't see how you calculated velocity at all. What did you get for the time to reach the center of the Earth? What's the period of this motion? What is dz/dt as a function of t? 3. Sep 28, 2011 ### craig.16 Doesn't matter, I worked it out now. Thanks for the reply anyway. Similar Discussions: SHM question involving a tunnel through earth	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8485392332077026, "perplexity": 504.36234273511946}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-39/segments/1505818692236.58/warc/CC-MAIN-20170925164022-20170925184022-00376.warc.gz"}
https://socratic.org/questions/how-do-you-write-the-point-slope-form-of-an-equation-for-a-lint-that-passes-thro	Algebra Topics # How do you write the point slope form of an equation for a lint that passes through (3,8) with m=2? Apr 24, 2017 See the solution process below: #### Explanation: The point-slope formula states: $\left(y - \textcolor{red}{{y}_{1}}\right) = \textcolor{b l u e}{m} \left(x - \textcolor{red}{{x}_{1}}\right)$ Where $\textcolor{b l u e}{m}$ is the slope and $\textcolor{red}{\left(\left({x}_{1} , {y}_{1}\right)\right)}$ is a point the line passes through. Substituting the slope and values of the point from the problem gives: $\left(y - \textcolor{red}{8}\right) = \textcolor{b l u e}{2} \left(x - \textcolor{red}{3}\right)$ ##### Impact of this question 389 views around the world You can reuse this answer	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 4, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9753295183181763, "perplexity": 1137.743623712215}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487649688.44/warc/CC-MAIN-20210619172612-20210619202612-00284.warc.gz"}
http://math.stackexchange.com/questions/295751/how-to-determine-the-sum-of-sum-k-1-infty-frackkk2	# How to determine the sum of $\sum_{k=1}^{\infty} \frac{k^k}{(k!)^2}$ $$\sum_{k=1}^{\infty} \frac{k^k}{(k!)^2}$$ The series converges by the ratio test but how would I find out the exact sum of the series. - Do you want the value to which the sum converges? or the limit of the sequence being summed? – amWhy Feb 5 '13 at 20:43 Sum, sorry I meant the sum. – MJBoa Feb 5 '13 at 20:44 no reason to think there is a nice closed-form expression for the sum. Why do you ask? – Will Jagy Feb 5 '13 at 20:47 Well, its on an old test from one of my classes, and the sum was asked for. I'm correct in saying it converges, at least? – MJBoa Feb 5 '13 at 20:49 The sum seems to converge, since $k^k \approx \frac{k!e^k}{\sqrt{2 \pi k}}$ using Stirling's approximation. Cancelling out $k!$ the summand becomes $\frac{e^k}{k!\sqrt{2 \pi k}} < \frac{e^k}{k!}$ for $k \geq 1$. since the latter sum clearly converges to $e^{e}-1$, by comparison test the former sum converges too.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9781623482704163, "perplexity": 287.96350381544204}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-35/segments/1440644064865.49/warc/CC-MAIN-20150827025424-00001-ip-10-171-96-226.ec2.internal.warc.gz"}
https://wimcouwenberg.wordpress.com/2016/07/27/mutual-information-and-gradients/	## Mutual information and gradients In this post we will see a that mutual information between functions (e.g. two images) can be expressed in terms of their gradient fields. First some definitions and background. Let $f{: [0, 1]^n} \to [0, 1]^m$ be a differential function for some $n \geq m$. In this post $f$ typically represents an image or a pair of images with $n=2$ and $m \in \{1, 2\}$. Let $D$ be the Jacobian of $f$. This is the $m \times n$ matrix with $\displaystyle{D_{ij}=\frac{\partial f_i}{\partial x_j}}$ for $1 \leq i \leq m$ and $1 \leq j \leq n$. The entropy $H(f)$ of $f$ is defined by the integral $H(f) = \displaystyle{\int_{[0,1]^n} \tfrac12 \log \lvert D D^{\mathrm{t}}\rvert \, \mathrm{d}V}$ where $\lvert \cdot \rvert$ is the determinant and $\mathrm{d}V$ the standard volume element. (In this post I will disregard any question of well-definedness of this integral.) To motivate this definition: if $n=m$ and $f$ is injective then $H(f)$ is the usual differential entropy of the push forward $f_{\ast}(\mathrm{d}V)$ of the standard volume form. If $m=1$ then the Jacobian $D = \nabla f$ equals the gradient of $f$ and $\lvert D D^{\mathrm{t}} \rvert = \lVert \nabla f \rVert^2$ so the entropy becomes $\displaystyle{H(f) = \int_{[0,1]^n} \log \lVert \nabla f \rVert \, \mathrm{d}V}$. Let $f = (f_1, \ldots, f_m)$ then the mutual information $I(f)$ of $f$ is defined by $\displaystyle{I(f) = \sum_{k=1}^m H(f_k) - H(f)}$. Mutual information is always non-negative. It expresses how much information is gained by knowing the joint value distribution of $f$ compared to knowing only the value distributions of the separate coordinates $f_1, \ldots, f_m$. In other words, mutual information is a measure of dependence between the coordinates: The higher the dependence the higher the mutual information while for independent coordinates the mutual information is $0$ (there is no information to be gained from their joint value distribution). The nice thing about mutual information is that it is invariant under any injective coordinate-wise distortion. In imaging related terms it is for example invariant under changes of gamma, gain and offset of the image. This is hugely beneficial in practical imaging applications where lighting conditions are never the same. Different images (the coordinates) may even have been produced with completely different sensing equipment. A key observation about mutual information is the following: $\displaystyle{\lvert D D^{\mathrm{t}} \rvert = v \cdot \prod_{k=1}^m \lVert \nabla f_k \rVert^2}$ for some function $v$ with values in $[0, 1]$ that depends only on the direction of the gradients but not their length. Moreover $v=0$ if and only if the gradients are linearly dependent and $v=1$ if and only if they are mutually orthogonal. Using this decomposition mutual information can be expressed as $\displaystyle{I(f) = \int_{[0,1]^n} -\tfrac12 \log(v) \, \mathrm{d}V}$. This confirms that mutual information is non-negative since $v\in[0,1]$ and therefore $\log(v) \leq 0$. I will conclude this post by looking at the specific case of a pair of 2-dimensional images so the case that $n=m=2$. Then the function $v$ has a simple explicit form. Let $\alpha$ be the angle between the gradients $\nabla f_1$ and $\nabla f_2$. Then $v = \sin^2(\alpha) = \frac{1-\cos(2\alpha)}{2}$. There are two pleasant observations to make: 1. Mutual information of a pair of images depends only on the double angle between their gradients. In particular it does not depend on the length or a sign change of either gradient. 2. The expression $\cos(2\alpha)$ is easy to compute as an inner product. The double angle can be accounted for by a simple rational transformation of the gradient. This will be explained in more detail in a next post. A next post will discuss the application of mutual information to image registration. It results in a method that is very efficient (based on FFT), is robust against image distortions and can also be applied to register (locate) a partial template image of any shape within a bigger scene. This entry was posted in Uncategorized. 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https://www.physicsforums.com/threads/questions-concerning-cosmological-constant.622406/	# Questions concerning Cosmological Constant 1. Jul 21, 2012 ### Rorkster2 I understand the begining basics of what it represents but how exactly does it tie into the inflation of space and what sort of energy is in a perfect vacuum? 2. Jul 21, 2012 ### tom.stoer The cosmological constant in Einstein's equations is simply a constant which (in absence of matter) causes a vacuum deSitter space time. Inflation caused by some inflaton matter field or "dark energy" as a source of accelerated expansion are mechanism caused by other (hypothetical) fields which mimic something like a cosmological constant. So instead of treating the constant as a constant it can be generated by the dynamics of other fields, it may vary in time which should explain why inflation had a beginning and an end etc. Both the inflaton and dark energy are more general mechanism than this cosmological constant. Current observations are in good agreement with the model of a simple cosmological constant. It is not known whether this constant is a constant to be introduced by hand (like Newtons constant), whether it can be explained by other fields or whether it is a constant + additional effects. Last edited: Jul 21, 2012 3. Jul 21, 2012 ### Naty1 Your question would be a sufficient basis for a lifetime career in cosmology..we have lots to learn! edit: And quantum mechanics, see next post. The cosmological constant remains rather mysterious as does dark energy and dark matter...Even Einstein got fooled initially. Tom's post captures many of the uncertainties. It is not perfectly 'settled science'....not that anything really is. What we observe is that the expansion of the universe is apparant, it's expansion has varied over time, and we are currently in a period of accelerated expansion. Read this recent discussion for some interesting insights: Expnding Universe Here I believe there is a bit of different usage in the discussion between the terms dark energy, cosmological constant, expansion and accelerated expansion in different posts....and a difference in their effects is explained. The Wikipedia discussion [DARK ENERGY] seems to me the equate dark energy and the cosmological constant. Note MarkM's post #18, and #34.....where the latter shows the relationship between vacuum energy and the cosmological constant. The cosmological seems to come along for 'free' with vacuum energy...also the 'ultimate free ride'....Whatever space is, it does not appear to be 'empty'. There are long discussions with these forums whether it is the mathematical 'metric' associated with our cosmological model [which describes distance] is all that is expanding or whether space itself is inflating/or expanding. Is iflation one time in one universe or eternal in an infinite number of universes? Is space a 'fabric' or just a geometric construct? Is space discrete or continuous or does that question not even make sense? All these have been the subject of discussions in these forums and are the subject of ongoing research papers. Last edited: Jul 21, 2012 4. Jul 21, 2012 ### Naty1 I kind of skipped that. Do you know what a 'perfect vacuum' is? I don't. But I do know there are a lot of 'different' vacuums...Wikipedia discusses them here: http://en.wikipedia.org/wiki/Perfect_vacuum#In_quantum_mechanics so the simple answer is: with every vacuum comes with some energy. I don't think anybody knows exactly what it is...but if so, somebody will correct me. One evidence MAY be the Casimer effect...but some think that has other origins. Let's just assume a good vacuum as in outer space. I think a 'perfect vacuum' would be in it's ground state, it's lowest energy configuration, but never zero. Quantum mechanics tells us that there are always fluctuations in fields and they are never quiescent, always bubbling....even though they may have zero average value. Space comes with some form of energy which is likely exactly what you are asking about. [Maybe one piece of that is dark energy??] One way to think about this is that at Planck scale, really,really tiny space and times, space and time get mixed up....there is always a 'froth'....an uncertainty. In fact Heisenberg uncertainty can also be utilized to described why there is always energy, variations in fields. But these theories do not REALLY describe exactly WHAT that energy is...only behaviors and rules we have come to observe. As a final perspective: space comes with horizons. Horizons 'cause' energy to pop out from 'nothing'....and even particles to appear! The Unruh effect and Hawking radiation are examples. If two observers are together, and one is inertial [free floating] and the other accelerating, they will count different numbers of vacuum particles and measure [ever so slightly different] temperatures! So space has 'vacuum energy' and expanding space [with horizons, like the Hubble sphere,for example] also comes with additional energy and particles [matter and radiation]. As you can tell, lots more to learn here too. 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http://chromotopy.org/?p=1271	On L-functions and algebraic K-theory Hi all! In this post, I’ll attempt to make some progress towards demystifying the relationship between L-functions and algebraic K-theory, with reference to a very simple example. I’ll say something about L-functions a little lower down, but the only thing I’ll say about K-theory in general is that it’s a way to get a spectrum from, say, a scheme. The deep connection between these two objects is supposed to be exemplified by the following very approximate and possibly incorrect statement. Ur-conjecture. [Lichtenbaum, Bloch-Kato, Beilinson, Scholbach…] Let $X$ be a geometric object, probably a scheme of a particular kind, and let $\zeta_X$ be the zeta function associated to $X$. Then for $n$ a sufficiently large positive integer, where $R(n)$ is some factor. Obviously this is completely meaningless until you know what $R(n)$ is, but people do have expressions for it (a relevant buzzword here is “Borel regulator”), about which I’ll say no more. The point is that there are formulae for values of L-functions at integers (and I’m sorry, but I’ll use “L-function” and “zeta function” interchangeably) involving torsion orders of K-groups. A conjecture of this form has been proven for $X$ the ring of integers of a totally real abelian field by piling Iwasawa theory and other manoeuverings onto the cases of the Quillen-Lichtenbaum conjecture proven by Voevodsky. It’s a humongous proof. In general, these conjectures are wide open and should be regarded with reverence. ## L-functions Before I really get going, though, I’m going to say a word about L-functions, which by convention are meromorphic functions on $\mathbb{C}$ with certain properties. I’m going to list these salient properties in bold, and illustrate them using the historically first L-function, which is of course the Riemann zeta $\zeta(s)$. We’ll later find that each of these properties has a purely homotopy-theoretic analogue on the K-theory spectrum. Dirichlet series. To an L-function $L$ is associated a Dirichlet series expansion with $a_n \in \mathbb{C}$. $\zeta(s)$ has $a_n = 1$ for all $n$. Abscissa of convergence. There is some $\sigma \in \mathbb{R}$, called the abscissa of convergence, such that the Dirichlet series converges to $L(s)$ for $\mathbf{Re}(s) > \sigma$. The abscissa of convergence of $\zeta$ is 1. Euler product expansion. There is an infinite product, typically over closed points of a scheme, of nice simple functions that also converges to $L(s)$ on some respectable proportion of $\mathbb{C}$. The Euler product expansion of $\zeta$ is Analytic continuation. This is implicit in the fact that I said an L-function was a meromorphic function on $\mathbb{C}$ in the first place, but if you take the more practical view that an L-function is defined by its Dirichlet series or Euler product, there’s some content here, and analytic continuation statements are usually hard to prove. Functional equation. An L-function has a (possibly twisted) reflective symmetry: there is some real number $\tau$, frequently equal to the abscissa of convergence, such that For the Riemann zeta, we have (lifted from Wikipedia) $\tau = 1$ and This just comes from the Archimedean L-factor (we’ll say nothing about that here) but there can be other things too. Analytic continuation and functional equation are usually proved together. You probably expect to be able to formulate a Riemann hypothesis for a general L-function, too, but we won’t get into that here. ## The simplest L-function I said the Riemann zeta was the historically first L-function, but it’s not the simplest. The simplest is the zeta function of a finite field $\mathbb{F}_q$, which is a single Euler factor: The number theorists probably use different notation for this, but I am a carefree rogue who does as he wills. $\Xi_q$ is manifestly meromorphic; it is its own Euler product; its Dirichlet series is given by expanding the above expression as a geometric series, with abscissa of convergence $0$; and its functional equation is This is the only example of an L-function which one can comfortably manipulate with one’s hands without substantial analytic ingenuity. It’s this basic example that we’ll spend much of the rest of this blog post studying. ## The K-theory of a finite field In a 1972 paper, Quillen proved one of my all-time favourite theorems by describing the homotopy type of the K-theory spectrum of a finite field. Theorem. [Quillen] The connected component $K(\mathbb{F}_q)_0$ of the $0$-space of the K-theory spectrum of $\mathbb{F}_q$ fits into a fibre sequence where $\psi^q$ is the $q$th Adams operation. This is the only example of a (global) K-theory spectrum which one can comfortably manipulate with one’s hands without substantial topological ingenuity. By Bott periodicity, we know the homotopy groups of BU: We also know that $\psi^q$ acts by multiplication by $q^m$ on $\pi_{2m} BU$. That means we can read off the positive-degree homotopy groups of $K(\mathbb{F}_q)$: Motivated by the ur-conjecture stated above, let’s define a function $f_q : \mathbb{N} \to \mathbb{C}$ by Suppose, as if possessed by a ghost, you wrote this function down without ever having heard of an L-function. You would still notice that is the restriction to $\mathbb{N}$ of an obvious complex-analytic function: $-\Xi_q$. This is the easiest case of the conjecture. Now what of the values of $\Xi_q$ at negative integers? We might hope that they have something to do with the negative homotopy groups of $K(\mathbb{F}_q)$, which is, after all, a spectrum. But as things stand, these negative homotopy groups are zero. This rift is signaling that the arithmetic incarnation of the bare K-theory spectrum is not the L-function; rather, it’s the Dirichlet series of the L-function, or the part of the L-function to the right of the abscissa of convergence. The topological object that we should really associate to the full L-function is the K(1)-localisation of the K-theory spectrum. To do this, we need to fix an ambient prime $p$, and let’s make sure it’s different from the characteristic of $\mathbb{F}_q$. ## The K(1)-localisation of the K-theory of a finite field I’m going to self-consciously write out all $L_{K(1)}$s in full. It’ll take up space, but hopefully it’ll aid clarity of notation. Here’s a slight variant of Quillen’s calculation; if you like, it arises by applying the Bousfield-Kuhn functor to the statement of Quillen’s theorem above. Proposition. The K(1)-localised K-theory spectrum $L_{K(1)} K(\mathbb{F}_q)$ fits into a fibre sequence Thus the homotopy groups of $L_{K(1)} K(\mathbb{F}_q)$, positive and negative, are given by Until further notice, when we talk about the “value” of an L-function at a point, we’ll mean up to multiplication by a rational number which is a p-adic unit. With this caveat, we now have the identity for all $n \in \mathbb{Z}$, except at $n = 0$, where things are untidy because $\Xi_q$ has a pole and both homotopy groups in the formula are infinite, and a regulator gets involved. That’s nice. We need to make a couple more observations about the homotopy groups of the localised K-theory spectrum. First, the natural map is an equivalence in degrees greater than $0$. (It’s also an equivalence in degree $0$, but I’d like to propose that we regard that as an accident.) This $0$ is the very same $0$ that occurs as the abscissa of convergence of $\Xi_q$, and it also shows up as the étale cohomological dimension of $\mathbb{F}_q$ minus 1 - a number I’d like to refer to as the Quillen-Lichtenbaum constant of $\mathbb{F}_q$. Second, the functional equation for $\Xi_q(n)$ is reflected in the homotopy groups of $L_{K(1)} K(\mathbb{F}_q)$ as follows. For any integer $m$, the $p$-adic valuation of $q^m - 1$ is the same as that of $q^{-m} - 1$. Thus for any odd integer $n$, we have an isomorphism Can this apparent duality be given a topological explanation? We’re going to do this by invoking a total of four or five different dualities, one of which is deep and will be totally blackboxed, so get ready. Let’s start tackling this by rearranging the above isomorphism slightly. Both of the groups there arise as quotients of $\mathbb{Z}_p$, so they come with a choice of generator: the image of $1$. That means that either group can be identified with its Pontryagin dual, so let’s do it on the left: There, now it looks more like a sensible duality. And the Pontryagin dual in there is a strong hint that Brown-Comenetz duality is involved. Briefly, the Brown-Comenetz dualising spectrum is a spectrum $I$ defined by the natural isomorphism for spectra $E$. To save pixels, we’ll write $IE$ for $\mathbf{Map}(E, I)$. Now our isomorphism reads But height 1 Gross-Hopkins duality states that for a K(1)-local spectrum E, Here $M$ is the first monochromatic slice functor, which for a K(1)-local spectrum is just the fibre of the map to its rationalisation. $M$ isn’t doing anything too destructive here, so let’s omit it for now and figure out its role in a moment. We’ve arrived at at least for odd $n \neq -1$. So we’d be home if $K(\mathbb{F}_q)$ were to be K(1)-locally Spanier-Whitehead self-dual. And it is! I learned this in Dustin Clausen’s thesis defence, but one can prove it by showing that exhibits $KU^\wedge_p$ as K(1)-locally self-dual up to a shift of 1, compatibly with Adams operations, and plugging this into the fiber sequence for $L_{K(1)}K(\mathbb{F}_q)$. And thus the duality is explained. Let’s circle back for a moment and say a quick word about the role of the monochromatic slice functor $M$. Since $L_{K(1)} K(\mathbb{F}_q)$ doesn’t have many torsion-free homotopy groups, the only effect of $M$ is to convert the $\mathbb{Z}_p$s in degrees $0$ and $-1$ into $\mathbb{Q}_p/ \mathbb{Z}_p$s in degrees $-1$ and $-2$. But that was exactly what was needed to get the original duality statement to work in those degrees, once we put in the Pontryagin dual. Thus the duality is not merely explained but embellished with a pristine little bow. ## The dictionary, and some questions As promised, we’ve given homotopy-theoretic interpretations of (almost) all the fundamental analytic properties of L-functions. Here’s a table to summarise: Analysis Topology L-function K(1)-localised K-theory spectrum Dirichlet series Bare K-theory spectrum Abscissa of convergence Quillen-Lichtenbaum constant Euler product ?? (But see below) Analytic continuation K(1)-localisation Functional equation Gross-Hopkins + Spanier-Whitehead duality (I apologise for the poor formatting - I’m not used to Markdown, and the latex support doesn’t seem to run to table environments). Some questions I don’t know how to answer at this point: • Let X be a Dedekind domain, L its field of fractions, and $k(x)$ the residue field at a closed point $x \in X$. Then there’s a cofiber sequence The left-hand arrow looks an awful lot like it’s trying to express K(X) as an Euler product over x of K(k(x)), but I don’t know how to make that precise. K(L) is a nuisance. • Now let X be a smooth proper variety over $\mathbb{F}_q$. There’s a spectral sequence with $E_2$ term given by étale cohomology of X with suitably Tate-twisted $\mathbb{Z}_p$ coefficients which abuts to the K(1)-local K-theory of X. Can we use this to relate the formula for the zeta function of X in terms of the characteristic polynomial of Frobenius on étale cohomology to a K-theoretic description of the special values? • What’s the analogue of $K(\mathbb{F}_q)$ for an archimedean L-factor? This might be related to work of Connes and Consani on cyclic homology with adelic coefficients. This section is tangential to the rest of the post, but I think it suggests some interesting avenues of investigation. Up until now we’ve been discussing L-functions defined on $\mathbb{C}$, but L-functions defined on $\mathbb{Z}_p$ are also of interest to number theorists. We’ll construct the p-adic analogue of an Euler factor, and see that it throws up some unexpected topology. In this section, we’ll assume that $p$ is odd. Things are probably substantially more awkward when $p = 2$. Let’s return (up to sign) to our function $f_q$, this time regarded as a function $\mathbb{N} \to \mathbb{Q}_p$: We’d like to extend $f_q$ to a continuous function $E_q: \mathbb{Z}_p \to \mathbb{Q}_p$. p-adic continuity sounds weak compared to complex analyticity, but it’s actually kind of hard to be p-adically continuous, and since $\mathbb{N}$ is dense in $\mathbb{Z}_p$, we’re going to have at most one choice of extension. In fact, we don’t have any choices, a fact which won’t surprise anyone who’s studied p-adic L-functions before. The proof of this is a byproduct of the construction of the real p-adic L-function. Let Observe that $S$ is already dense in $\mathbb{Z}_p$, so if we can find a continuous function $\mathbb{Z}_p \to \mathbb{Q}_p$ that agrees with $f_q$ on $S$, that’s the best we can do, whether the two functions agree on the whole of $\mathbb{N}$ or not. Here’s the crux: There’s a continuous function known as the Iwasawa logarithm, and a continuous function known as the p-adic exponential. They have most of the properties you’d expect of a pair of functions called $\exp$ and $\log$, except that the identity only holds when $x$ is congruent to $1$ mod $p$. In general, where the integer $r$ and the $(p-1)$st root of unity $\omega$ are uniquely determined by the requirement that $p^r \omega x \equiv 1 \mod p$. So we can follow our hearts and define but we have to live with the fact that $E_q(n) = f_q(n)$ only when $n \in S$. For general $n \in \mathbb{N}$, for a suitable $(p-1)$st root of unity $\omega$. Fine. Where’s the topology? What spectrum is $E_q$ telling us about? At integers outside $S$, the values of $E_q$ don’t match up with the homotopy groups of $L_{K(1)} K(\mathbb{F}_q)$ anymore. Instead, for all integers $n \neq 0$, where M is a K(1)-local spectrum defined as the fiber of $\psi^{\omega q} - 1 : KU^\wedge_p \to KU^\wedge_p$. So, a couple more questions: • What’s special about M? Why is it any better than $L_{K(1)}K(\mathbb{F}_q)$? What other spectra are “p-adically continuous” in this sense? • Constructing p-adic L-functions more serious than $E_q$ usually requires some fairly heavy p-adic analysis. Can we partly bypass this by proving p-adic continuity results for the Picard-graded homotopy groups of certain K(1)-local spectra and using the orders of these homotopy groups? I believe some such continuity results already exist, in the folklore if not in the literature. Thanks for reading this many words! Let me finish up by remarking that some of the stuff in this post carries through to K-theory spectra of rings of integers, and using work of Dwyer and Mitchell as a bridge, one can obtain results on the special values of Kubota-Leopoldt-type p-adic zeta functions. If I can think of enough interesting stuff to say about that, it might be the subject of a future post. I think the Chromotopy platform doesn’t currently support blog comments, but if you have any comments or questons, please email me at sglasman at math dot mit dot edu. Update: Dustin Clausen emailed me with some helpful comments about the Spanier-Whitehead self-duality of $L_{K(1)} K(\mathbb{F}_q)$, resulting in some alterations in the text. The K-Theory of Endomorphisms For the purposes of this post, I am going to assume that you are all familiar with the basics of Algebraic $K$-theory. If you’re not, just treat it as a black box, a gadget which takes in one of : • a (simplicial) ring (spectrum) • an augmented (simplicial) bimodule over a ring • an exact category • a Waldhausen category • a small, stable $\infty$-category and spits out a spectrum $K(\mathcal{C})$, such that $K(\mathcal{C})$ does the job of ”storing all Euler characteristics” or “additive invariants”. An important example is when $\mathcal{C}$ is the category of finitely generated projective modules over a ring $R$, and then we call this $K(R)$. It is a generally-accepted fact that the $K$-theory of rings is generically a very difficult and often useful thing to compute (knowledge of the $K$-groups of $\mathbb{Z}$, for example, would be quite valuable to many). In an ideal world, we would be able to understand the functor on the category of algebras augmented over a given ring $R$, but this ends up being fairly intractable. One might hope that it would be easier to look at what happens when restricted to free augmented algebras, that is looking at the functor on (flat) $R$-bimodules where $T_R (M)$ is the tensor algebra on $M$. This is pretty tough, but it turns out that we can do it. Let’s take a baby step first, and try to understand the K-theory of the “linearized” tensor algebra. The 1st Goodwillie derivative of the identity functor on augmented $R$-algebras is given by where $I$ is the augmentation ideal of $A$. In the case where $A = T_R (M)$ for some $R$-bimodule $M$, then the Goodwillie derivative $P_1 (id)(T_R (M)) = R \oplus M$, where $R \oplus M$ is the the square-zero extension of $R$ by $M$, the ring given by demanding that $M^2 = 0$. In studying the $K$-theory of square-zero extensions, then, we are also studying the $K$-theory of the linearization of the tensor algebra functor (note how many simplifications we have already done!). A different strategy for dealing with this computational difficulty is to try and understand how the $K$-theory of a ring changes as we “perturb” the ring. To do this, we look at “parametrized $K$-theory” (or the “$K$-theory of parametrized endomorphisms”): Definition. Let $R$ be a ring, and let $M$ be an $R$-bimodule. We define the parametrized $K$-theory of $R$ with coefficients in $M$, $K(R;M)$, to be the $K$-theory of the exact category of pairs $(P,f)$ where $P$ is a finitely-generated projective $R$-module and $f: P \rightarrow P \otimes_{R} M$ is a map of $R$-modules. We think about $K(R;M)$ as being the $K$-theory of endomorphisms with coefficients that are allowed to be in $M$. If our picture of a finitely-generated projective $R$-module $P$ is as living as a summand of a rank $n$ free $R$-module, then an element of the above exact category is an $n \times n$ matrix with entries in $M$ that commutes with the projection map $R^n \rightarrow R^n$ defining $P$. Why should we look towards endomorphisms as perturbations? Well, the picture is supposed to be the following: Let $R$ be a ring and $M$ an $R$-bimodule. This is the same data as a sheaf $\mathcal{F}_M$ over $\operatorname{Spec}(R)$, and we would like to think of a deformation of this over, say, $R[t]/t^2$ as a flat sheaf $\mathcal{F}_{M'}$ over $R[t]/t^2$ (a module) that restricts to $\mathcal{F}_M$ over $\operatorname{Spec}(R)$. which corresponds to an element of $\operatorname{Ext}^1_R (M,M)$, or a derived endomorphism. The idea is that an extension of $M$ corresponds to a deformation of $M$, which is a reasonable perspective. The above definition is not immediately seen to be relevant to our original stated desire to study perturbations, but in the investigations of Dundas and McCarthy of stable $K$-theory, the $K$-theory of endomorphisms naturally comes up. In this paper they prove the following theorem: Theorem. For $R$ a ring and $M$ a discrete $R$-bimodule) where $R \oplus M$ is again the the square-zero extension of $R$ by $M$. We think of $R \oplus M$ as a perturbation of $R$ by $M$, as the elements that we are adding on (those coming from the direct summand $M$) are “so small’’ that they multiply to zero. This result relates the “perturbation’’ approach to understanding $K$-theory to our earlier potential approach of understanding the free objects in the category of augmented $R$-algebras. That’s great, but we’d like to actually have some idea of what these things are. To get a hint at what we should be looking for, we go back to what was classically studied by Almkvist et al. ### $K$-Theory of Endomorphisms: The Classical Story Definition. Let $R$ be a ring and consider the category $End(R)$, whose objects are pairs $(P,f)$ with $P$ a finitely-generated projective $R$-module and $f:P \rightarrow P$ an endomorphism. The morphisms in this category are commutative diagrams of the appropriate type. Now, we might ask what possible “additive invariants’’ there are on this category, having in mind a few examples. The key (and, as it turns out, universal) one is the following: Example. [The Characteristic Polynomial] Let $(P,f) \in End(R)$, then the characteristic polynomial of $f$ is given by This can also be obtained in the usual way as a determinant. The important property of the characteristic polynomial is that if we have a commutative diagram in $End(R)$ with exact rows, then meaning that the characteristic polynomial takes short exact sequences of endomorphisms to products (which are sums in the abelian group where the characteristic polynomials lie). We now know that a search for additive invariants is a desire to compute $K$-theory, and so we define: Definition. $K_0 (End(R))$ is defined to be the free abelian group on isomorphism classes of objects in $End(R)$ modulo the subgroup generated by the relations $[(P,f)] = [(P',f')] + [(P'',f'')]$ if there is a commutative diagram with the rows exact. There is a natural splitting coming from thinking of the category of finitely generated projective $R$-modules as living in $End(R)$ as the guys with $0$ endomorphisms. Of course, $End(R)$ is an exact category, and we could define higher $K$-groups for this as well. $K_0 (End(R))$ is the repository for additive invariants, in that it has the following universal property: Proposition. Let $F$ be a map from the commutative monoid of isomorphism classes of objects in $End(R)$ to an abelian group $A$, such that $F$ splits short exact sequences as above. Then $F$ factors through $K_0 (End(R))$, or $K_0(End(R))$ is the initial abelian group for which short exact sequences of endomorphisms split. What does this look like, though? Well, let’s go back to the characteristc polynomial: Theorem. The map is an isomorphism where is the multiplicative group of fractions with constant term 1. The moral of this is that the characteristic polynomial encodes all of the additive information about an endomorphism (the trace is a special case of this, of course, being read off by the constant term of the characteristic polynomial). Something interesting is the following: Proposition. The inclusion $\tilde{W}(R) \rightarrow W(R)$ exhibits $\tilde{K}_0 (R)$ as a dense $\lambda$-subring of $W(R)$, where $W(R)$ are the big Witt vectors of $R$, modelled as power series with constant term $1$. This means that we might as well think of the big Witt vectors as being limits of characteristic polynomials of endomorphisms, which is the starting point for the line of thought that led to the Lindenstrauss and McCarthy results. It is also important to mention what some of the uses of this equivalence are: • Calculations that may be difficult to perform on Witt vectors might be easy if we think of them as coming from characteristic polynomials of endomorphisms; • Operations that exist on Witt vectors get turned into operations on K-theory, and vice-versa: • The $n$-th ghost map is given by $gh_n ([P,f] = tr(f^n))$; • The Frobenius map is given by $F_n ([P,f]) = [P, f^n]$; • The Verschiebung map is given by $[V_n ([P,f]) = [P^{\oplus n}, v_n f]$, where $v_n f$ is represented by a shift in the first $n-1$ factors and action by $f$ in the last factor. This represents an $n$-th root of $f$, in that $V_n^n$ looks like applying $f$ to all of the blocks of $P^{\oplus n}$. ### Lindenstrauss-McCarthy and Topological Witt Vectors The first thing to do to start generalizing the previous construction is to allow for a “parametrization’’, like we said before. Definition. Let $R$ be a ring and $M$ an $R$-bimodule. $End(R;M)$ is the category whose objects are pairs $(P,f)$ with $P$ a finitely generated projective $R$-module and $f: P \rightarrow P \otimes_R M$ a map of $R$-modules. As in $End(R)$, morphisms are commutative diagrams. $End(R;M)$ inherits an exact/Waldhausen structure from considering what’s happening in the “base’’, and so it makes sense to define parametrized $K$-theory as There is a natural map from $End(R;M)$ to the category of finitely generated projective $R$-modules that forgets the endomorphism, and so we can reduce and consider We can also consider $M$ to be simplicial by geometrically realizing, and this assembles to a functor which is a good setting to do Goodwillie calculus. The remarkable result of Lindenstrauss-McCarthy is the following: Theorem. The functors $\tilde{K} (R; -), W(R;-)$ from simplicial $R$-bimodules to spectra have the same Taylor tower, where $W(R;-)$ is a “topological Witt vectors’’ construction. Now, this is only remarkable if I tell you what this $W(R;-)$ functor is, so let’s do that. Lindenstrauss and McCarthy describe this using the (notationally cumbersome) language of FSPs instead of spectra or spectral categories, which makes everything a bit hard to read. The idea is a bit simpler, however. Construction. Let $R$ be a ring, $M$ a bimodule. We define a bunch of spectra $U^n (R;M)$ with $C_n$ action by letting $U^n (R;M)$ be the derived cyclic tensor product (over $R$) of $n$ copies of $M$. There is an evident $C_n$ action on $U^n (R;M)$ given by permuting the factors, and moreover when $n \mid N$, we have natural restriction maps between the fixed points We then define Example. $U^1 (R;M) \simeq THH(R;M)$ as normally defined, and if $R = M$ then we have that $W(R;M) \simeq TR(R;M)$. This functor $W(R;M)$ is meant to be an attempt to define $TR$ in the absence of the cyclic symmetry that is present when $M = R$. The remarkable thing is that this actually works! The nomenclature is justified by the following $p$-typical statement: Theorem. (Hesselholt-Madsen) $\pi_0 (TR(R;p)) \simeq W_{(p)}(R),$ where the latter is a suitable version of $W(R;M)$ that takes a homotopy limit over only $U^{p^n}(R;M)^{C_{p^n}}$. The virtue of this setup is that we have explicit understanding of what the layers of the Taylor tower of $W(R;-)$ look like, as well as various splitting theorems, e.g. we have the following fundamental cofibration sequence’’: Theorem. There is a homotopy fiber sequence where $W^{n}(R;M)$ are the functors obtained by truncating the homotopy limit defining $W(R;M)$. Moreover, we have that $W^{n}(R;M)$ is the $n$th polynomial approximation of $W (R;M)$, so this is the fiber sequence that computes the layers of the Taylor tower. ### Generalizations The problem with the Lindenstrauss-McCarthy story is that it is only proven for discrete rings, but all of the constructions floating around can just as well be made when $R$ is a connective ring spectrum and $M$ is a (simplicial) $R$-bimodule. One of the great tricks that one can do is to use the resolution of connective ring spectra by simplicial rings. That is, given a connective ring spectrum $R$, there is a $id$-Cartesian cube with initial vertex $R$, and such that all other vertices are canonically stably equivalent to the Eilenberg-MacLane spectrum of a simplicial ring. The following fact lets us promote results about simplicial rings to those about connective ring spectra: Theorem. If $\chi$ is an $(id)$-Cartesian $k$-cube of connective ring spectra, then the cube $K(\chi)$ is $(k+1)$-Cartesian.</span> Idea for proving generalization: Show that the functors $W(-;-), \tilde{K}(-;-)$ have a similar property, so that the equivalence given by Lindenstrauss and McCarthy can be promoted to one for connective ring spectra. This works just fine, and we arrive at: Theorem. (P.) The Lindenstrauss-McCarthy result holds for connective ring spectra. Unfortunately, this is mostly just a one-off trick, and we are still looking for a good conceptual understanding of what the Lindenstrauss-McCarthy equivalence really means. It largely goes back to the Dundas-McCarthy theorem mentioned earlier, which can also be proven for connective ring spectra, but fails to admit a generalizable proof. The issue that arises there is that their proof relies heavily on the linear models of $K$-theory when working over discrete rings, and these simply faily to work when we move to other contexts (their proof involves the addition of maps, a simplicial model of $K$-theory, etc.). If we had a good way of dealing with the coherence issues in defining the Dundas-McCarthy maps, then we might be able to generalize the proof and understand what’s going on, but that’s something that has yet to be done. Hypothetical abelian varieties from K-theory I heard an idea tossed around recently that I’d like to share with you all. I worry that it might be a little half-baked, as I’ve only heard about it recently and so maybe haven’t spent enough time sketching out the edges of it. Maybe writing this will help. Throughout, $k$ is a perfect field of positive characteristic $p$. Where Lie theorists study Lie algebras, formal algebraic geometers study (covariant) Dieudonné modules. The essential observation is that the sorts of formal Lie groups appearing in algebraic topology are commutative and one-dimensional, meaning that their associated Lie algebras are one-dimensional vector spaces with vanishing brackets, and so it is unreasonable to think that it would carry much interesting information about its parent formal Lie group. To correct for this, one takes the collection of all curves on the formal group (i.e., without reducing to their linear equivalence classes, as one does when building the Lie algebra) and remembers enough structure stemming from the group multiplication that this assignment becomes an equivalence. Such collections of curves form modules, called “Dieudonné modules”, over a certain ground ring, called the Cartier ring, which is built upon the ambient field $k$. The three relevant pieces of structure are the actions of homotheties, of the Frobenius, and of the Verschiebung, which were all described way back in this post on Witt vectors. (In the notation of that post, we’re interested just in $F_p$ and $V_p$.) Altogether, this gives a formula for the Cartier ring: Say that a Dieudonné module is formal when it is: finite rank; free as a $\mathbb{W}(k)$-module; reduced, meaning that it’s $V$-adically complete; and uniform, meaning that the natural map $M / VM \to V^k M / V^{k+1} M$ is an isomorphism. Then, the Dieudonné functor on finite height formal groups lands in the subcategory of formal Dieudonné modules, and there it restricts to an equivalence. Actually, more is true: the Dieudonné module of a $p$-divisible group can be made sense of, and Dieudonné modules which are free of finite rank (but without reducedness or uniformity) are equivalent to $p$-divisible groups. Dieudonné modules are thrilling because they are just modules, whereas $p$-divisible groups are these unwieldy ind-systems of finite group schemes. The relative simplicity of the data of a Dieudonné module allows one to compute basic invariants very quickly: Theorem: Take $\mathbb{G}$ to be a $p$-divisible group and $M$ its Dieudonné module. There is a natural isomorphism of $k$-vector spaces $T_0 \mathbb{G} \cong M / VM$. In the case that $\mathbb{G}$ is one-dimensional, then the rank of $M$ as a $\mathbb{W}(k)$-module agrees with the height $d$ of $\mathbb{G}$. Moreover, if $\gamma$ is a coordinate on $\mathbb{G}$ (considered as a curve), then $\{\gamma, V\gamma, \ldots, V^{d-1} \gamma\}$ forms a basis for $M$. There is also something like a classification of the simple Dieudonné modules over $\bar{\mathbb{F}}_p$ (where, among other things, the étale component of a p-divisible group carries little data): Theorem (Dieudonné): For $m$ and $n$ coprime and positive, set Additionally, allow the pairs $(m, n) = (0, 1)$ to get and $(m, n) = (1, 0)$ to get (All these modules $G_{m,n}$ have formal dimension $n$ and height $(m+n)$.) For any simple Dieudonné module $M$, there is an isogeny $M \to G_{m,n}$, i.e., a map with finite kernel and cokernel. Moreover, up to isogeny every Dieudonné module is the direct sum of simple objects. Every abelian variety $A$ comes with a p-divisible group $A[p^\infty]$ arising from its system of $p$-power order torsion points. This connection is remarkably strong; for instance, there is the following theorem: Theorem (Serre–Tate): Over a $p$-adic base, the infinitesimal deformation theories of an abelian variety $A$ and its $p$-divisible group $A[p^\infty]$ agree naturally.[1] For this reason and others, the $p$-divisible group of an abelian variety carries fairly strong content about the parent variety. On the other hand, it is not immediately clear which $p$-divisible groups arise in this way. Toward this end, there is a symmetry condition: Lemma (“Riemann–Manin symmetry condition”): As every abelian variety is isogenous to its Poincaré dual and the corresponding (Cartier) duality on $p$-divisible groups sends ($V$ to $F$ and hence) $G_{m,n}$ to $G_{n,m}$, these summands must appear in pairs in the isogeny type of the Dieudonné module of $A[p^\infty]$. As a simple example, this gives the usual categorization of elliptic curves: an elliptic curve is $1$-dimensional, hence has $p$-divisible group of height $2$. One possibility, called a supersingular curve, is for the Dieudonné module to be isogenous to $G_{1,1}$; this is a $1$-dimensional formal group of height $2$ and it satisfies the symmetry condition. The only other possibility, called an ordinary curve, is for the Dieudonné module to be isogenous to $G_{1,0} \oplus G_{0,1}$; this is the sum of a $1$-dimensional formal group of height $1$ with an étale component of height $1$, and it too satisfies the symmetry condition. A remarkable theorem is that the converse of the symmetry lemma holds as well: Theorem (Serre, Oort; conjectured by Manin): If a Dieudonné module $M$ satisfies the above symmetry condition, then there exists an abelian variety whose $p$-divisible group is isogenous to $M$. Both proofs of this theorem are very constructive. Serre’s proof explicitly names abelian hypersurfaces whose $p$-divisible groups are of the form $G_{m,n} \oplus G_{n,m}$, for instance. Now, finally, some input from algebraic topology. The Morava $K$-theories of Eilenberg–Mac Lane spaces give a collection of formal groups which can be interpreted in the following way: Theorem (Ravenel–Wilson): For $% $, the $p$-divisible group associated to $K(n)^* K(\mathbb{Q}/\mathbb{Z}, m)$ is (in a suitable sense) the $m$th exterior power of the $p$-divisible group $K(n)^* K(\mathbb{Q}/\mathbb{Z}, 1)$. It is smooth, has formal dimension $\binom{n-1}{m-1}$, and has height $\binom{n}{m}$. (Additionally, it is zero for $m > n$.) Theorem (Buchstaber–Lazarev): For $% $, the same $p$-divisible group has Dieudonné module isogenous to the product of $\frac{1}{n_0} \cdot \binom{n}{m}$ copies of $G_{n_0-m_0,m_0}$, where $m_0/n_0$ is the reduced fraction of $m/n$. The conclusion of Buchstaber and Lazarev is that this means that these $p$-divisible groups almost never have realizations as abelian varieties, since they mostly don’t satisfy the symmetry condition. The only time that they do is something of an accident: when $n$ is even and $m = n/2$, then the corresponding Dieudonné module is isogenous to that of a large product of copies of a supersingular elliptic curve. However, Ravenel observed that Pascal’s triangle is symmetric: Observation (I heard this from Ravenel, but surely Buchstaber and Lazarev knew of it): The sum of all of the Dieudonné modules satisfies the Riemann-Manin symmetry condition. This is an interesting observation. In light of the comments at the end of the Buchstaber–Lazarev paper, one wonders: why privilege $m = n/2$? But, even more honestly, why privilege $m = 1$ in our study of chromatic homotopy theory? A recurring obstacle to our understanding of higher-height cohomology theories has been the disconnection from the picture of globally defined abelian varieties. Could there be a naturally occurring abelian variety whose $p$-divisible group realizes the large ($2^{n-1}$-dimensional!) formal group associated to the $K$-theoretic Hopf ring of Eilenberg–Mac Lane spaces? The explicit nature of the solutions to Manin’s conjecture show us that, yes, it is certainly possible to write down large products of hypersurfaces to give a positive answer to this question with the words “naturally occurring” deleted. This alone isn’t very helpful, however, and so to pin down what we might be even talking about there are a number of smaller observations that might help: • If such an abelian variety existed, there would be a strange filtration imposed on its $p$-divisible group arising from the degrees of the Eilenberg–Mac Lane spaces. The Riemann–Manin symmetry condition tells us that these $G_{m,n}$ and $G_{n,m}$ factors must come in pairs, but in almost all situations, one pair appears on one side of the middle-dimension $n/2$ and the other appears on the other side. What could this mean in terms of the hypothetical abelian variety? • Relatedly, this pairing arises from the “$\circ$-product” structure on the level of Hopf algebras, or as a sort of Hodge-star operation on the level of Dieudonné modules. What sort of structure on the abelian variety would induce such an operation, and in such an orderly fashion? • There should be accessible examples of these hypothetical varieties at low heights. For instance, the one associated to height $1$ via mod-$p$ complex $K$-theory is (canonically, not merely isogenous to) a sum $G_{1,0} \oplus G_{0,1}$ — i.e., it comes from an ordinary elliptic curve. Can we identify which elliptic curve — and in what sense we can even ask this question? Is there a naturally occurring map from the forms of $\mathbb{G}_m$ to the ordinary locus on the (noncompactified?) moduli of elliptic curves? What about forms of $\hat{\mathbb{G}}_m$? What if we select a supersingular elliptic curve instead — is there an instructive assignment to abelian varieties whose Dieudonné module has isogeny type $G_{1,0} \oplus G_{1,1} \oplus G_{0,1}$? (On the face of it, this last bit doesn’t look so helpful, but maybe it is.) [1] - Presumably this can be expressed by saying that the map from a moduli of abelian varieties to a moduli of $p$-divisible groups is formally étale, but no one seems to say this, so maybe I’m missing something. I’ve been making an effort to learn some arithmetic geometry recently. I started with local class field theory, which was mind-blowing. When I was a first year, someone sat me down and instructed me that I must take a course in complex geometry to become competent — and they were right, and it was a wonderful course, and I’m really glad I got that advice. I have no idea how (especially as I’ve been bumbling about with formal groups for so long!) it slipped past me that local class field theory is another one of these core competencies, and really one of the great achievements of twentieth century mathematics. I gave a kind of measly talk about this a month ago, when I was trying to stir up interest in a reading group. The notes are a little batty, but they’re fun enough, and you can find them here. Speaking of mind-blowing things, last week there was a week-long workshop on perfectoid spaces at MSRI, which I attended between one-third and one-half of. There are video lectures available on the MSRI website; at the very least everyone should watch Scholze’s introduction, just to get a sense of what all the fuss is about, and then ideally both of Weinstein’s lectures, which were excellent and very much adjacent to the subject of this post — this blog, really. And, as an uninspired parting remark, we topologists do have access to the pro-system	{"extraction_info": {"found_math": true, "script_math_tex": 418, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8873714804649353, "perplexity": 343.3280046794384}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-36/segments/1471982293692.32/warc/CC-MAIN-20160823195813-00269-ip-10-153-172-175.ec2.internal.warc.gz"}
https://nips.cc/Conferences/2019/ScheduleMultitrack?event=15880	Spotlight Optimal Sparse Decision Trees Xiyang Hu · Cynthia Rudin · Margo Seltzer Wed Dec 11th 04:55 -- 05:00 PM @ West Exhibition Hall C + B3 Decision tree algorithms have been among the most popular algorithms for interpretable (transparent) machine learning since the early 1980's. The problem that has plagued decision tree algorithms since their inception is their lack of optimality, or lack of guarantees of closeness to optimality: decision tree algorithms are often greedy or myopic, and sometimes produce unquestionably suboptimal models. Hardness of decision tree optimization is both a theoretical and practical obstacle, and even careful mathematical programming approaches have not been able to solve these problems efficiently. This work introduces the first practical algorithm for optimal decision trees for binary variables. The algorithm is a co-design of analytical bounds that reduce the search space and modern systems techniques, including data structures and a custom bit-vector library. We highlight possible steps to improving the scalability and speed of future generations of this algorithm based on insights from our theory and experiments.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8219229578971863, "perplexity": 1516.4037691762387}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347423915.42/warc/CC-MAIN-20200602064854-20200602094854-00317.warc.gz"}
http://www.ams.org/joursearch/servlet/PubSearch?f1=msc&onejrnl=proc&pubname=one&v1=35J60&startRec=120	American Mathematical Society My Account · My Cart · Customer Services · FAQ Publications Meetings The Profession Membership Programs Math Samplings Policy and Advocacy In the News About the AMS You are here: Home > Publications AMS eContent Search Results Matches for: msc=(35J60) AND publication=(proc) Sort order: Date Format: Standard display Results: 120 to 140 of 140 found Go to page: 1 2 3 4 5 [120] Kirk E. Lancaster. Existence and nonexistence of radial limits of minimal surfaces . Proc. Amer. Math. Soc. 106 (1989) 757-762. MR 969523. Abstract, references, and article information View Article: PDF This article is available free of charge [121] Jeanne Trubek. Asymptotic behavior of solutions to $\Delta u+Ku\sp \sigma=0$ on ${\bf R}\sp n$ for $n\geq 3$ . Proc. Amer. Math. Soc. 106 (1989) 953-959. MR 987615. Abstract, references, and article information View Article: PDF This article is available free of charge [122] Tilak Bhattacharya. Radial symmetry of the first eigenfunction for the $p$-Laplacian in the ball . Proc. Amer. Math. Soc. 104 (1988) 169-174. MR 958061. Abstract, references, and article information View Article: PDF This article is available free of charge [123] Takaŝi Kusano, Ezzat S. Noussair and Charles A. Swanson. Existence of decaying entire solutions of a class of semilinear elliptic equations . Proc. Amer. Math. Soc. 104 (1988) 1141-1147. MR 929405. Abstract, references, and article information View Article: PDF This article is available free of charge [124] Jenn-Fang Hwang. Phragm\'en-Lindel\"of theorem for the minimal surface equation . Proc. Amer. Math. Soc. 104 (1988) 825-828. MR 964864. Abstract, references, and article information View Article: PDF This article is available free of charge [125] Juan J. Manfredi. $p$-harmonic functions in the plane . Proc. Amer. Math. Soc. 103 (1988) 473-479. MR 943069. Abstract, references, and article information View Article: PDF This article is available free of charge [126] Robert C. McOwen. Point singularities and conformal metrics on Riemann surfaces . Proc. Amer. Math. Soc. 103 (1988) 222-224. MR 938672. Abstract, references, and article information View Article: PDF This article is available free of charge [127] Chang Shou Lin and Wei-Ming Ni. A counterexample to the nodal domain conjecture and a related semilinear equation . Proc. Amer. Math. Soc. 102 (1988) 271-277. MR 920985. Abstract, references, and article information View Article: PDF This article is available free of charge [128] R. Jensen, P.-L. Lions and P. E. Souganidis. A uniqueness result for viscosity solutions of second order fully nonlinear partial differential equations . Proc. Amer. Math. Soc. 102 (1988) 975-978. MR 934877. Abstract, references, and article information View Article: PDF This article is available free of charge [129] Philip W. Schaefer. Some maximum principles in semilinear elliptic equations . Proc. Amer. Math. Soc. 98 (1986) 97-102. MR 848884. Abstract, references, and article information View Article: PDF This article is available free of charge [130] William P. Ziemer. A Poincar\'e-type inequality for solutions of elliptic differential equations . Proc. Amer. Math. Soc. 97 (1986) 286-290. MR 835882. Abstract, references, and article information View Article: PDF This article is available free of charge [131] Kirk E. Lancaster. Nonexistence of some nonparametric surfaces of prescribed mean curvature . Proc. Amer. Math. Soc. 96 (1986) 187-188. MR 813836. Abstract, references, and article information View Article: PDF This article is available free of charge [132] Chris Cosner and Klaus Schmitt. A priori bounds for positive solutions of a semilinear elliptic equation . Proc. Amer. Math. Soc. 95 (1985) 47-50. MR 796444. Abstract, references, and article information View Article: PDF This article is available free of charge [133] Fang-Hua Lin. On the elliptic equation $D\sb i[a\sb {ij}(x)D\sb jU]-k(x)U+K(x)U\sp p=0$ . Proc. Amer. Math. Soc. 95 (1985) 219-226. MR 801327. Abstract, references, and article information View Article: PDF This article is available free of charge [134] Neil S. Trudinger. On an interpolation inequality and its applications to nonlinear elliptic equations . Proc. Amer. Math. Soc. 95 (1985) 73-78. MR 796449. Abstract, references, and article information View Article: PDF This article is available free of charge [135] Nichiro Kawano, Takaŝi Kusano and Manabu Naito. On the elliptic equation $\Delta u=\varphi (x)u\sp \gamma$ in ${\bf R}\sp 2$ . Proc. Amer. Math. Soc. 93 (1985) 73-78. MR 766530. Abstract, references, and article information View Article: PDF This article is available free of charge [136] Wei Ming Ni. On a singular elliptic equation . Proc. Amer. Math. Soc. 88 (1983) 614-616. MR 702285. Abstract, references, and article information View Article: PDF This article is available free of charge [137] E. M. Harrell and B. Simon. Schr\"odinger operator methods in the study of a certain nonlinear P.D.E . Proc. Amer. Math. Soc. 88 (1983) 376-377. MR 695279. Abstract, references, and article information View Article: PDF This article is available free of charge [138] Charles V. Coffman. An alternate variational principle for $\Delta u-u+\mid u\mid \sp{r-1}{\rm sgn}\,u=0$ . Proc. Amer. Math. Soc. 87 (1983) 666-670. MR 687637. Abstract, references, and article information View Article: PDF This article is available free of charge [139] H. Rhee. Spherically symmetric entire solutions of $\Delta \sp{p}u=f(u)$ . Proc. Amer. Math. Soc. 69 (1978) 355-356. MR 0481494. Abstract, references, and article information View Article: PDF This article is available free of charge [140] Peter Hess. On a unilateral problem associated with elliptic operators . Proc. Amer. Math. Soc. 39 (1973) 94-100. MR 0328336. Abstract, references, and article information View Article: PDF This article is available free of charge Results: 120 to 140 of 140 found Go to page: 1 2 3 4 5	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8919673562049866, "perplexity": 1536.6974735608662}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-36/segments/1471982924728.51/warc/CC-MAIN-20160823200844-00112-ip-10-153-172-175.ec2.internal.warc.gz"}
http://ned.ipac.caltech.edu/level5/March01/Battaner/node24.html	### 4.2.3 Recent developments After these important papers in which the basic scenario was outlined and the general assumptions justified, it was necessary to develop more detailed models, mainly numerical and N-body simulations due to the high complexity of the various physical processes involved. A schematic list of models and reviews is now given, which should be completed with references therein: Cole (1991), White and Frenk (1991), Navarro and White (1993), Kauffmann, Guidernoni and White (1994), Cole et al. (1994), Navarro, Frenk and White (1996, 1997), Lacey and Cole (1993), Avila-Reese, Firmani and Hernandez (1998), Avila-Reese et al. (1999), Sensui, Funato and Makino (1999), Salvador-Sole, Solanes and Manrique (1998), Baugh et al. (1999), Subramanian, Cen and Ostriker (1999), Steinmetz (1999), van den Bosch (1999) and a large series of papers, reflecting the importance of the topic. Models can be classified as "semi-analytical" (in which some processes are given a simplified treatment assuming simple recipes, based on either previous numerical calculation or on theoretical ideas), numerical simulations (e.g. hydrodynamical simulations, collision-less simulations) N-body simulations (the most widely used) and even analytical. Some hybrid models are difficult to classify in this scheme. It is first necessary to adopt a cosmological model, the most popular one being the "standard" CDM (with = 1, h = 0.5, for instance) or the CDM (more in consonance with current values, = 0.3, = 0.7, h= 0.65). A primordial fluctuation spectrum must often be adopted, usually a power law P(k) kn, with n ranging from 0 to -1.5 (for example), where k is the wave number. Another important parameter used by most models is . In general, the variance is defined as < > 1/2; then is the present value for a scale-length of 8 Mpc. This parameter is adopted "a priori" taking into account the present large-scale structure, rather than considering a real free parameter. Usual values adopted are 0.6 for the standard CDM and 1 for the CDM. Other parameters characterize the calculation methods. For instance the initial redshift, the number of particles in N-body simulations and the box in Mpc3 in which the calculations are performed. The so called "Virgo consortium" (Jenkins et al. 1997) is able to handle 2563 particles and a large volume of the order of 60 Mpc. Parameters controlling the resolution of the simulation and the efficiency with which gas cools have a higher influence on the results (Kay et al. 1999). These models not only deal with the formation of halos, but also with the ability of gas to form stars, with matter and energy outputs, mainly due to supernova explosions, the evolution of the baryonic component, the explanation of the Hubble Sequence, how spirals merge to produce spirals and so on. From our point of view, the rotation of galaxies strongly depends on the structure of the halos, which is determined in the first stage of the computations. The latest evolution of visible galaxies is, paradoxically, the most difficult to understand and to model. For instance, the Initial Mass Function (IMF) is largely unknown and yet is decisive in galactic evolution. The hierarchical process of merging, the formation and internal structure of dark matter halos is said to be the best known process. This could be due, in part, to the relative simplicity of the process, but also to the evident fact that it is easier to make predictions about the unobservable. In general, even if some observable facts remain insufficiently explained, these families of theoretical models provide a very satisfactory basis to interpret any evolutionary and morphological problem.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9274084568023682, "perplexity": 1198.2961403815173}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-49/segments/1416400380574.41/warc/CC-MAIN-20141119123300-00030-ip-10-235-23-156.ec2.internal.warc.gz"}
https://rd.springer.com/article/10.1007/s11229-018-01968-y?error=cookies_not_supported&code=b3ba2da3-0fe1-436c-baf5-05db032eae7c	# Correction to: Deduction and definability in infinite statistical systems The Original Article is available ## Correction to: Synthese https://doi.org/10.1007/s11229-017-1497-6 1. 1. Prop. 1 is false as stated. The proof implicitly assumes that all Cauchy sequences in $$X^$$ are bounded. The definition of completeness should be amended as follows to make the proposition true: a locally convex vector space X with topology generated by a family of semi-norms L is said to be complete if every bounded Cauchy net converges to an element of the space. Note that this definition differs from some standard definitions; it might more properly be called bounded completeness. This definition should be understood to apply throughout the paper. 2. 2. Prop. 6 is false. Equation (1) provides a sufficient, but not necessary, condition for a W-algebra $$\mathfrak {R}$$ to be the bidual of a C-algebra, i.e. $$\mathfrak {R}\cong \mathfrak {A}^{}$$. To see this, notice that Eq. (1) implies the predual $$\mathfrak {R}_$$ is weak* closed in $$\mathfrak {R}^$$, which implies that $$\mathfrak {R}$$ is reflexive. Thus, for a counterexample, one need only choose a non-reflexive W-algebra $$\mathfrak {R}$$, of which there are many. Since the rest of the paper pertains to biduals, the definition of wholeness should be amended as follows: call a W-algebra $$\mathfrak {R}$$ whole iff there is some C-algebra $$\mathfrak {A}$$ such that $$\mathfrak {R}\cong \mathfrak {A}^{*}$$. On this definition, the remaining propositions in §5.3 are true. The only proposition whose proof makes use of Prop. 6 is Prop. 7. As such, the proof of Prop. 7 in the appendix is not valid. However, it can be recovered as follows: Let $$\mathfrak {R}$$ be a W-algebra and $$\mathfrak {A}$$ and $$\mathfrak {B}$$ be two C-algebras such that $$\mathfrak {A}^{}\cong \mathfrak {R}\cong \mathfrak {B}^{}$$. Cor. 1.13.3 of Sakai (1971) implies that $$\mathfrak {R}$$ has a unique predual, and hence a unique normal state space. This entails that the state spaces of $$\mathfrak {A}$$ and $$\mathfrak {B}$$ are affine homeomorphic and that this homeomorphism preserves global orientations (We know that these state spaces are globally oriented by Thm. 5.54 of Alfsen and Shultz (2001, p. 222)). It follows from Cor. 5.72 of Alfsen and Shultz (2001, p. 234) that $$\mathfrak {A}$$ and $$\mathfrak {B}$$ are -isomorphic. ## References 1. Alfsen, E., & Shultz, F. (2001). State spaces of operator algebras. Boston, MA: Birkhauser. 2. Sakai, S. (1971). C-algebras and W-algebras. New York: Springer. ## Author information Authors ### Corresponding author Correspondence to Benjamin H. Feintzeig. The original article can be found online at https://doi.org/10.1007/s11229-017-1497-6. ## Rights and permissions Reprints and Permissions	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9571380615234375, "perplexity": 510.9418470889879}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178350717.8/warc/CC-MAIN-20210225041034-20210225071034-00211.warc.gz"}
https://stacks.math.columbia.edu/tag/02P4	## 42.1 Introduction In this chapter we discuss Chow homology groups and the construction of Chern classes of vector bundles as elements of operational Chow cohomology groups (everything with $\mathbf{Z}$-coefficients). We start this chapter by giving the shortest possible algebraic proof of the Key Lemma 42.6.3. We first define the Herbrand quotient (Section 42.2) and we compute it in some cases (Section 42.3). Next, we prove some simple algebra lemmas on existence of suitable factorizations after modifications (Section 42.4). Using these we construct/define the tame symbol in Section 42.5. Only the most basic properties of the tame symbol are needed to prove the Key Lemma, which we do in Section 42.6. Next, we introduce the basic setup we work with in the rest of this chapter in Section 42.7. To make the material a little bit more challenging we decided to treat a somewhat more general case than is usually done. Namely we assume our schemes $X$ are locally of finite type over a fixed locally Noetherian base scheme which is universally catenary and is endowed with a dimension function. These assumptions suffice to be able to define the Chow homology groups $\mathop{\mathrm{CH}}\nolimits _(X)$ and the action of capping with Chern classes on them. This is an indication that we should be able to define these also for algebraic stacks locally of finite type over such a base. Next, we follow the first few chapters of [F] in order to define cycles, flat pullback, proper pushforward, and rational equivalence, except that we have been less precise about the supports of the cycles involved. We diverge from the presentation given in [F] by using the Key lemma mentioned above to prove a basic commutativity relation in Section 42.27. Using this we prove that the operation of intersecting with an invertible sheaf passes through rational equivalence and is commutative, see Section 42.28. One more application of the Key lemma proves that the Gysin map of an effective Cartier divisor passes through rational equivalence, see Section 42.30. Having proved this, it is straightforward to define Chern classes of vector bundles, prove additivity, prove the splitting principle, introduce Chern characters, Todd classes, and state the Grothendieck-Riemann-Roch theorem. There are two appendices. In Appendix A (Section 42.68) we discuss an alternative (longer) construction of the tame symbol and corresponding proof of the Key Lemma. Finally, in Appendix B (Section 42.69) we briefly discuss the relationship with $K$-theory of coherent sheaves and we discuss some blowup lemmas. We suggest the reader look at their introductions for more information. We will return to the Chow groups $\mathop{\mathrm{CH}}\nolimits _(X)$ for smooth projective varieties over algebraically closed fields in the next chapter. Using a moving lemma as in [Samuel], , and and Serre's Tor-formula (see or ) we will define a ring structure on $\mathop{\mathrm{CH}}\nolimits _*(X)$. See Intersection Theory, Section 43.1 ff. In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 2, "x-ck12": 0, "texerror": 0, "math_score": 0.9649207592010498, "perplexity": 326.6251209398252}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662604495.84/warc/CC-MAIN-20220526065603-20220526095603-00205.warc.gz"}
https://www.arxiv-vanity.com/papers/cond-mat/0609048/	# Breakdown of a topological phase: Quantum phase transition in a loop gas model with tension Simon Trebst Microsoft Research, Station Q, University of California, Santa Barbara, CA 93106 Philipp Werner Department of Physics, Columbia University, 538 West 120th Street, New York, NY 10027 Matthias Troyer Theoretische Physik, Eidgenössische Technische Hochschule Zürich, CH-8093 Zürich, Switzerland Kirill Shtengel Department of Physics and Astronomy, University of California, Riverside, CA 92521 Chetan Nayak Microsoft Research, Station Q, University of California, Santa Barbara, CA 93106 Department of Physics and Astronomy, University of California, Los Angeles, CA 90095 February 7, 2021 ###### Abstract We study the stability of topological order against local perturbations by considering the effect of a magnetic field on a spin model – the toric code – which is in a topological phase. The model can be mapped onto a quantum loop gas where the perturbation introduces a bare loop tension. When the loop tension is small, the topological order survives. When it is large, it drives a continuous quantum phase transition into a magnetic state. The transition can be understood as the condensation of ‘magnetic’ vortices, leading to confinement of the elementary ‘charge’ excitations. We also show how the topological order breaks down when the system is coupled to an Ohmic heat bath and discuss our results in the context of quantum computation applications. Topological phases are among the most remarkable phenomena in nature. Although the underlying interactions between electrons in a solid are not topologically invariant, their low-energy properties are. This enhanced symmetry makes such phases an attractive platform for quantum computation since it isolates the low-energy degrees of freedom from local perturbations – a usual cause of errors ToricCode . Tractable theoretical models with topological phases in frustrated magnets ToricCode ; magnets , Josephson junction arrays Ioffe02 ; Motrunich02 , or cold atoms in traps ColdAtoms have been proposed. However, such phases have not, thus far, been seen experimentally outside of the quantum Hall regime. Is it because topological phases are very rare and these models are adiabatically connected only to very small regions of the phase diagrams of real experimental systems? In this paper, we take a first step towards answering this question. We begin with the simplest exactly soluble model of a topological phase ToricCode , whose Hamiltonian is given below in Eq. (1). This model describes a frustrated magnet with four-spin interactions similar to cyclic ring exchanges. It is closely related to the quantum dimer model DimerModel for frustrated magnets, which can be realized in Josephson junction arrays Ioffe02 . We consider perturbations of the soluble model that, when sufficiently large, drive the system out of the topological phase. The question is, how large? A small answer would imply that this topological phase is delicate and occupies a small portion of the phase diagram. This might explain the paucity of experimentally-observed topological phases. Instead, we find that ‘sufficiently large’ is a magnetic field of order one () in units of the basic four-spin plaquette interaction. Our numerical simulations demonstrate, for the first time, several key signatures of the phase transition out of the topological phase, including the finite-size degeneracy splitting of the topological sectors, the condensation of ‘magnetic’ excitations, and the confinement of ‘electric’ charges. We also consider perturbing the system by coupling it to an Ohmic heat bath. When coupled to such a bath, a quantum mechanical degree of freedom can undergo a transition from coherent to incoherent behavior Leggett87 . Recently, the effects of such a coupling on quantum phase transitions, at which divergent numbers of quantum mechanical degrees of freedom interact, have also been studied Werner04 . In both contexts, the coupling to the heat bath tends to make the system more classical. Coherent quantum oscillations are suppressed, while broken symmetry phases – which are essentially classical – are stabilized. A topological phase is quantum mechanical in nature. We find that coupling the heat bath to the kinetic energy, i.e. plaquette flip operator, does not destroy such a phase. However, when the dissipation is strong, the gap becomes very small, and the topological phase may be too delicate to observe or use at reasonable temperatures. On the other hand, if the heat bath is coupled to the classical state of each plaquette, the topological phase is destroyed through a Kosterlitz-Thouless transition at a dissipation strength of order one. These results can be recast in quantum information language: the ground states in different topological sectors are the different basis states of an encoded quantum memory. Quasiparticle excitations are states outside of the code subspace. The stability of the topological phase, as measured by an energy gap within a topological sector (essentially the energy cost for a pair of quasiparticles), translates into an error rate for topological qubits. At zero temperature, errors are due to the virtual excitation of pairs of quasiparticles, assuming that the system is shielded from perturbations at frequencies higher than . Such virtual processes lead to a splitting between topological sectors , where is the system size and is a characteristic velocity. As the temperature is increased, the thermal excitation of particles eventually becomes more important and the error rate is 111In a special case of a topological phase considered in Chamon05 , thermal activation always dominates quantum tunneling.. (The actual concentration of excitations leading to unrecoverable errors was studied in Wang03 .) ## The model – HTC=−A∑v∏j∈\scriptsize vertex(v)σzj−B∑p∏j∈\scriptsize plaquette(p)σxj, (1) where the are quantum spins on the edges of a square lattice with vertices on a torus. Since all terms in Hamiltonian (1) commute with each other, the model can be solved exactly ToricCode . The ground-state manifold can be described as a quantum loop gas where the loops consist of chains of up-pointing -spins and the loop fugacity is . On the torus there are four degenerate ground states which can be classified by a winding number parity along a cut in the or direction. Here we study the effect of perturbing the Hamiltonian (1) with a loop tension which can be introduced either by a longitudinal magnetic field or local Ising interaction of the form H=HTC−h∑iσzi−J∑⟨ij⟩σziσzj, (2) where () is the strength of the magnetic field (Ising interaction). These are the dominant perturbations expected in a physical implementation, e.g. in a Josephson junction implementation Ioffe02 ; Motrunich02 they arise from electric potential perturbations or Coulomb interactions between neighboring quantum dots. We discuss this model in the limit of a large charge gap, i.e. , where it becomes equivalent to the ‘even’ Ising gauge theory IGT . The low-energy sector has no free charges and any state is described by a collection of loops that can be obtained from a reference state (e.g. all ) by a sequence of plaquette flips. Let us introduce a new plaquette spin operator with eigenvalues where is the number of times a given plaquette has been flipped, counting from the reference state. Then , where and are the plaquettes separated by the edge . The plaquette flip term in Eq. (1) becomes . In the new variables, Hamiltonian (2) becomes equivalent to the transverse field Ising model (with both nearest and next-nearest neighbor Ising interactions) in a basis restricted to loop states. Independent of the choice of basis states the system orders at a critical field strength determined by continuous-time quantum Monte Carlo simulations TIM . The transverse field Ising model for the plaquette spins can be mapped to a classical (2+1)-dimensional Ising model: Hcl=−Kτ∑τ,pSp(τ)Sp(τ+Δτ)−K∑τ,⟨p,q⟩Sp(τ)Sq(τ), (3) where the imaginary time is discretized in steps of and . The magnetic field and the local Ising interaction introduce a nearest and next-nearest neighbor interaction between the classical spins , respectively. Since the exact nature of this Ising interaction does not play a role in the following we do not discuss the case of next-nearest neighbor couplings in detail. The coupling along real-space coordinates is then given by and along imaginary time by . The model (3) describes the well-known continuous magnetic phase transition of the 3D Ising model. For isotropic interactions, , the critical coupling has been determined with high precision to be Ferrenberg:91 . Setting this gives a critical loop tension with isotropic couplings at . The magnetic susceptibility diverges at the transition and the magnetization shows a corresponding kink, as shown in Fig. 2. Although this is not a symmetry-breaking transition, the analogous transition driven by next-nearest interaction is a continuous quantum phase transition from a topologically ordered quantum state to a broken symmetry state ContinuousPhaseTransition . The magnetic transition can also be understood in terms of the condensation of ‘magnetic vortices’, plaquettes with . While for the original Hamiltonian (1), the gap to these excitations is , they become gapless and condense at the critical loop tension, as shown in Fig. 3. The gap has been estimated from measurements of the imaginary time correlation length as which we have calculated applying continuous-time quantum Monte Carlo simulations using the ALPS looper code ALPS ; LooperCode ). The thermal excitation of pairs of magnetic vortices occurs with probability , also shown in Fig. 3. ## Topological order – The breakdown of topological order at the phase transition can be seen from the energy splitting between the ground-states for the various topological sectors. When winding parities are used as basis states for a quantum memory, this splitting causes phase errors. (The absence of ‘electric’ charges precludes any transitions between different winding parities so bit flip errors cannot occur.) In the topological phase, the virtual excitation of quasiparticles leads to a small splitting between the topological sectors. In the classically-ordered phase, on the other hand, the energy splitting should scale with , which corresponds to the energy cost of a loop in the ordered ground state. As the winding parity is conserved by imaginary time spin-flip operations, we can simulate the system in one of the topological sectors by defining an initial spin configuration that corresponds to the respective limit for large loop tension. Fig. 4 shows the calculated splitting for various system sizes in the vicinity of the critical loop tension. At the phase transition, the behavior qualitatively changes from power-law scaling for strong loop tension to an exponential suppression in the topological phase for small loop tension as discussed above. A more quantitative picture arises from the finite-size scaling analysis of the energy splitting between the {even-odd}- and {even-even}-parity sectors shown in Fig. 5. For the critical loop tension we find a power-law scaling with an exponent . Below the critical value the scaling turns into exponential scaling as expected for the topological phase. ## Confinement transition – For the loop gas Hamiltonian (1) the elementary electric charge excitations (end-points of an open loop) are deconfined. For strong loop tension, however these excitations are expected to become confined, thereby eliminating all open loops. We can study this confinement transition in our simulations of model (3) by artificially introducing pairs of electric charge excitations for the sampled loop configurations. This allows us to measure the confinement length as the square root of the average second moment of the distance between the two excitations, which for a torus with even extent has to be normalized by a factor . The measured confinement length shown in Fig. 6 clearly demonstrates that electric charges remain deconfined for the full extent of the topological phase and that the confinement transitions occurs simultaneously with the magnetic phase transition. At the critical loop tension the confinement lengths for various system sizes cross which demonstrates that the confinement length diverges with the same critical exponents as the magnetic correlation length and there is only one length scale describing the phase transition. This can be seen from the following finite-size scaling argument: We may write the finite-size scaling behavior for the confinement length as where () is the critical exponent of the correlation (confinement) length respectively and . With we obtain for the confinement length . At the phase transition the existence of a crossing point then implies . For our model without dynamical electric charges, this measure of the confinement of test charges is closely related to the calculation of a Wilson loop expectation value. In the presence of dynamical electric charges, this becomes trickier; Polyakov loops have been used as an order parameter for the finite temperature transition of the 3D Ising gauge modelLGT . ## Dissipation – Finally, we discuss the effect of dissipation when Hamiltonian (1) is coupled to an Ohmic heat bath. Since our model excludes dynamical electric charges, we do not consider coupling a heat bath directly to . Instead, we first examine coupling a heat bath to so that a ‘phonon’ is created when a plaquette flips. This type of dissipation could occur in a Josephson junction model Schoen:90 or in a spin model through the spin-phonon coupling. The standard procedure Caldeira for a linear spectral density (‘Ohmic’ dissipation) results in an effective action for independent Ising chains with long-range couplings in an external longitudinal magnetic field (which here means parallel to ). We note that as a consequence of the Lee–Yang theorem Lee:52 , there can be no singularities of the respective partition function at any real non-zero longitudinal field, ruling out the existence of a quantum phase transition for this model. In particular, this implies that the magnetic gap remains finite for any dissipation strength! An entirely different behavior arises if dissipation is coupled such that it stabilizes the ‘classical’ state of the system. Coupling the bath to either or stabilizes the classical state of a single spin or a plaquette, respectively. We consider the latter as it should be more effective at damping quantum fluctuations, although we expect the former to have similar physics. The same procedure as above then leads to a model for decoupled Ising chains given by Hcl=−Kτ∑τ,pSp(τ)Sp(τ+Δτ)−α2∑τ<τ′,p(πNτ)2Sp(τ)Sp(τ′)sin2(πNτ\|τ−τ′\|), (4) where the parameter measures the dissipation strength. This model has been well studied Dissipation and is known to exhibit a Thouless-type phase transition into a classically disordered, fluctuationless phase. The critical value of this transition depends weakly on the cutoff in the long-range interaction; in our simulations . At the transition the magnetic gap vanishes, in sharp contrast to the previous case. Due to the long range interactions introduced by the dissipative coupling, the spin-spin correlations will asymptotically decay as Griffiths:67 . It is therefore a non-trivial task to define a correlation-time and hence to estimate the excitation gap. For one observes an exponential decay of the correlation function onto the asymptotic behavior, and subtracting the -contribution allows one to estimate , which is found to grow linearly with . For , this procedure can no longer be used. In this region we estimate from the asymptotic decay of the correlations, proportional to . These s grow approximately exponentially in the region . Alternatively, one could define a correlation-time from the crossover scale where the short-time behavior crosses over to the asymptotic form. This definition in the spirit of a ‘Josephson length’ Chakravarty:95 yields (up to a normalization factor) the same results as the extracted from the asymptotic decay. The gap estimated from the inverse correlation function, as well as the error probability is plotted in Fig. 7. We find that the error probability remains negligibly small below the crossover value . ## Outlook – We have shown that the topological phase which governs the toric code model ToricCode actually exists in an extended region of phase space around the soluble point. It is stable against deviations of the system Hamiltonian from the ideal one and also the coupling of the system to its environment. In general, this demonstrates that a system does not necessarily have to be particularly fine-tuned to reach a topological phase. The paucity of their experimental observations may thus be due not to some intrinsic delicateness of such phases, but rather to the experimental subtlety involved in identifying them. In future work, these conclusions need to be tested in other, more exotic topological phases which support universal topological quantum computation Freedman01 . We thank E. Ardonne, L. Balents, S. Chakravarty, and A. Kitaev for stimulating discussions. ## References • (1) A. Yu. Kitaev, Ann. Phys. 303, 2 (2003). • (2) R. Moessner and S. L. Sondhi, Phys. Rev. Lett 86, 1881 (2001); C. Nayak and K. Shtengel, Phys. Rev. B 64, 064422 (2001); L. Balents, M. P. A. Fisher, and S. Girvin, Phys. Rev. B 65, 224412 (2002); M. Freedman, et al. Ann. Phys. 310, 428 (2004); M. A. Levin and X.-G. Wen, Phys. Rev. B 71, 045110 (2005); P. Fendley and E. Fradkin, Phys. Rev. B 72, 024412 (2005). • (3) L. B. Ioffe et al., Nature 415, 503 (2002). • (4) O. I. Motrunich and T. Senthil, Phys. Rev. Lett. 89, 277004 (2002). • (5) L.-M. Duan, E. Demler, and M. D. Lukin, Phys. Rev. Lett. 91, 090402 (2003); V. Gurarie, L. Radzihovsky, and A. V. Andreev, Phys. Rev. Lett. 94, 230403 (2005). • (6) D. S. Rokhsar and S. A. Kivelson, Phys. Rev. Lett. 61, 2376 (1988). • (7) A. Leggett at al., Rev. Mod. Phys. 59, 1 (1987). • (8) P. Werner et al., Phys. Rev. Lett. 94, 047201 (2005); P. Werner et al., J. Phys. Soc. Jpn. Suppl. 74, 67 (2005). • (9) C. Chamon, Phys. Rev. Lett. 94, 040402 (2005). • (10) C. Wang, J. Harrington, and J. Preskill, Ann. Phys. 303, 31 (2003). • (11) R. Moessner, S. L. Sondhi, and E. Fradkin, Phys. Rev. B65, 024504 (2002). • (12) H. W. J. Blöte and Y. Deng, Phys. Rev. E66, 066110 (2002). • (13) A. M. Ferrenberg and D. P. Landau, Phys. Rev. B44, 5081 (1991). • (14) E. Ardonne, P. Fendley, and E. Fradkin, Ann. Phys. 310, 493 (2004). • (15) F. Alet et al., J. Phys. Soc. Jpn. Suppl. 74, 30 (2005). See also http://alps.comp-phys.org. • (16) S. Todo and K. Kato, Phys. Rev. Lett. 87, 047203 (2001). • (17) M. Caselle, M. Hasenbusch, and M. Panero, JHEP 01, 057 (2003). • (18) G. Schön and A. D. Zaikin, Phys. Rep. 198, 237 (1990). • (19) A. O. Caldeira and A. J. Leggett, Ann. Phys. 149 374 (1983). • (20) T. D. Lee and C. N. Yang, Phys. Rev. 87, 410 (1952). • (21) D. J. Thouless, Phys. Rev. 187, 732 (1969); J. Fröhlich and T. Spencer, Commun. Math. Phys. 84, 87 (1982); J. Bhattacharjee et al., Phys. Rev. B24, 3862 (1981); E. Luijten and H. Messingfeld, Phys. Rev. Lett. 86, 5305 (2001). • (22) R. B. Griffiths, J. Math. Phys. 8, 478 (1967). • (23) S. Chakravarty and J. Rudnick, Phys. Rev. Lett. 75 501 (1995). • (24) M. H. Freedman, Found. Comput. Math. 1, 183 (2001), and references therein.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8678984642028809, "perplexity": 776.9586786095649}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243991648.40/warc/CC-MAIN-20210514060536-20210514090536-00077.warc.gz"}
http://physics.stackexchange.com/questions/57561/why-does-plancks-law-for-black-body-radiation-have-that-bell-like-shape	# Why does Planck's law for black body radiation have that bell-like shape? I'm trying to understand Planck's law for the black body radiation, and it states that a black body at a certain temperature will have a maximum intensity for the emission at a certain wavelength, and the intensity will drop steeply for shorter wavelengths. Contrarily, the classic theory expected an exponential increase. I'm trying to understand the reason behind that law, and I guess it might have to do with the vibration of the atoms of the black body and the energy that they can emit in the form of photons. Could you explain in qualitative terms what's the reason? - Note that a function which must be smooth, normalizable and everywhere non-negative is going to have one or more peaks ... – dmckee Mar 21 '13 at 13:29 @dmckee hehe that's clever but sadly doesn't help me in the physical side :) – clabacchio Mar 21 '13 at 13:57 No, there isn't any physics in it. Still, the argument is very general and can be used as a handwavy motivation for lots of different systems having maxima in various parameters. It's just that you then have to understand the physics that sets any particular maximum. – dmckee Mar 21 '13 at 14:12 @dmckee true, and indeed understanding why the maximum exists is the "easy" part. The harder (to me) is what are the concurrent phenomena that create that curve. – clabacchio Mar 21 '13 at 14:27 @clabacchio This can be understood better if you think of it as a resonance phenomenon. The various porcesses in the hot body, which are well known, do not occur with equal probability/rate. Given the temperature of the hot body, some processes are more probable than orhers, and there is a particular process that dominates the emission spectrum. This is what you get from the basic equation $\lambda_{max}T=c$, where c is a known constant. – JKL Mar 21 '13 at 16:44 The Planck distribution has a more general interpretation: It gives the statistical distribution of non-conserved bosons (e.g. photons, phonons, etc.). I.e., it is the Bose-Einstein distribution without a chemical potential. With this in mind, note that, in general, in thermal equilibrium without particle-number conservation, the number of particles $n(E)$ occupying states with energy $E$ is proportional to a Boltzmann factor. To be precise: $$n(E) = \frac{g(E) e^{-\beta E}}{Z}$$ Here $g(E)$ is the number of states with energy $E$, $\beta = \frac{1}{kT}$ where $k$ is the Boltzmann constant, and $Z$ is the partition function (i.e. a normalization factor). The classical result for $n(E)$ or equivalently $n(\lambda)$ diverges despite the exponential decrease of the Boltzmann factor because $g(E)$ grows unrealistically when the quantization of energy levels is not accounted for. This is the so-called ultraviolet catastrophe. When the energy of e.g. photons is assumed to be quantized so that $E = h\nu$ the degeneracy $g(E)$ does not outstrip the Boltzmann factor $e^{-\beta E}$ and $n(E) \longrightarrow 0$ as $E \longrightarrow \infty$, as it should. This result is of course due to Planck, hence the name of the distribution. It is straightforward to work this out explicitly for photons in a closed box or with periodic boundary conditions (e.g. see Thermal Physics by Kittel). I hope this was not too technical. To summarize, the fundamental problem in the classical theory is that the number of accessible states at high energies (short wavelengths) is unrealistically large because the energy levels of a "classical photon" are not quantized. Without this quantization, the divergence of $n(E)$ (equivalently, of $n(\lambda)$) would imply that the energy density of a box of photons is infinite at thermal equilibrium. This is of course nonsensical. - Joshua has beaten me to an answer, but I'll still post this since it's written at a simpler level. The reason you get a maximum because there are two effects that oppose each other. The number of modes per unit frequency rises as frequency squared, so as long as the energy of the modes is well below kT the energy is proportional to frequency squared. This is why the black body spectrum initially rises approximately as frequency squared. However the probability that a mode is excited falls exponentially as soon as the energy of the mode is greater than kT, so as the frequency goes to infinity the emitted radiation falls to zero. The net result of the two effects is that the emission first rises then falls again, and that's why there is a maximum in the middle. - This (your response) is probably a more direct answer to his question. I.e., the increase in the degeneracy initially dominates and is then eventually overcome by the exponential suppression. This is a really generic behavior. – Joshua Barr Mar 21 '13 at 10:05 If I can ask, what's the reason for the squared increase of the modes with less than kT energy? – clabacchio Mar 21 '13 at 10:28 @clabacchio Put simply: In this case, there are more ways a particle can exist with a larger energy. To see this qualitatively, try considering a (quantum mechanical) particle in a 3D box. The quantitative result is different for photons than a massive particle, but the qualitative physics is the same. Note the dimensionality is crucial here. More information here. – Joshua Barr Mar 21 '13 at 10:59	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8918620944023132, "perplexity": 321.1273346510922}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-48/segments/1448398462686.42/warc/CC-MAIN-20151124205422-00104-ip-10-71-132-137.ec2.internal.warc.gz"}
https://www.physicsforums.com/threads/sum-of-squares.123541/	# Sum of Squares 1. Jun 11, 2006 ### Dragonfall Which squares are expressible as the sum of two squares? Is there a simple expression I can write down that will give me all of them? Some of them? Parametrization of the pythagorean triples doesn't seem to help. 2. Jun 11, 2006 ### StatusX What do you mean by "parameterization of pythagorean triples"? If it's what I think you mean, I don't see why this wouldnt give you enough information for what you want to do. 3. Jun 11, 2006 ### shmoe 0 is a square, so really all of them. Excluding this trivial case, if c^2 can be written as c^2=a^2+b^2 where a and b are non zero, then we can divide by common factors to get d^2=e^2+f^2, where the terms are relatively prime. Do you know any characterization of integers that can be written as sums of relatively prime squares (if not, what about primes)? Then you'd know c^2 would have to have a divisor of this form (conversely having a divisor of this form will ensure a representation). 4. Jun 12, 2006 ### Dragonfall I worded the question wrong. I wanted to ask "given a square, how do I know if it can be written as the sum of two squares (except 0)". I got it now. Similar Discussions: Sum of Squares	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8911657929420471, "perplexity": 569.1337122375838}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-13/segments/1490218189462.47/warc/CC-MAIN-20170322212949-00528-ip-10-233-31-227.ec2.internal.warc.gz"}
http://www.airports-worldwide.com/articles/article0701.php	Surface of revolution By Wikipedia, the free encyclopedia, http://en.wikipedia.org/wiki/Surface_of_revolution A portion of the curve x=2+cos z rotated around the z axis A surface of revolution is a surface created by rotating a curve lying on some plane (the generatrix) around a straight line (the axis of rotation) that lies on the same plane. Examples of surfaces generated by a straight line are the cylindrical and conical surfaces. A circle that is rotated about a (coplanar) axis through the center generates a sphere. If the axis is coplanar and outside the circle it generates a toroidal surface. ## Area formula If the curve is described by the parametric functions x(t), y(t), with t ranging over some interval [a,b], and the axis of revolution is the y axis, then the area A is given by the integral $A = 2 \pi \int_a^b x(t) \ \sqrt{\left({dx \over dt}\right)^2 + \left({dy \over dt}\right)^2} \, dt,$ provided that x(t) is never negative. This formula is the calculus equivalent of Pappus's centroid theorem. The quantity $\left({dx \over dt}\right)^2 + \left({dy \over dt}\right)^2$ comes from the Pythagorean theorem and represents a small segment of the arc of the curve, as in the arc length formula. The quantity x(t) is the path of (the centroid of) this small segment, as required by Pappus's theorem. If the curve is described by the function y = f(x), axb, then the integral becomes $A=2\pi\int_a^b y \sqrt{1+\left(\frac{dy}{dx}\right)^2} \, dx$ for revolution around the x-axis, and $A=2\pi\int_a^b x \sqrt{1+\left(\frac{dx}{dy}\right)^2} \, dy$ for revolution around the y-axis. These come from the above formula. For example, the spherical surface with unit radius is generated by the curve x(t) = sin(t), y(t) = cos(t), when t ranges over [0,π]. Its area is therefore $A = 2 \pi \int_0^\pi \sin(t) \sqrt{\left(\cos(t)\right)^2 + \left(\sin(t)\right)^2} \, dt = 2 \pi \int_0^\pi \sin(t) \, dt = 4\pi.$ For the case of the spherical curve with radius r, $y(x) = \sqrt{r^2 - x^2}$ rotated about the x-axis $A = 2 \pi \int_{-r}^{r} \sqrt{r^2 - x^2}\,\sqrt{1 + \frac{x^2}{r^2 - x^2}}\,dx$ $= 2 \pi \int_{-r}^{r} r\,\sqrt{r^2 - x^2}\,\sqrt{\frac{1}{r^2 - x^2}}\,dx$ $= 2 \pi \int_{-r}^{r} r\,dx$ $= 4 \pi r^2\,$ ## Rotating a function To generate a surface of revolution out of any 2-dimensional scalar function y = f(x), simply make u the function's parameter, set the axis of rotation's function to simply u, then use v to rotate the function around the axis by setting the other two functions equal to f(u)sinv and f(u)cosv conversely. For example, to rotate a function y = f(x) around the x-axis starting from the top of the xz-plane, parameterize it as $\vec r(u,v)=\langle u,f(u)\sin v,f(u)\cos v\rangle$ for $u\in x$ and $v\in[0,2\pi]$ . ## Geodesics on a surface of revolution Geodesics on a surface of revolution are governed by Clairaut's relation. ## Applications of surfaces of revolution The use of surface of revolutions is essential in many fields in physics and engineering. When certain objects are designed digitally, revolutions like these can be used to determine surface area without the use of measuring the length and radius of the object being designed. Published - July 2009	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 13, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.827829122543335, "perplexity": 489.27281107841986}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-09/segments/1487501170651.78/warc/CC-MAIN-20170219104610-00650-ip-10-171-10-108.ec2.internal.warc.gz"}
http://math.stackexchange.com/questions/284072/distributional-differential-equation-somehow-related-to-compact-support-distrib	# Distributional differential equation, somehow related to compact support distributions I've been mulling over a problem from Friedlander's Introduction to Distribution Theory for a few days now: in Chapter 3 (on distributions with compact support), it asks to solve the differential equation $x\partial u-\lambda u=0$ for arbitrary $\lambda\in\mathbb{C}$. I've seen it stated that a distribution $u\in\mathbb{R}^n$ is homogeneous of degree $\lambda$ iff it satisfies Euler's equation, i.e. $\sum x_i\partial_i u=\lambda u$, but these just refer to Gel'fand and Shilov's book which I don't have access to. Moreover, this approach doesn't seem the author's intended one, given the chapter this problem is in. My method was basically to try and use "integrating factors", but of course we have only defined multiplication by smooth functions, which $x_+^\lambda$ is generally not. I've basically seen it without proof via online searching that $u=Ax_+^\lambda+Bx_-^\lambda$ is the general answer, but I can't seem to crack it, or find anything in this problem with compact support. Any help would be much appreciated. - ## 1 Answer I'll outline several ideas that come into mind and give some remarks concerning the validity of the reasoning. First, I assume that we work in $\Bbb R$. Let's for instant suppose that $\lambda< -1$. We take a test function $\phi$, multiply it by $x^{-\lambda-1}$ (therefore, it's still a test function), and then apply to it both parts of our equation. After some easy modifications, we arrive to $$\langle(x^{-\lambda}u)',\phi\rangle=0,$$ which yields $$x^{-\lambda}u=const.$$ Now we have a particular solution (notably, so called $\mathrm{pf.}x^{\lambda}_+$ and $\mathrm{pf.}x^{\lambda}_+$, partie finie d'Hadamard in France, don't know the name in English). There're also general solutions, i.e. solving $$x^{-\lambda}u=0.$$ It's easy to see that the support of $u$ in this case (cf. your reference book, the corresponding lemma should be there; or just prove, it's an exercise, really) is exactly $\{0\}$. By another classic theorem, the family of solutions is described by $$\sum_{j=0}^{[\lambda]}c_j\delta_{0}^{(j)},$$ where $c_j$ are constants and $[\lambda]$ is the integer part of $\lambda$. What doesn't work this smoothly in other cases: if $\lambda$ is complex, then $x^\lambda$ is a multivalued function. Then again, we can't directly apply the same method if $\lambda\ge-1$. You can explicitly find the solutions for $\lambda=0$ and $\lambda=-1$. On the other hand, if a differential equation has a solution in a classical sense, then this solution is a solution in the sense of distributions. -	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.943148136138916, "perplexity": 202.7707760168524}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-30/segments/1469257829972.19/warc/CC-MAIN-20160723071029-00165-ip-10-185-27-174.ec2.internal.warc.gz"}
http://mathoverflow.net/questions/4125/if-spec-z-is-like-a-riemann-surface-whats-the-analogue-of-integration-along-a/4130	# If Spec Z is like a Riemann surface, what's the analogue of integration along a contour? Rings of functions on a nonsingular algebraic curve (which, over $\mathbb{C}$, are holomorphic functions on a compact Riemann surface) and rings of integers in number fields are both examples of Dedekind domains, and I've been trying to understand the classical analogy between the two. As I understand it, $\operatorname{Spec} \mathcal{O}_k$ should be thought of as the curve / Riemann surface itself. Is there a good notion of integration in this setting? Is there any hope of recovering an analogue of the Cauchy integral formula? (If I am misunderstanding the point of the analogy or stretching it too far, please let me know.) - For a variety $X$ over a finite field, I guess one can take $\ell$-adic sheaves to replace differential forms. Then the local integral around a closed point $x$ (like integral over a little loop around that point) is the trace of the local Frobenius $\operatorname{Frob}_x$ on the stalk of sheaf, the so-called naive local term. Note that $\operatorname{Frob}_x$ can be regarded as an element (or conjugacy class) in $\pi_1(X)$, "a loop around $x$". The global integral would be the global trace map $$H^{2d}_c(X,\mathbf{Q}_{\ell})\to\mathbf{Q}_{\ell}(-d),$$ and the Tate twist is responsible for the Hodge structure in Betti cohomology (or the $(2\pi i)^d$ one has to divide by). The Lefschetz trace formula might be the analog of the residue theorem in complex analysis on Riemann surfaces. For the case of number fields, each closed point $v$ in $\operatorname{Spec} O_k$ still defines a "loop" $\operatorname{Frob}_v$ in $\pi_1(\operatorname{Spec} k)$ (let's allow ramified covers. One can take the image of $\operatorname{Frob}_v$ under $\pi_1(\operatorname{Spec} k)\to\pi_1(\operatorname{Spec} O_k)$, but the target group doesn't seem to be big enough). For global integral, there's the Artin-Verdier trace map $H^3(Spec\ O_k,\mathbb G_m)\to\mathbb{Q/Z}$ and a "Poincaré duality" in this setting, but I don't know if there is a trace formula. The fact that 3 is odd always makes me excited and confused. So basically I think of trace maps (both local and global) as counterpart of integrals. Correct me if I was wrong. - This is interesting, but a little over my head. If you don't mind, could you explain how one thinks about the contour integral in this abstract setting, i.e. in terms of the fundamental group? – Qiaochu Yuan Nov 4 '09 at 19:58 A loop in this setting is a map from a scheme with a "cyclic" fundamental group. Finite fields have this property, so their spectra can be viewed as circles. For Spec Z, the only interesting loops we see are the canonical maps from spectra of finite fields. One has an analogy between integration of differentials and parallel transport along connections, so we are determining how a vector bundle (our l-adic sheaf) is transformed as we follow flat sections around a "circle". – S. Carnahan Nov 5 '09 at 2:37 "The fact that 3 is odd always makes me excited and confused." Fantastic. – Cam McLeman May 8 '11 at 18:56 I think that you have understood the analogy correctly, and you have pinpointed one of its weaknesses. Although number fields are like one dimensional functional fields in many ways, one of the differences is that the vector space of Kahler differentials for a number field has dimension 0, not 1. Here Kahler differentials are the vector space generated by symbols $dx$, subject to the relations $d(x+y) = dx + dy$ and $d(xy) = x dy + y dx$. Therefore, there is nothing like differentials, and nothing we can integrate. But it is possible that there is some more sophisticated way to solve this problem. (Maybe using Arakelov geometry?) I'm looking forward to reading the other answers. - This certainly does have a bit of the Arakelov smell about it. Maybe if you include the infinite places there's some different notion of Kahler differential? – Ben Webster Nov 4 '09 at 18:40 That makes sense; it's related to the fact that the abc conjecture is much harder than the Mason-Stothers theorem. I think a lot of people would want to see a notion of differential in the number field setting for this reason. – Qiaochu Yuan Nov 4 '09 at 18:45 I think David's module of differentials on R should include the rule that dc = 0 when c is a constant: here the ring of constants is some specified subring R_0 of R. (So e.g. when R = C[t] one takes R_0 = C; otherwise, the differentials are way bigger than you want.) So we see the problem: David's module is Omega_{Z/Z}, and the relative dimension is 0, while for Omega_{C[t]/C} the relative dimension is 1. In order to have a good notion of differentials you'd need a "field of constants" "inside" Spec Z, which would have... oh, I dunno.... one element? – JSE Nov 5 '09 at 1:31 Yup! JSE's comment is exactly right, and gives a great way of seeing why we want a field with one element. – David Speyer Nov 5 '09 at 1:38 Lichtenbaum talked about that and interesting applications last December: institut.math.jussieu.fr/projets/tn/STN/11/lichtenbaum-11.pdf – Thomas Riepe May 8 '11 at 15:05 For a number $f$ the local-global formula written as $$f_2\cdot f_3\cdot \dotsb \cdot f_{\mathbb Q} = 1$$ where $f_p$ is the inverse power of $p$ that is equal to $f$ in the local field of $p$-adic numbers and $f_{\mathbb{Q}}$ is $f$ should provide a reasonable analogue (in other words, integration over all holes gives $0$ on a closed surface). (An example for $f = 75: 1 \times 1/3 \times 1/25 \times 1 \times \dotsb \times 75 = 1$) A better formulation would involve adeles: the adele ring $\mathbb{A}_{\mathbb Q}$ is a semi-restricted product of $p$-adic rationals and rationals themselves. There is a map $\mathbb Q\to\mathbb A_{\mathbb Q}$ and every element in the image has the property that the product over all places gives $1$. - I think we have to be careful about the two senses of the word "residue" here. Either way my hope was that one could recover the residue as part of a more general computation. – Qiaochu Yuan Nov 4 '09 at 18:43 I'm rewriting the answer and making it better. I remember things one by one, but there should definitely be a good answer! – Ilya Nikokoshev Nov 4 '09 at 18:54 I take the product formula as an analog of the fact that a principal divisor on a curve has degree 0, so it is a (at least special case of the) residue theorem: the local integral at all points add up to the integral over the boundary, and there's no boundary if we also include \infty! – shenghao Nov 4 '09 at 19:59 That's absolutely what I mean, yes. – Ilya Nikokoshev Nov 4 '09 at 20:56	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9425798058509827, "perplexity": 354.3218542153966}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609526311.33/warc/CC-MAIN-20140416005206-00099-ip-10-147-4-33.ec2.internal.warc.gz"}
http://physics.stackexchange.com/questions/35443/why-should-space-be-empty?answertab=votes	# Why should space be empty? [duplicate] Possible Duplicate: How vacuous is intergalactic space? The emptiness of space is explained in many articles... But, space does contain some matter due to these possible reasons: • During the explosion of supernovae, some elements are created. Not all supernovae nucleosynthesis are successful. So, there should be some scattering of matter into outer space. • Cosmic rays are coming from interstellar space (and even from sun) in all directions. Would all of the particles have enough efficiency to reach their destination (I mean, take earth or any other interstellar object)?. Some protons or neutrons could be scattered into space. • Even the gaseous molecules in atmosphere of celestial bodies have possibility to reach escape velocity and go into space. Are my assumptions correct? If so, then space must contain elements up to some extent. Isn't it? Or is it due to the infiniteness of space, that these scattered particles are ignored? - ## marked as duplicate by David Z♦Sep 17 '12 at 17:52 Still mixing too many idea in each question. "there should be some wastage of matter." No, it's not wasted, it just remains as lighter bit. "Would all of the particles have enough efficiency to reach their destination" The don't have 'destinations', they just have momenta. Some will get 'somewhere' (by which you presumably mean somewhere interesting) others will never get to anywhere in particular, but those places will still be somewhere in a strict sense of the word. – dmckee Sep 5 '12 at 14:46	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8035472631454468, "perplexity": 1387.41992787392}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1429246651873.94/warc/CC-MAIN-20150417045731-00147-ip-10-235-10-82.ec2.internal.warc.gz"}
http://tex.stackexchange.com/questions/130354/latex-image-not-showing-up	# LaTeX - Image not showing up I am writing up some coursework using LaTeX, and I wanted to import a diagram, however when I did the image did not appear in the compiled pdf. The image has been exported as a eps file, and also as a pdf, both of which had the same result. (Although the PDF also required me to provide bounding box figures, as latex couldn't fetch them from the file). I am including the graphic like this: \documentclass[10pt]{article} \usepackage[dvips]{geometry,color,graphicx} \geometry{a4paper} \begin{document} \begin{figure}[ht!] \centering \includegraphics[width=300pt]{figure.eps} \caption{caption} \label{label} \end{figure} \end{document} - ## migrated from stackoverflow.comAug 27 '13 at 18:18 This question came from our site for professional and enthusiast programmers. Welcome to TeX.sx! Your post was migrated here from Stack Overflow. Please register on this site, too, and make sure that both accounts are associated with each other (by using the same OpenID), otherwise you won't be able to comment on or accept answers or edit your question. – Werner Aug 27 '13 at 18:20 Is figure.eps in the present working directory of the LaTeX file you are compiling? – dustin Aug 27 '13 at 18:20 Remove the extension (*.eps) and make sure it is in the same directory as the TeX file. – Dan Aug 27 '13 at 18:20 What options are you passing to the graphicx package? – Andrew Swann Aug 27 '13 at 18:21 delete [dvips] and .eps then your example should work with latex (using the eps version) or pdflatex (using the pdf version) If it doesn't work it is presumably a problem with the image, but as a test use \fbox{\includegraphics{figure}} so latex puts a visible box around the space where it thinks the image is, if that box is wrong, suspect the boundingbox in the eps file. – David Carlisle Aug 27 '13 at 18:59 Delete [dvips] and .eps then your example should work with latex (using the eps version) or pdflatex (using the pdf version) If it doesn't work it is presumably a problem with the image, but as a test use \fbox{\includegraphics{figure}} so latex puts a visible box around the space where it thinks the image is, if that box is wrong, suspect the boundingbox in the eps file. The dvips driver can include EPS files but not PDF and the pdftex engine used by pdflatex can include PDF files but not EPS. If you use [dvips] explicitly LaTeX will set things up for dvips so it will not work if you process the document with pdflatex. If you omit the option and the file extension, then a default option is taken depending on the engine used ([dvips] if you use latex and [pdftex] if you use pdflatex. In each case a default list of file extensions is defined, so if you do not put an explicit .eps extension in \includegraphics but have both .eps and .pdf available, latex will try figure.eps and pdflatex will try figure.pdf - As a bonus is it possible if you could explain why that works? – handuel Aug 27 '13 at 19:12 @handuel see update – David Carlisle Aug 27 '13 at 19:18 Thanks, that's perfect – handuel Aug 27 '13 at 19:20	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9733520150184631, "perplexity": 2409.708853771806}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-06/segments/1422115869647.75/warc/CC-MAIN-20150124161109-00245-ip-10-180-212-252.ec2.internal.warc.gz"}
https://scaron.info/robot-locomotion/zmp-support-area.html	# ZMP support area When a legged robot walks over a regular terrain, we can simplify its dynamics to a reduced model like the linear inverted pendulum where the center of mass (CoM) is controlled via the zero-tilting moment point (ZMP) of contact forces with the ground. To be physically feasible, the ZMP must lie in a support area, also known as support polygon, which is roughly "the area between the feet" in simple cases. In more complex cases, such as a hand on a wall or a knee on the ground, the area is a bit more complex but it is still computable. ## Modeling The constrained equations of motion describe the dynamics of our articulated system without simplification: \begin{equation} \begin{array}{rcl} \bfM(\bfq) \qdd + \qd^\top \bfC(\bfq) \qd & = & \bfS^\top \bftau + \bftau_g(\bfq) + \bftau_\mathit{ext} + \bfJ(\bfq)^\top \bff \\ \bfF(\bfq) \bff & \leq & \bfzero \end{array} \end{equation} The configuration $$\bfq$$ consists of its $$n$$ actuated degrees of freedom and 6 unactuated floating base coordinates. Contact forces from the environment are stacked in the vector $$\bff$$ and constrained by the matrix $$\bfF(\bfq)$$ of Coulomb friction cones. The interest in switching to a simplified model like the linear inverted pendulum (LIP) is to reduce the dimension of this equation. The constrained equations of motion for the LIP are line-by-line similar to those of the whole-body: \begin{equation} \begin{array}{rcl} \bfpdd_G & = & \omega^2 (\bfp_G - \bfp_Z) \\ \bfA(\bfp_G) \bfp_Z & \leq & \bfb(\bfp_G) \end{array} \end{equation} The configuration $$\bfp_G$$ consists of the the 2D horizontal position of the center of mass. Contact forces from the environment are aggregated in the position $$\bfp_Z$$ of the ZMP and constrained by the second line of half-space inequalities. The matrix $$\bfA(\bfp_G)$$ and vector $$\bfb(\bfp_G)$$ define the support area, which is a polygon as soon as we take into account actuation limits of the robot. They can be computed from the contact geometry and friction information encoded in $$\bfJ(\bfq)$$ and $$\bfF(\bfq)$$ by polyhedral projection. Further technical details on how to do this are given in Section IV of this paper. ## Simplification on flat floors In pre-2010 robotics papers, we often see the ZMP support area defined as the "convex hull of ground contact points", where $$\bfA$$ and $$\bfb$$ do not depend on $$\bfp_G$$ any more and can be readily computed from ground contact points by a convex hull algorithm (a particular case of polyhedral projection). This simplification is only valid if we make two assumptions: • All ground contact points are coplanar. • The coefficient of friction between robot feet and the ground is infinite. If we follow the general algorithm from the previous section, which takes the CoM position $$\bfp_G$$ as an input, it is not clear how the ZMP support area becomes independent from $$\bfp_G$$ under these two conditions. Let us detail this step by step. ### Infinite friction Let us consider for simplicity of calculations that our flat ground contact plane is horizontal (othogonal to gravity). In the absence of friction constaints, our friction cone inequalities are reduced to contact unilaterality constraints: \begin{equation} \forall \textrm{ contact } i, \ f_i^z > 0 \end{equation} This means we can exert arbitrary horizontal forces $$f_i^x$$ and $$f_i^y$$ at every contact point $$i$$. ### Newton-Euler equations Our contact forces are bound to the CoM and ZMP by Newton and Euler equations. First, Newton equation gives us: \begin{equation} \sum_{\textrm{contact } i} \bff_i = \bff = \frac{m g}{h} (\bfp_G - \bfp_Z) \end{equation} with $$m$$ the total mass, $$g$$ the gravity constant and $$h$$ the CoM height above ground. Second, Euler equation (with no angular momentum at the center of mass since we are in a pendular model) gives us: \begin{equation} \sum_{\textrm{ contact } i} \bfp_i \times \bff_i = \bfp_G \times \bff \end{equation} Let's unwrap it to understand better what it says. On flat floor, we have $$z_i = 0$$ for all contact points $$i$$, so that the vector cross products expand to: \begin{equation} \begin{array}{rcl} \sum_{i} y_i f_i^z & = & y_G f^z - h f^y \\ \sum_{i} -x_i f_i^z & = & -x_G f^z + h f^x \\ \sum_{i} (x_i f_i^y - y_i f_i^x) & = & x_G f^y - y_G f^x \end{array} \end{equation} Since we are not limited by friction, the third equation always has a solution and we can trim it out. In the other two equations, we can replace $$\bff$$ by its expression $$mg (\bfp_G - p_Z) / h$$ from the linear inverted pendulum model, for instance: \begin{equation} \begin{array}{rcl} y_G f^z & = & y_G m g (h - 0) / h = y_G m g \\ h f^y & = & m g (y_G - y_Z) \\ y_G f^z - h f^y & = & m g y_Z \\ mg y_Z & = & \sum_i {f_i^z} y_i \end{array} \end{equation} and similarly for the other equation. Defining $$\alpha_i = \frac{f_i^z}{mg}$$, we get: \begin{equation} \begin{array}{rcl} \bfp_Z & = & \sum_{i} \alpha_i \bfp_i \\ \forall i, \alpha_i & \geq & 0 \end{array} \end{equation} which is the vertex representation of the convex hull of ground contact points $$\bfp_i$$. ## To go further ZMP support areas can be derived in any multi-contact configurations, but in this case they depend (nonlinearly) on the position of the center of mass. Simplified ZMP support areas can be applied for walking over both horizontal and inclined. For instance, they have been used by biped robots to climb stairs and quadruped robots to trot over inclined surfaces. Joint torque limits can also be included to compute actuation-aware ZMP support polygons. ## Discussion You can use Markdown with $\LaTeX$ formulas in your comment.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.988291323184967, "perplexity": 1266.6816250203653}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030335059.31/warc/CC-MAIN-20220927225413-20220928015413-00442.warc.gz"}
https://neatpour.com/employer-address-ufkgmi/viewtopic.php?tag=b7ba9e-variational-principle-ansatz	# variational principle ansatz \| variational principle. Kiryl Pakrouski, Quantum 4, 315 (2020). : (x) = Ae x 2 parameter A = 4 r 2 ˇ from normalization condition (ii)calculate hHi= hTi+ hVi hTi= ~2 2m hVi= m!2 8 On how to solve these kind of integrals, see Ref. In the picture below, I've illustrated my point. 0521803918 - Variational Principles and Methods in Theoretical Physics and Chemistry Robert K. Nesbet Frontmatter More information. Published by IEE. In particular, we study the two matrix model with action tr [m2 2 (A2 1+A 2 2)− 1 4 [A1,A ]2] which has not been exactly solved. The first of these is the variational principle. 5 Variational Principles So far, we have discussed a variety of clever ways to solve differential equations, but have given less attention to where these differential equations come from. • Adapt — remix, transform, and build upon the material. In practice, the prepared quantum state is indirectly assessed by the value of the associated energy. Okay I think I've nailed the point into the floor enough by now. the variational parameters equal to zero. 51 Downloads; 8 Citations; Abstract. You are free to: • Share — copy or redistribute the material in any medium or format. A variational principle is developed for fractional kinetics based on the auxiliary-field formalism. The problem is that Variational methods certainly means the general methods of Calculus of variations.This article is just one example of these methods (perhaps not even the sole example even within quantum mechanics). Recently, the variational principle and associated Levy Ansatz have been proposed in order to obtain an analytic solution of the fractional Fokker-Planck equation. So a natural question to ask is, ‘‘what's our best guess for the free energy of the actual system’’? One of the central issues in the use of principal component analysis (PCA) for data modelling is that of choosing the appropriate number of retained components. A quick comment about notation: When we write it means that we're considering the average of some observable O in the trial ensemble; that is, it answers the question ‘‘what would the average of O be if the system were actually the trial Hamiltonian?’’ Operationally, is calculated using the probability weights of the trial Hamiltonian, by calculating, (Notice the subscript ‘‘tr’’ on the partition function and Hamiltonian here.). Variational principle for fractional kinetics and the Lévy Ansatz Sumiyoshi Abe Department of Physical Engineering, Mie University, Mie 514-8507, Japan Abstract A variational principle is developed for fractional kinetics based on the auxiliary-field formalism. Adaptive Derivative-Assembled Pseudo-Trotter ansatz Variational Quantum Eigensolver (ADAPT-VQE) ADAPT-VQE is an algorithm where the structure of the ansatz is determined in an adaptive manner. Applying the variational principle to (1+1) dimensional relativistic quantum ﬁeld theories Jutho Haegeman UGent, Department of Physics and Astronomy, Krijgslaan 281 S9, B-9000 Gent, Belgium E-mail: [email protected] J. Ignacio Cirac Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Str. The variational theorem states that for a Hermitian operator H with the smallest eigenvalue E0, any normalized jˆi satisﬂes E0 • hˆjHjˆi: Please prove this now without opening the text. Generalized variational mechanics began in the 1950s with the breakthrough works of Reissner [2] ontwo-ﬁeldvariationalprinciplesforelasticityproblems, in which the displacement u i and stress ˙ ij are consid-eredindependentﬁelds. The variational principle ensures that this expectation value is always greater than the smallest eigenvalue of $$H$$. We formulate an optimization problem of Hamiltonian design based on the variational principle. However, the study of dynamical properties therewithin resorts to an ansatz, whose validity has not yet been theoretically proven. %� And my best guess for is the one that makes as close to possible. In the last decade, physical and geometrical investigations about the relationship between horizon thermodynamics and gravitational dynamics suggest that gravity could be an emergent phenomenon. The strategy of the variational principle is to use a problem we can solve to approximate a problem we can't. The origin of the Hartree–Fock method dates back to the end of the 1920s, soon after the discovery of the Schrödinger equation in 1926. The suppression of nonphysical quasiparticle reflections from the boundary of the nonuniform region is … Variational quantum algorithm for nonequilibrium steady states Nobuyuki Yoshioka, Yuya O. Nakagawa, Kosuke Mitarai, and Keisuke Fujii Phys. 2(��^��4�q�� 4�/{�+�R�؞��=i�� Ԅ#�%7]�k꧃B,b��4��V/��N��,��6s��\|�BX��wI�U��(\�S�eϨ�w��}��:"M��M�Yoi��F�LBm(��E�s�L��zJ�(U'U��d��. Finally, minimize the variational free energy by setting its derivative w.r.t. It is applied to the Fokker-Planck equation with spatiotemporal fractionality, and a variational solution is obtained with the help of the Lévy Ansatz. Variational Principle Techniques and the Properties 117 While the total energy for the trial wave function in terms of the variational parameter α is ( ) φφ φ φ α H E ˆ = 2 2 1 4 3 2 3 α+ mωα− m h. (30) On minimizing E(α) with respect to α results 0 4 3 4 3 2 = 2 − 2 = α ω α m d m dE h or h The matrix product state ansatz works directly in the thermodynamic limit and allows for an efficient implementation (cubic scaling in the bond dimension) of the variational principle. Variational Principle Approach to General Relativity Chakkrit Kaeonikhom Submitted in partial fulﬂlment of the requirements for the award of the degree of Bachelor of Science in Physics B.S. The strategy of the variational principle is to use a problem we can solve to approximate a problem we can't.. More preciesly, suppose we want to solve a hard system with a Hamiltonian .Our plan of attack is to approximate it with a different ‘‘trial Hamiltonian’’ which has the same general ‘‘flavor’’ as the actual Hamiltonian, but (in contrast) is actually solvable. You could also call a different name such as a ‘‘variational ansatz’’ or a ‘‘guess of the solution shape’’ or even ‘‘a random shot in the dark.’’ The main point is that the the trial Hamiltonian should be a solvable problem that's similar to the actual problem at hand. The variational principle of quantum mechanics states that the average measured value of an observable with respect to a state is at least the observable operator’s minimum eigenvalue. In recent work, we have developed a variational principle for large N multi-matrix models based on the extremization of non-commutative en-tropy. 8 The Variational Principle 8.1 Approximate solution of the Schroedinger equation If we can’t ﬁnd an analytic solution to the Schroedinger equation, a trick known as the varia-tional principle allows us to estimate the energy of the ground state of a system. See Chapter 7 of the textbook. Variational Principles in Classical Mechanics by Douglas Cline is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License (CC BY-NC-SA 4.0), except where other-wise noted. Then we study the equivalence and di erence of the variational principles and the derived evolution equations in Sec.3. Here, we test the simplest variational ansatz for our entropic varia-tional principle with Monte-Carlo measurements. Hooray, we've learned the variational principle. ten Bosch, A.J. It is because variational principles have constantly produced more and more profound physical results, many of which underlie contemporary theoretical physics. The key new idea in his approach was the use … The least energy dissipation principle is well known in various linear systems such as viscous ﬂow in Newtonian ﬂuid, and electric current in ohmic devices. For instance, suppose we'd like to understand the general d … If an object is viewed in a plane mirror then we can trace a ray from the object to the eye, bouncing o the mirror. Here, we test the simplest variational ansatz for our entropic variational principle with Monte-Carlo measurements. 107 0 obj ), Connection to Quantum Mechanics and trial wavefunctions. Suppose we are given a Hilbert space and a Hermitian operator over it called the Hamiltonian H. Ignoring complications about continuous spectra, we look at the discrete spectrum of H and the corresponding eigenspaces of each eigenvalue λ (see spectral theorem for Hermitian operators for the mathematical background): This is achieved by confining the nonuniformity to a (dynamically expandable) finite region with fixed boundary conditions. Variational principles in fluid dynamics may be divided into two categories. It is applied to the Fokker-Planck equation with spatiotemporal fractionality, and a variational solution is obtained with the help of the Lévy Ansatz. Free Energy Principles And Variational Methods In Applied Mechanics 3rd Edition PDF Book Thanks for telling us about the problem. ;��y�"%��4�E�;.�H��Z�#�3QH��u�m�?��6�{]7%M�פw�{^�s�i�V6F)2;��DT9eJ@���j�" ��39n� ��_��2 _��E��.��3F��q��G�i ڟ�6��Н ��&��^s8�;5kÑF�v ~��H�>`��PL��5G%��M+ua�u��ŝ��n�ٿ��A�΁D�@!1 �鋢1v6t2�;�88�f��e�'�"�� S^\$��M�x� �� @7�_�Y�2��YL&��"�t��CC�~\|�A. Notice that no matter what I choose for the parameter, the variational free energy is always bigger than the actual free energy . A. Variational Principles For the purposes of this paper, let us deﬁne a state selective variational principle as a smooth function of a wave function ansatz’s variables with the following property: if the ansatz is capable of exactly describing the individual Hamiltonian eigenstate of interest, Like Hartree-Fock, our approach is deterministic, state-specific, applies a variational principle to a minimally correlated ansatz, produces energy stationary points, relaxes the orbital basis, has a Fock-build cost-scaling, and can serve as the foundation for correlation methods such as perturbation theory and coupled cluster theory. Page generated 2020-09-20 15:48:00 PDT, by. No matter how good a guess your variational free energy is, it will always be greater than or equal to the actual free energy ; that is. This class of ansätze is inspired by the theory of quantum optimal control and leads to an improved convergence of VQAs for some important problems such as the Fermi-Hubbard model at half-filling, and show that our variational circuits can approximate the ground state of this model with significantly higher accuracy and for larger systems. Download BibTex. Given a Hamiltonian the method consists Variational principles have always played an impor-tantroleinboththeoreticalandcomputationalmechan-ics [1–33]. The Variational Principle. Variational quantum eigensolver with fewer qubits ... one can exponentially increase the bond dimension of the tensor network variational ansatz on a quantum computer. Bronsted and Rockafellar [6] h ave used it to obtain subdifferentiability properties for convex functions on Banach spaces, and Browder [7] has applied it to nonconvex subsets of Banach spaces. One of the key points today is that interacting systems are very difficult to solve in general. Variational calculations for Hydrogen and Helium Recall the variational principle. A variational ansatz for momentum eigenstates of translation-invariant quantum spin chains is formulated. Iterate until convergence. Variational method → Variational method (quantum mechanics) – I think that the move in 2009 was, unfortunately, a clear mistake. << /Filter /FlateDecode /Length 2300 >> We now move to more physical statements about the behavior of the solutions of the TISE. 1 Introduction. Variational Principal Components Christopher M. Bishop Microsoft Research 7 J. J. Thomson Avenue, Cambridge, CB3 0FB, U.K. [email protected] Cite Icon Cite. This bound allows us to use classical computation to run an optimization loop to find this eigenvalue: Use a classical non-linear optimizer to minimize the expectation value by varying ansatz parameters $$\vec{\theta}$$. I'm not sure if I'll get around to finishing up the rest of this page…for now just go on to the next page about non-interacting spins. (PDF) Variational ansatz-based quantum simulation of imaginary … Singlet Unitary Coupled Cluster Ansatz 이권학, 이준구 한국과학기술원 전기 및 전자공학부 [email protected], [email protected] Singlet Unitary Coupled Cluster Ansatz for Quantum Chemistry Simulation Using Variational Method Gwonhak Lee, June-Koo Kevin Rhee School of Electrical Engineering, KAIST 요 약 stream Next, calculate the variational free energy . The best variational solution we can find is the one that gets as close as possible to the actual Hamiltonian. The matrix product state ansatz works directly in the thermodynamic limit and allows for an efficient implementation (cubic scaling in the bond dimension) of the variational principle. There was a funny look on his face, like, ‘‘Oh, you're expecting me to teach you something?’’ Well, yes, we would like you to teach us some statistical mechanics! [5]. 3. The rst variational principle was formulated about 2000 years ago, by Hero of Alexandria. Practically speaking, our strategy is to start with a whole family of possible trial Hamiltonians, and then just pick the one whose variational free energy is the smallest. We summarise the results in Table1under various conditions. The variational principle is a useful tool to have in our pocket because it lets us leverage the Hamiltonians which we actually can solve to solve Hamiltonians which we can't. Douglas Hartree's methods were guided by some earlier, semi-empirical methods of the early 1920s (by E. Fues, R. B. Lindsay, and himself) set in the old quantum theory of Bohr. Our approach combines the P representation of the density matrix and the variational principle for open quantum system. The rst variational principle was formulated about 2000 years ago, by Hero of Alexandria. The Ritz method is a direct method to find an approximate solution for boundary value problems.The method is named after Walther Ritz, although also commonly called the Rayleigh-Ritz method.. (I've left out the parameter for simplicity). For instance, our family of trial Hamiltonians might be all possible 2D Ising models. Our approach combines the P representation of the density matrix and the variational principle for open quantum system. The steady-state density matrix of the lattice system is constructed via a purified neural-network Ansatz in an extended Hilbert space with ancillary degrees of freedom. h jO^j i h j i 1 (1) With j inormalized, the equation simpli es to h jO^j i 1 (2) 2. Comparison of Unitary Coupled Cluster Ansatz Methods for the Variational Quantum Eigensolver Ethan Hickman ([email protected]), Aaron M. Roth, Yingyue Zhu University of Maryland CMSC 657 December 12, 2019 Abstract The variational quantum eigensolver (VQE) is a hybrid quantum-classical algorithm Variational principles and generalized variational principles for nonlinear elasticity with finite displacement. Often this is based on a similar problem that has an exact solution. We describe how to implement the time-dependent variational principle for matrix product states in the thermodynamic limit for nonuniform lattice systems. The variational principle Theory Example: One-dimensional harmonic oscilator How to do this using the variational principle... (i)pick a trial function which somehow resembles the exact ground state w.f. the variational principle is an extension of Rayleigh’s principle of the least energy dissipation [7]. Christopher Bishop; Proceedings Ninth International Conference on Artificial Neural Networks, ICANN'99 \| January 1999. Variational Principles and Lagrangian Mechanics Physics 3550, Fall 2012 Variational Principles and Lagrangian Mechanics Relevant Sections in Text: Chapters 6 and 7 The Lagrangian formulation of Mechanics { motivation Some 100 years after Newton devised classical mechanics Lagrange gave a di erent, considerably more general way to view dynamics. Variational Methods of Approximation The concept behind the Variational method of approximating solutions to the Schrodinger Equation is based on: a) An educated guess as to the functional form of the wave function. In general, a parameterized ansatz wavefunction will be in a superposition of eigenstates of the Hamiltonian. Authors; Authors and affiliations; Chien Wei-zang; Article. conditions (a) = (b) = 0: How do they look like for the rigid body equation? Additional examples and problems can be found in the following books of the author: 1. We have a lot of choices; picking and gives us one possible trial Hamiltonian; picking and gives us another possibility, etc., and the variational principle tells us that our best guess for and is the choice that minimizes . Reddy J. (Physics) Fundamental Physics & Cosmology Research Unit The Tah Poe Academia Institute for Theoretical Physics & Cosmology Department of Physics, Faculty of Science Naresuan University March 15, 2006. variational principles: as the approximate ansatz becomes more and more ﬂexible, we are guaranteed to recover the exact eigenstate eventually. 1, Garching, D-85748, Germany See Chapter 16 of the textbook. The key concepts of the algorithm are demonstrated for the nonlinear Schr\"odinger equation as a canonical example. Novel adaptive derivative-assembled pseudo-trotter (ADAPT) ansatz approaches and recent formal … where we can pick the parameters and that enter into the Hamiltonian. Here I've plotted how depends on the parameter in the trial Hamiltonian. A. Variational Principles For the purposes of this paper, let us deﬁne a state selective variational principle as a smooth function of a wave function ansatz’s variables with the following property: if the ansatz is capable of exactly describing the individual Hamiltonian eigenstate of interest, ON THE VARIATIONAL PRINCIPLE 325 The proof of this theorem is based on a device due to Bishop and Phelps [4]. … Variational calculation for Helium Recall the variational principle. We present a method to perform a variational analysis of the quantum master equation for driven-disspative bosonic fields with arbitrary large occupation numbers. (Refer Section 3 - Applications of the Variational Principle). Reduced variational principles: Euler-Poincar eIII Theorem (Poincar e(1901-02): Geometric Mechanics is born) Hamilton’s principle for rigid body action S = R t 1 t0 L(R;R_ )dt = 0 is equivalent to Sred = Z t 1 t0 l()dt = 0; with 2R3 and for variations of the form = _ + ; and bdry. If an object is viewed in a plane mirror then we can trace a ray from the object to the eye, bouncing o the mirror. Moreover, we construct circuits blocks which respect U(1) and SU(2) symmetries of the physical system and show that they can signiﬁcantly speed up the training process and alleviate the gradient vanishing problem. The variational principle allows us to reframe an unknown problem in terms of a known problem; it tells us how we can ‘‘guess’’ the closest possible answer in terms of a ‘‘trial’’ solution. When Prof. Kivelson walked into class today, he looked a bit taken by surprise. Please refer the reference for details. A variational ansatz for momentum eigenstates of translation-invariant quantum spin chains is formulated. (I don't even know if I'll get around to writing the rest of the sections…I have a life too, you know! variational principles and extending the principles to the general stochastic evolution of mixed states. Abstract. There's a whole bunch of different 's that we can pick, and our best choice is because it minimizes . Rev. Given a variational ansatz for a Hamiltonian we construct a loss function to be minimised as a… Variational Principle for the Many Body Density Matrix. Honestly, it's much more important to understand the logic behind a variational argument than to know how to prove it…so of all the sections on this page, the Motivation and Overview section is most important. By design, the variational quantum eigensolver (VQE) strives to recover the lowest-energy eigenvalue of a given Hamiltonian by preparing quantum states guided by the variational principle. The ambition of this book is to describe some of their physical applications. Variational Principle. It is applied to the Fokker-Planck equation with spatio-temporal fractionality, and a variational solution is obtained with the help of the L\'evy Ansatz. Thepreviousliterature,howev-er,consideredonlydis Among the others, Padmanabhan’s theory of “emergent gravity” focus on the concept of spacetime as an effective macroscopic description of a more fundamental microscopic theory … The recent proof by Guerra that the Parisi ansatz provides a lower bound on the free energy of the Sherrington-Kirkpatrick (SK) spin-glass model could have been taken as offering some support to the validity of the purported solution. Let $$\psi$$ be a properly normalized trial solution to the previous equation. The McLachlan variational principle for real-time dynamics governs the equations of motion of variational parameters, where the variational ansatz is automatically generated and dynamically expanded along the time-evolution path, such that the \McLachlan distance", which is a measure of the simulation accuracy, remains below a xed Mechanics.In this study project, the Variational Principle has been applied to several scenarios, with the aim being to obtain an upper bound on the ground state energies of several quantum systems, for some of which, the Schrodinger equation cannot be easily solved. The variational principle ensures that this expectation value is always greater than the smallest eigenvalue of $$H$$. This bound allows us to use classical computation to run an optimization loop to find this eigenvalue: Use a classical non-linear optimizer to minimize the expectation value by varying ansatz parameters $$\vec{\theta}$$. Review of Equations of Solid Mechanics 47 2. A variational principle is developed for fractional kinetics based on the auxiliary-field formalism. And this is precisely the focal point where variational QMC and deep learning meet—the former provides the loss function in the form of the variational principle, while the latter supplies a powerful wave function ansatz in the form of a deep neural network. Share	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8597283363342285, "perplexity": 764.3839538816006}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243991269.57/warc/CC-MAIN-20210516105746-20210516135746-00479.warc.gz"}
https://infoscience.epfl.ch/record/55576	Infoscience Journal article # Assessment of climate change impacts on Alpine discharge regimes with climate model uncertainty The present study analyzes the uncertainty induced by the use of different state-of-the-art climate models on the prediction of climate change impacts on the runoff regimes of 11 mountainous catchments in the Swiss Alps having current glaciation rates between 0 and 50 %. The analyzed climate change scenarios are the result of 19 Regional Climate Model (RCM) runs obtained for the period 2070 – 2099 based on two different green house gas emission scenarios (the A2 and B2 scenarios defined by the Intergovernmental Panel on Climate Change) and on three different coupled Atmosphere-Ocean General Circulation Models (AOGCMs) called HadCM3, ECHAM4/OPYC3 and ARPEGE/OPA. The hydrological response of the studied catchments to the climate scenarios is simulated through a conceptual reservoir-based precipitation-runoff transformation model called GSM-SOCONT. For the glacierized catchments, the glacier surface corresponding to these future scenarios is updated through a conceptual glacier surface evolution model. The obtained results show that all climate change scenarios induce in all catchments an earlier start of the snowmelt period leading to a shift of the hydrological regimes and of the maximum monthly discharges. The mean annual runoff undergoes in most cases a significant decrease. For the glacierized catchments, the simulated regime modifications are mainly due to the increase of the mean temperature and corresponding impacts on the snow accumulation and melting processes. The hydrological regime of the catchments located at lower altitudes is more strongly affected by the changes of the seasonal precipitation. For a given emission scenario, the simulated regime modifications of all catchments are highly variable for the different RCM runs. This variability is induced by the driving AOGCM but also to a large part by the inter-RCM variability. The differences between the different RCM runs is so important that the predicted climate change impacts for the two emission scenarios A2 and B2 are overlapping.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8349596858024597, "perplexity": 2605.235335692158}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-47/segments/1510934805914.1/warc/CC-MAIN-20171120052322-20171120072322-00299.warc.gz"}
http://www.chegg.com/homework-help/questions-and-answers/300-g-block-dropped-onto-relaxed-vertical-spring-aspring-constant-k-27-n-cm-block-becomes--q138466	"A 300 g block is dropped onto a relaxed vertical spring that has aspring constant of k= 2.7 N/cm. The block becomes attached tothe spring and compresses the spring 12 cm before momentarilystopping. (a) While the spring is being compressed, what work is done onthe block by the gravitational force on it? I used Wg=mgd=.3528J (b) What work is done on the block by the spring force whilethe spring is being compressed? Wspring = 1/2k(xinitial^2-xfinal^2) .0135 N/m (0-1.44m) -1.944J (c) What is the speed of the block just before it hitsthe spring? (Assume that friction is negligible.) Wspring=-1/2kd^2, Kfinal - Kinitial = Wspring,K=1/2mv^2. so Kfinal - Kinitial = -1/2kd^2 0-1/2mv^2=-1/2kd^2 v=sqrt((k*d^2)/m) v=.36 m/s (d) If the speed at impact is doubled, what is themaximum compression of the spring? Using the previous equation solved ford. d=.076 m. So far I've gotten the first two right and the second twowrong. Can anybody help? Edit: I always forget that I can find answerboard linksto even problems in the Homework help section. /physics-topic-5-72900-0.aspx This helped me get the third one right (3.257m/s) and now I just need to work a little more on the last one.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9094159603118896, "perplexity": 1773.7880390172334}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1429246639325.91/warc/CC-MAIN-20150417045719-00106-ip-10-235-10-82.ec2.internal.warc.gz"}
http://relativity.livingreviews.org/Example/LivRevRelTemplate/LivRevRelse7.html	7 Figure Environments 7.1 Supported Environments To display PostScript figures in the document, we recommend the graphicx or epsf packages. They are activated by adding the lines \usepackage{epsf} \usepackage{graphicx} in the header of the article. These packages can be used to display both PostScript and Encapsulated PostScript figures. We discourage authors from using the psfig package, as this may cause converter problems. If you are using the graphicx package, you might also submit other image formats (preferably PDF). 7.2 Images A straightforward way to include an image is demonstrated here: Figure 1 is generated using the command sequence in exImage.tex. exImage.tex Here figure.eps (see Line 3 of exImage.tex) is the name of the PostScript file. In Line 3 one can also set the scaling factor (here 0.8, which means scaling to 80% of the original size). The option [h] in Line 2 instructs LATEX to position the figure right at the place where the figure environment stands in the source with reference to the surrounding text paragraphs. Other options are [t] or [b] which position the figure at the top or bottom of the page, resp. Please add a caption to every figure. Labels for referencing the figure are also obligatory. Recently, we added support for the graphics package. This allows you to insert a figure using the well-known includegraphics command. You need to insert the usepackage{graphicx} after the usepackage{epubtk}. Figure 2 is generated using the command sequence in exImage2.tex. exImage2.tex Note that you should not provide a suffix for the figure in the \includegraphics command. You need to provide proper versions of the graphics, i.e. a rasterized version as the first parameter of epubtkImage and a pdf, png or jpg for inclusion in the pdf. Furthermore, it is possible to build a figure out of several independant subfigures without first combining them to a single figure: The source code for this example is available in exImage3.tex: exImage3.tex	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9490883350372314, "perplexity": 1610.176527400669}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-42/segments/1413507443869.1/warc/CC-MAIN-20141017005723-00139-ip-10-16-133-185.ec2.internal.warc.gz"}
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http://export.arxiv.org/abs/1905.06051	math.PR (what is this?) # Title: Transfer of regularity for Markov semigroups Abstract: We study the regularity of a Markov semigroup $(P_t)_{t>0}$, that is, when $P_t(x,dy)=p_t(x,y)dy$ for a suitable smooth function $p_t(x,y)$. This is done by transferring the regularity from an approximating Markov semigroup sequence $(P^n_t)_{t>0}$, $n\in\mathbb{N}$, whose associated densities $p^n_t(x,y)$ are smooth and can blow up as $n\to\infty$. We use an interpolation type result and we show that if there exists a good equilibrium between the blow up and the speed of convergence, then $P_{t}(x,dy)=p_{t}(x,y)dy$ and $p_{t}$ has some regularity properties. Subjects: Probability (math.PR) Cite as: arXiv:1905.06051 [math.PR] (or arXiv:1905.06051v1 [math.PR] for this version) ## Submission history From: Lucia Caramellino [view email] [v1] Wed, 15 May 2019 09:37:17 GMT (32kb) Link back to: arXiv, form interface, contact.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8140192627906799, "perplexity": 1237.6106181079713}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579251799918.97/warc/CC-MAIN-20200129133601-20200129163601-00480.warc.gz"}
https://www.techtud.com/chapter/12161/12172/23668	GATE 2006 question on Group The set {1,2,3,5,7,8,9} under multiplication modulo 10 is not a group. Given below are four possible reasons. Which one of them is false? (A) It is not closed (B) 2 does not have an inverse (C) 3 does not have an inverse (D) 8 does not have an inverse Things you need to know Now, Let S = {1,2,3,5,7,8,9} (A) is True. Because, (87)%10 = 6 ∉ S (B) is True. Because, inverse 'e' should exit such that (2e) % 10 = 1, but such 'e' is not in S. (C) is False. Because, (37)% 10 = 1, Hence, inverse of 3 = 7 (D) is True. Because, inverse 'e' should exit such that (8e) % 10 = 1, but such 'e' is not in S. Ans. (C)	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8363452553749084, "perplexity": 2972.5141583894642}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655878639.9/warc/CC-MAIN-20200702080623-20200702110623-00015.warc.gz"}
https://www.aanda.org/articles/aa/full_html/2010/05/aa13042-09/aa13042-09.html	Subscriber Authentication Point Free Access Issue A&A Volume 513, April 2010 A20 12 Extragalactic astronomy https://doi.org/10.1051/0004-6361/200913042 16 April 2010 A&A 513, A20 (2010) ## The unusual N IV]-emitter galaxy GDS J033218.92-275302.7: star formation or AGN-driven winds from a massive galaxy at z = 5.56 E. Vanzella1 - A. Grazian2 - M. Hayes3 - L. Pentericci2 - D. Schaerer3,4 - M. Dickinson6 - S. Cristiani1 - M. Giavalisco7 - A. Verhamme5 - M. Nonino1 - P. Rosati8 1 - INAF - Osservatorio Astronomico di Trieste, via G.B. Tiepolo 11, 40131 Trieste, Italy 2 - INAF - Osservatorio Astronomico di Roma, via Frascati 33, 00040 Monteporzio Roma, Italy 3 - Geneva Observatory, University of Geneva, 51 Ch. des Maillettes, 1290 Versoix, Switzerland 4 - Laboratoire d'Astrophysique de Toulouse-Tarbes, Université de Toulouse, CNRS, 14 avenue E. Belin, 31400 Toulouse, France 5 - Department of Physics, University of Oxford. Denys Wilkinson Building, Keble Road, Oxford, UK 6 - NOAO, PO Box 26732, Tucson, AZ 85726, USA 7 - Astronomy Department, University of Massachusetts, Amherst MA 01003, USA 8 - European Southern Observatory, Karl-Schwarzschild-Strasse 2, 85748 Garching, Germany Received 31 July 2009 / Accepted 15 December 2009 Abstract Aims. We investigate the nature of the source GDS J033218.92-275302.7 at redshift 5.56. Methods. The spectral energy distribution of the source is well-sampled by 16 bands photometry from UV-optical (HST and VLT), near infrared, near infrared (VLT) to mid-infrared (Spitzer). The detection of a signal in the mid-infrared Spitzer/IRAC bands 5.8, 8.0 - where the nebular emission contribution is less effective - suggests that there is a Balmer break, the signature of an underlying stellar population formed at earlier epochs. The high-quality VLT/FORS2 spectrum shows a clear Ly emission line, together with semi-forbidden N IV] 1483.3-1486.5 also in emission. These lines imply a young stellar population. In particular, the N IV] 1483.3-1486.5 feature (if the source is not hosting an AGN) is a signature of massive and hot stars with an associated nebular emission. Conversely, it may be a signature of an AGN. The observed SED and the Ly emission line profile were modeled carefully to investigate the internal properties of the source. Results. From the SED-fitting with a single and a double stellar population and from the Ly modeling, it turns out that the source seems to have an evolved component with a stellar mass of 1010 and age 0.4 Gyr, and a young component with an age of 0.01 Gyr and star formation rate in the range of 30-200 . The limits on the effective radius derived from the ACS/z850 and VLT/Ks bands indicate that this galaxy is denser than the local ones with similar mass. A relatively high nebular gas column density is favored from the Ly line modeling ( 1021 cm-2). A vigorous outflow ( 450 km s-1) has been measured from the optical spectrum, consistent with the Ly modeling. From ACS observations it turns out that the region emitting Ly photons is spatially compact and has a similar effective radius (0.1 kpc physical) estimated at the 1400 Å rest-frame wavelength, whose emission is dominated by the stellar continuum and/or AGN. The gas is blown out from the central region, but, given the mass of the galaxy, it is uncertain whether it will pollute the IGM to large distances. We argue that a burst of star formation in a dense gas environment is active (possibly containing hot and massive stars and/or a low luminosity AGN), superimposed on an already formed fraction of stellar mass. Key words: galaxies: formation - galaxies: evolution ## 1 Introduction In the past few years, dedicated space-borne and ground-based observatories and refined techniques have allowed us to discover and analyze galaxies at increasingly large distances. It is common practice in observational cosmology to select efficiently star-forming galaxies (e.g. Lyman break galaxies, LBGs, or Lyman alpha emitters, LAEs) and active galactic nuclei (AGN) up to redshift 6.5 (e.g. Taniguchi et al. 2005; Steidel et al. 1999; Dickinson et al. 2004; Giavalisco et al. 2004a; Ando et al. 2006; Cristiani et al. 2004; Fan et al. 2003; Fontanot et al. 2007; Bouwens et al. 2006) or spheroidal/fossil massive galaxies up to redshift 4, whose apparent morphology has recently introduced a sub class of ultradense objects (e.g. Buitrago et al. 2008; Daddi et al. 2005; Cimatti et al. 2008; van Dokkum et al. 2008). High-redshift galaxies have been studied through deep multi-wavelength surveys, with the aim of maximizing the information on the energetic output of the sources spanning a wide range of the electromagnetic spectrum from X-ray to radio wavelengths. This approach has shown its efficiency of constraining luminosity functions up to redshift 6-7 and down to few percent of L* (e.g. Bouwens et al. 2007,2008), stellar mass functions (for masses higher than ) up to redshift 6 (Stark et al. 2009; Fontana et al. 2006; Eyles et al. 2005; Yan et al. 2005; Eyles et al. 2007; Fontana et al. 2009; Stark et al. 2007; Labbé et al. 2006) and morphology evolution (Taniguchi et al. 2009; Ferguson et al. 2004; Conselice et al. 2008). Detailed studies of internal the properties of high-redshift galaxies, such as information about hot stars, dust, ionized gas in regions, and the large-scale outflows of neutral and ionized interstellar material (ISM) are now becoming feasible up to redshift 4-5 (e.g. Ando et al. 2007; Vanzella et al. 2009; Ouchi et al. 2008; Shapley et al. 2003). Useful information from the Ly profile modeling of high-redshift galaxies with radiative transfer codes is producing interesting constraints on the dynamical and physical state of the ISM and ionizing sources (e.g. Verhamme et al. 2008; Schaerer & Verhamme 2008). From this point of view, thanks to the combination of depth, area, and multivawelength coverage, the Great Observatories Origins Deep Survey project (see Dickinson et al. 2003b; Giavalisco et al. 2004b, for a review about this project) is ideal for studing galaxies at high redshift and the connection between photometric, spectroscopic, and morphological - size properties (e.g. Ravindranath et al. 2006; Pentericci et al. 2007; Vanzella et al. 2009; Conselice et al. 2008; Pentericci et al. 2009), and their relation with the environment (e.g. Elbaz et al. 2007). Along with enabling a systematic study of normal galaxies, multi-frequency surveys over large areas and depth also allow us to discover rare objects. In particular, a new class of objects showing prominent N IV] 1486 emission have recently been reported. Such a feature is rarely seen at any redshift. A small fraction (1.1%, 1.7 < z < 4) of the QSO sample extracted from the SDSS fifth data release is nitrogen rich, showing N IV] 1486 or N III] 1750 emission lines and N V] 1240 also in emission (typically stronger than the rest of the population (Jiang et al. 2008). Similarly, Glikman et al. (2007) discuss the discovery of two low-luminosity quasars at redshift 4 with Ly and C IV lines, moderately broad N IV] 1486 emission, and an absent N V] 1240 line. In these particular cases, the blinding intensity of the central engine is reduced, allowing study of the properties of the host galaxy. Fosbury et al. (2003) report on an lensed galaxy at redshift 3.357 (the Lynx arc) whose spectrum shows N IV] 1486, O III] 1661, 1666, C III] 1907, 1909, as well as the absence of the N V] 1240 line. Their modeling of the spectrum favors a hot (T 80 000 K) blackbody over an AGN as the ionizing source. Alternatively, Binette et al. (2003) suggest an obsured AGN as a photoionizing source of the Lynx arc. Villar-Martin et al. (2004) propose a population of Wolf-Rayet (WR) stars as the ionization source for the same galaxy, with an age below 5 Myr that contributes to a fast enrichment of the interstellar medium. In this scenario the stars involved are much colder than those proposed in Fosbury et al. (2003). In the present work we report on the source GDS J033218.92-275302.7 at redshift 5.563 located within the GOODS southern field, for which extensive information (photometry, spectroscopy and morphology) is available. The galaxy has been discovered during the ESO/FORS2 spectroscopic survey (Vanzella et al. 2006). We focus our attention on this source because it shows several unique characteristics. First the high S/N spectrum exibits a relatively bright N IV] 1486 feature in emission, a unique example among the more than 100 spectra of high z starburst galaxies that were obtained from the GOODS/FORS2 campaign (Vanzella et al. 2008). Second, while the spectrum shows a bright Ly line indicating a young stellar component, the photometry shows a prominent Balmer break indicating that there is also an evolved component. Finally, the bright IRAC flux suggests that this is a massive galaxy, especially interesting if one considers its very high redshift. Indeed the object was already noted by several authors; for example, Fontanot et al. (2007) selected it as an AGN candidate on the basis of morphological and color considerations, but discarded it from the sample because of its peculiar galaxy-like optical spectrum. Wiklind et al. (2008) report it among the 11 candidates with photometric redshifts in the range 4.9 < z < 6.5, dominated by an old stellar population, with ages 0.2-1.0 Gyr and having very high stellar masses, in the range (0.5-5) . Also Stark et al. (2007) report for this source a stellar mass highr than . Similarly Pentericci et al. (2009) note that this is one of a handful of bright Ly emitting LBGs at high redshift with an evolved population, indicating that not all Lyman alpha emitters are young primeval galaxies. As noted recently by Schaerer & de Barros (2009) and Raiter et al. (2010), strong nebular emission lines may bias the result of the SED-fitting of high-redshift galaxies (e.g. [O III] 4959-5007, H). Indeed the apparent photometric breaks might actually be produced in some cases by the boost of some of the lines. Something similar may be happening in this case, therefore it is important to quantify the strength of these lines and their influence on global photometry. In the present work we perform dedicated SED-fitting allowing single and multiple stellar populations, and important information is extracted from the Ly profile modeling. Together with the morphological appearance, constraints have been placed on the stellar mass density, ages, gas, dust content, and outflows. The work is structured as follow. In Sect. 2 a summary of the photometric, spectroscopic, and morphological properties is given, and in Sect. 3 the possible scenarios are discussed about the nature of the source. Section 4 describes the SED and Ly modeling, and in Sect. 5 we discuss the results. Section 6 concludes the work. In the following the standard cosmology is adopted (H0 = 70 km s-1 Mpc-1, = 0.3, = 0.7). If not specified, magnitudes are given on the AB system. ## 2 Source GDS J033218.92-275302.7 Source GDS J033218.92-275302.7 is located within the southern field of the Great Observatories Origins Deep Survey. The multi-wavelength observations consists of deep U, R (VLT), B435, V606, i775 and z850 (HST), Js, H, Ks (VLT), 3.6, 4.5, 5.8, 8.0, and 24 (Spitzer) bands. Moreover observations in the X-ray and radio domains are available from Chandra and the Very Large Array, respectively (Miller et al. 2008; Luo et al. 2008). A considerable part of the spectroscopic information of that field has been collected by the VLT/FORS2 spectrograph, which has produced about one thousand redshift determinations (with a resolution of 13 Å, at 8600 Å), between redshift 0.5 and 6.2, in particular more than one hundred LBGs have been confirmed at redshift beyond 3.5 (Vanzella et al. 2008,2006,2005). The VLT/FORS2 spectroscopic survey has been complemented in the redshift interval 1.6 < z < 3.5 and at z < 1 by the VLT/VIMOS spectroscopic survey, which is producing more than 5000 spectroscopic identifications (Balestra et al. 2010; Popesso et al. 2008). Such measurements increase the spectroscopic information available from previous works (e.g. VVDS Szokoly et al. 2004; Le Fèvre et al. 2005). Source GDS J033218.92-275302.7 with z850 = 24.61 0.03 was selected as a V606-band dropout and was confirmed to be at redshift 5.563 (redshift of the Ly line, Vanzella et al. 2006). ### 2.1 UV spectral properties Figure 1: 1-dimensional spectrum of the galaxy discussed in the present work. The Ly and N IV] 1486 emission lines are evident, together with the detection of the continuum and the IGM attenuation. The inner box shows a zoom of the Ly line, the asymmetry and the red tail are visible (the latter is marked with a solid segment, 40 Å long, or 1500 ). The transmissions of the ACS V606, i775, and z850 are also shown. The exposure time was 14 400 s. Open with DEXTER The main spectral features are the Ly emission line (EW 60 Å rest frame), the break of the continuum blueward of the line, and the semi-forbidden emission line N IV] 1486, a doublet 1483.3-1486.5 Å (see Fig. 1 and Table 1 for a physical quantities summary). The detection of the N IV] 1486 in emission is unusual for LBGs. However, this atomic transition has been identified by Fosbury et al. (2003) in the Lynx arc and in a sub class of QSOs (e.g., Glikman et al. 2007; Baldwin et al. 2003). In the following the main properties of the UV spectrum are described: 1. As shown in the top panel of Fig. 2 an emission line at 9742 Å is detected. The spectral resolution is in principle sufficient to resolve the double profile of the two N IV] 1486 components, i.e. 1483.3 Å and 1486.5 Å. We interpret this feature as the detection of one of the two components. A first possibility is that this line is the 1483.3 Å component: in this case the redshift turns out to be higher than the observed Ly redshift. The Ly emission from LBGs is commonly observed to be redshifted relative to the systemic velocity traced by other, non-resonant emission lines or stellar absorption lines (Shapley et al. 2003; Tapken et al. 2007; Verhamme et al. 2008; Vanzella et al. 2009). Therefore, it would be unusual if, in this object, the N IV] redshift were higher than that from Ly. The other possibility is that we are detecting the 1486.5 Å component at redshift 5.553, suggesting a high-density limit and a velocity offset (i.e. a presence of an outflow) between Ly and N IV] lines of +457 km s-1 (dz = 0.01). A further proof of this possibility is that the redshift and outflow estimated from the spectrum are consistent with the results from the Ly profile modeling discussed below (see Sect. 4.2). Therefore, in the following we assume the line to be the 1486.5 Å component. Table 1: Summary of the physical quantities derived from the spectral features and morphological analysis. 2. There is no detection for the SiIV 1393.8-1402.8 doublet, either in emission or absorption, and similarly for the N V 1240-1243 doublet; their luminosity limits are reported in Table 1 (see also Fig. 2). 3. From the 2-dimensional spectrum, the FWHM of the spatial profiles of the Ly and the continuum (by collapsing columns along the Y-axis, see Fig. 3) are fully comparable, 0.7 arcsec. This is also compatible with the seeing during observations, 0.7 arcsec. More interestingly, a better constraint on the Ly extension comes from the ACS i775 band. As shown in Fig. 1, the i775 band is mainly probing the UV emission region between the Ly line and 1300 Å (less than 100 Å rest-frame), and the part blueward of the Ly line is strongly attenuated by the IGM absorption. Since the Ly equivalent width is 60 Å, it turns out that the ACS i775 image is dominated by the Ly emission. In Sect. 2.3 we show that the morphological properties derived from the i775 band (probing the Ly line) and the z850 band (not containing the Ly line and probing the emission at 1400 Å) are similar, e.g. the effective radii of the two are the same order. This implies that the spatial extension of the Ly line is similar to the emitting region at 1400 Å rest-frame. Figure 3 shows the contour plots of the 2-dimensional Ly region. Various sigmas above the noise fluctuation are reported from 1 to 20. It is evident that the asymmetric shape in the wavelength domain (that extends at least 40 Å (3)). If the Ly emission arises from a simple expanding shell of material, then it is expected to have less of velocity width at its outer extremes than along the line of sight to the central region, where the transverse velocity component tends to zero. Figure 2:Three zoomed regions of the 1D spectrum (red dotted line is the solid black line smoothed over two pixels). Dotted plots show the rescaled sky spectrum. Filled circles mark the position of the lines at the redshift 5.553 (case in which only the N IV] 1486.5 component is detected). Dotted open circles denote the other (but less probable) case in which the N IV] 1483.3 component is detected (z=5.568). Red filled triangles mark the positions at the observed Ly redshift, z=5.563. Top: the positions of the N IV] 1486 and C IV] features are shown. In the sky spectrum two sky emission lines are marked: (1) 10082.46 Å and (2) 10124.01 Å (also shown in the 2-dimensional spectrum of Fig. 8). Middle: the region where N V is expected to lie is shown. The open blue triangle indicates the estimated intrinsic Ly redshift from the Ly modeling. Bottom: the region around the SiIV feature is shown. In the middle ( 8150 Å) and bottom ( 9170 Å) cases, the sky-lines contamination is minimal. Open with DEXTER Figure 3: Contour plot of the 2-dimensional Ly region, spatial direction (pixels, 1 pixel = 1.5 kpc) versus wavelength (Å, 20 Å corresponds to 751 ). The spectral interval of 7960-8280 Å and spatial extension of 22.5 kpc are shown. Thin dotted and continuous black lines mark 0, 5, 10, 20 above the mean signal. Thick lines mark 1 (red) and 3 (blue). Horizontal dotted lines indicate the FWHM of the profiles (spatial extension) by collapsing columns in the Ly region and on the continuum region. Open with DEXTER ### 2.2 Photometric properties Figure 5 shows the overall SED (black squares) and Table 2 summarizes the multi-band photometry of the source collected from different instruments mounted on ground and space-based telescopes. Magnitudes, errors and 1- lower limits (l.m.) are derived from the MUSIC catalog, (Grazian et al. 2006; Santini et al. 2009). There are other two WFI U bands observations not reported in the table, U35 and U38, with a slightly different filter shape. Their lower limits are 27.84 and 26.75, respectively, much shallower than the limit provided by the VLT . Despite the high redshift of the source discussed here, its (i775 - z850) color is 0.59, significantly bluer than the typical threshold of 1.3 adopted to select galaxies beyond redshift 5.5 (e.g. Bouwens et al. 2007; Dickinson et al. 2004). This comes from the contribution of the Ly emission to the flux in the i775 filter (see Fig. 1), that decreases the (i775 - z850) color by about 0.7 mag (as shown in Vanzella et al. 2009). For this reason it has been selected as a V606-band dropout source. It is also an R-band dropout source if referred to the ground-based photometry, see Table 2. As shown in Fig. 4, the source has been detected in the V606 band with a magnitude of 27.63 0.22, showing an attenuation of 94% to respect the 1400 Å emission (z850 band), which is consistent with the average IGM transmission at this redshift (e.g. Songaila 2004). Figure 4: Cutouts of the source GDS J033218.92-275302.7. From left to right: the B435, V606, i775, and z850 HST/ACS bands. The box side is 1.0 arcsec (6 kpc proper at the redshift of the source). Drawn from the V2.0 ACS catalog http://archive.stsci.edu/prepds/goods/. The last right box is the ISAAC/Ks band. The box side is 3 arcsec. Open with DEXTER Apart from the IGM attenuation that influences the bands bluer than the z850, the main feature of the SED is the discontinuity detected between VLT/ISAAC J, H, and Ks bands ( rest-frame) and the Spitzer/IRAC channels ( rest-frame), see Fig. 5. Such a discontinuity is consistent with estimates available in the literature for the same object (FIREWORKS, Wuyts et al. 2008; Stark et al. 2007; Wiklind et al. 2008; Raiter et al. 2010). The typical uncertainties (1) span the range 0.03 to 0.11 going from the HST/ACS, VLT/ISAAC and Spitzer channels 1 and 2, for the last two Spitzer bands (5.6 and 8 ) the errors increase to 0.2/0.3 mag. Figure 5: The resulted template fitting over the MUSIC multi-band catalog of the GDS J033218.92-275302.7. Left: single stellar population modeling. Blue solid line is the fit adopting the maximum ratio [O III]/[O II] (prescription Single/[O III] max'' in Table 3), and red dashed line is the fit with the Schaerer & de Barros (2009) method (Single SB09'' in the same table), see text for details. Right: double stellar population modeling. Green dashed line shows the evolved component (age 0.4 Gyr), the thin cyan line the young contribution (age 0.01 Gyr), and black thick line the best fit composition of the two. Open with DEXTER Table 2: Summary of the photometric information (magnitudes and 1- errors) for our source. Figure 6: The effective radius measured with the SExtractor algorithm (pixel units) as a function of the z850 magnitude for a sample of stellar-like sources selected having S/G classifier larger than 0.97 (crosses). Solid line is the median value and the dotted lines the 1 percentiles. The open red circles are stars confirmed spectroscopically. The filled blue circle indicates the source described in the present work. Open with DEXTER ### 2.3 Morphological properties Image cutouts of the isolated source GDS J033218.92-275302.7 in the B435, V606, i775, z850 (HST/ACS), and Ks (VLT/ISAAC) bands are shown in Fig. 4, where each box is 1.0 arcsec wide for the ACS figures (6 kpc proper at the redshift of the source), while it is 3 arcsec on a side for the ISAAC Ks image. We considered the ACS/z850 and the ISAAC/Ks bands to derive basic morphological quantities (see Table 1). • z850 band ( 1400 Å). The uncorrected-PSF effective radius (Re) available from the ACS/GOODS public catalog v2.0 is 3.37 pixels. The same quantity derived from a sample of 80 stellar-like sources (SExtractor star/galaxy index larger than 0.97, 1.0 = star, 0.0 = extended source) with z850 magnitude in the range 23.5-25 gives a median value and 1 percentiles of 2.590-0.098+0.091, implying that this source is not a stellar-like object (see Fig. 6). The same result is obtained for the i775 band (6 from the median value of the stars). We note that the SExtractor star/galaxy index of the source is quite high, 0.83, but lower than the typical value of the stars. Therefore, even though the present galaxy is clearly a compact source in the UV rest-frame, it is marginally resolved both in the i775 and z850 bands. To derive PSF-corrected morphological parameters, we ran the GALFIT program (Peng et al. 2002) in both bands. The morphological shape of the source is not particularly complicated so a good fit is reached (reduced = 0.591) by adopting a simple Gaussian profile (Sèrsic model with n = 0.5) and leaving the Re, the axis ratio B/A, the coordinates X,Y, the magnitude, and the position angle as free parameters (SExtractor estimates were used as a first guesses). In the left panel of Fig. 7 the z850 band image of the galaxy is shown, and the residuals map provided by GALFIT, as a result of the subtraction of the best-fit model from the original galaxy, do not show significant structures (middle panel of the same figure). An effective radius Re = 0.62 0.05 pixels (0.11 0.01 kpc physical) and B/A = 0.61 0.14 have been obtained. We explored how the variation in Re affects the residuals fixing n=0.5 and B/A=0.61. GALFIT was run 80 times by varying the Re from 0.025 to 2.0 pixels with a step of 0.025. Its behavior is shown in the right panel of Fig. 7 where the minimum of the residuals for Re 0.650 is clear, still with acceptable values between Re 0.3-1.0 (pixels). It is worth noting that, if we perform the same fit in the i775 band, the best Re is slightly smaller than the z850 band (0.425 pixels, i.e. 0.013 arcsec or 0.08 0.01 kpc physical), implying an even more compact region for the Ly emission. Figure 7: The ACS z850 image of the source ( left, 100 100 pixels) and the residual image after subtraction of the model derived from GALFIT ( middle). In the right panel the behavior of the residuals (standard deviation calculated on a 20 20 pixels area centered on the source) as a function of the Re (fixed during GALFIT runs) is shown. The minimum value corresponds to Re 0.65 pixels, i.e. 0.02 arcsec or 0.12 kpc proper. The dotted line indicates the median value of the background residuals (standard deviation calculated in the blank regions). Open with DEXTER • Ks band ( 3300 Å). To approach the optical rest-frame wavelengths, we exploit the information derived from the near infrared (NIR hereafter) observations (VLT/ISAAC Ks band). The resolution from the ground is not comparable to that obtained from the space; however, we note that the present object has an FWHM fully consistent with the seeing at the epoch of observations of that particular region of the GOODS field (0.50 arcsec), i.e., it is not resolved. Assuming a Gaussian shape with an FWHM of 0.50 arcsec, it turns out that the is 0.9 kpc physical. For the Gaussian profile the Re = (FWHM/2)/1.738, i.e. the radius containing half of the total light. Since the source is not resolved, the derived Re is an upper limit at this wavelength. There are three other sources (with secure redshift, quality A'') in the FORS2 sample that show a similar z850 UV morphology of the source discussed here (with a SExtractor Re smaller than 3.5 pixels (0.105 arcsec)). One is the QSO GDS J033229.29-275619.5 (Fontanot et al. 2007) fully compatible the stellar PSF and two other LBGs: GDS J033217.22-274754.4 at redshift 3.652 already described in Vanzella et al. (2008) with a double structure of the Ly profile and GDS J033240.38-274431.0 at redshift 4.120, which shows a continuum with clear Ly emission and no other features. ## 3 Possible scenarios for GDS J033218.92-275302.7 ### 3.1 A chance superposition? If we interpret the emission line detected at = 9742 Å as a [O II] 3727 foreground emitter superimposed on a background and brighter higher-z (z=5.563) LBG, its redshift would be 1.614. It is well known that in the GOODS-S field there is an overdensity structure at that redshift, z=1.61 (e.g. Kurk et al. 2009; Castellano et al. 2007; Vanzella et al. 2008). However we can exclude this possibility on the basis of the ACS morphology, shows a compact and circular shape (see Fig. 4) and from the ultradeep U band observations carried out by the VLT/VIMOS instrument (Nonino et al. 2009), provide an upper limit of 30 AB at 1 (also it has not been detected in the ACS B435 band image). The source should be detectable in the blue if there is star formation activity traced by the [O II] 3727 line. Moreover, assuming a flat continuum at that 1 limit (30 AB), the rest-frame [O II] 3727 equivalent width would be larger than 104 Å. Even though we consider this possibility largely unlikely, we note that an example of strong [O II] 3727 emitter has been reported by Stern et al. (2000). ### 3.2 Is it an AGN? The spectral range up to 10 164 Å allows us to detect possible emission lines testing for the presence of an AGN (e.g. N V] 1240-1243, SiIV 1393.8-1402.8, and C IV] 1548.2-1550.8). Those features are routinely detected in spectra of the most obscured AGNs (e.g. Polletta et al. 2008,2006). Figure 8: Extracted VLT/FORS2 2-dimensional spectrum of the galaxy discussed in the present work (the spectral interval 9403-10 164 Å is shown). As a check of the wavelength calibration the skyline position derived from the FORS2 spectrum are reported (the sky line measurement performed by VLT/UVES and Keck/HIRES are indicated within parenthesis www.eso.org/observing/dfo/quality/UVES/pipeline/sky_spectrum.html). The expected positions of the C IV] feature are also shown with thick arrows: case (A) at the redshift of the observed N IV] 1486 (z=5.553) and case (B) at the redshift of the Ly line (z=5.563); see text for details. Open with DEXTER #### 3.2.1 Line emission in the UV • V 1240-1243 feature. Observationally, the N V emission is often present in the AGN case. The FORS2 spectrum allow us to measure emission features down to . No N V line is detected (see Fig. 2). • IV 1548.2-1550.8 feature. Unfortunately, the C IV line is at the very red limit of the observed spectrum, which in this particular slit position is determined by the detector end. Figures 8 and 2 (top panel) show the 2-dimensional and the 1-dimensional spectra zoomed at the red edge, respectively. On one hand, if the redshift of the C IV feature is higher than 5.565 (z(C IV) > 5.565, higher than the Ly redshift), then the doublet is completely out. Alternatively, if z(C IV) < 5.554, then the doublet is completely in. In general, LBGs show the observed Ly redshift as higher than the other spectral features because of its asymmetry which typically arises from backscattering of the receding material (e.g. Vanzella et al. 2009; Shapley et al. 2003). Similarly, in the AGN case, a blueshift of the C IV feature with respect to the observed Ly peak is typically observed, of several hundred kilometers per second. In particular a blueshift of 600 100 with respect to the Ly line is measured in the SDSS QSO composite spectrum (e.g. see Table 4 and Fig. 9 of Vanden Berk et al. 2001). If this is the case, the C IV feature should fall in the available spectrum, and its luminosity limit is at 2 of the background noise fluctuation. The limits and luminosities estimates on Ly, N V, Si IV, and C IV are reported in Table 1. #### 3.2.2 X-ray emission A correlation between 2-10 keV X-ray luminosity and [O III] 5007 or H emission line luminosities is observed for galaxies at z < 1 (e.g., Panessa et al. 2006; Silverman et al. 2008). Assuming that this correlation also holds at higher redshifts, then we may use the X-ray luminosity to derive a constraint on the fluxes for AGN-powered emission lines in the IRAC bands. From the current 2 Ms observations, there is no detection at the position of the source (3 limit of 1043 erg/s at 3-13 keV rest-frame, Luo et al. 2008). This limit roughly corresponds to an upper limit for both H and [O III] 5007 luminosities of 1040.5-42.5 and , respectively. Such values are affected by large uncertainties in the assumed relations (intrinsic scatter) and the limit derived from the 2 Ms image. However, if compared to the (at least) one magnitude jump between the VLT/ISAAC and Spitzer/IRAC magnitudes, these estimations suggest that, besides a line contribution to the IRAC magnitudes, there is also a significant contribution from stellar emission beyond 5000 Å rest-frame, i.e., of a relatively evolved stellar population (see Sect. 4). For example a line luminosity of 7 is needed to boost the 4.5 AB magnitude from 24 to 23 (adopting a bandwidth of 10 100 Å, Fazio et al. 2004). #### 3.2.3 A rare class of QSOs: an open possibility It is interesting to compare our N IV] 1486 emitter spectrum with the composite spectra of quasars available in the literature. This has already been done by several authors (e.g. Baldwin et al. 2003; Glikman et al. 2007; Jiang et al. 2008). None of the published average quasar spectral templates show any trace of N IV] 1486 emission. Nevertheless, focusing the attention on this spectral feature, Bentz et al. (2004) compile a sample of 6650 quasars in the range 1.6 < z < 4.1 showing the N IV] 1486 line (other than the N III] 1750), and more recently an updated work by Jiang et al. (2008) (on SDSS data release 5) reported that such objects are 1.1% of the total SDSS quasar sample. They also note that for this small fraction, the N V 1240 and Ly are much stronger than the rest of the population. We recall that our source does not show the N V 1240 line. More interestingly and similar to our findings, Glikman et al. (2007) have discovered two low luminosity QSOs at redshift 4 showing Ly, N IV] 1486 and C IV 1548-1550 emissions, but no detection of N V 1240. In one case the equivalent width of the N IV] 1486 is larger than the C IV one (240 Å vs. 91 Å), while it is the opposite for the other (24 Å vs. 91 Å). Our source has a luminosity of M145 = -22.1 (AB) and shows a clear N IV] 1486 emission with an equivalent width of 22 Å and FWHM 400 . The Ly line shows a narrow component with a measured FWHM of 600 . As performed in Glikman et al. (2007), since the blue side of the line profile is absorbed, we forced the symmetry in the line by mirroring the red side of the line profile over the peak wavelength and computed the Gaussian fit. The narrow-line component increases to 750 . The broad-line feature (indicated with a segment in the innner box of Fig. 1) gives an FWHM of 3500 . This would put the source in the QSO regime (velocity width larger than 1000 ). Therefore, the present source may be consistent with the interpretation of a low-luminosity quasar in which the host starburst galaxy is visible (similarly to Glikman et al. 2007). The study of stellar populations of low-luminosity AGNs (e.g. low-luminosity Seyfert galaxies, low-ionization nuclear emission line regions, LINERs, and transition-type objects, TOs) has been addressed for the local Universe (e.g., González Delgado et al. 2004), but this is still a poorly explored regime at higher redshift. While it is beyond the scope of the present work to explore the link between the coevolution of (circumnuclear) starburst activity and the central black hole accretion, we simply note that both AGN and star-formation required gas to fuel them, and it happens on different temporal and spatial scales, on sub-parsec and typically above few hundred parsecs (up to several kilo-parsecs) regions, respectively (e.g., Davies et al. 2007; Chen et al. 2009). In the present case, the size of the UV emitting region is compact, but still resolved in the z850 ACS image (as shown in Sect. 2). In summary, the presence of an AGN - in a rare evolutionary stage - may be indicated by the N IV] 1486 and broad Ly features, even though N V 1240, Si IV 1394-1493, and (possibly) C IV 1548-1550 are not detected. Table 3: SED modeling: results. ### 3.3 A multi-burst galaxy in a peculiar stage of evolution? The source GDS J033218.92-275302.7 has already been analyzed in Wiklind et al. (2008), who classify it as a pure'' balmer break galaxy (their ID ). The discontinuity detected between the Ks and 3.6 bands is interpreted as a signature of the Balmer break, suggesting a relatively evolved age of stellar populations with a significant stellar mass already in place (age of 0.7 Gyr and M* 7-8 ). A similar conclusion has been reached by Stark et al. (2007), who find an even higher stellar mass of (their ID 32_8020). However, most probably the observed ( ) color is contaminated by emission lines in the 3.6 band, e.g. [O III] 4959-5007. A similar boost to the flux in the 4.5 band may come from the H emission line. It was also selected as H emitter by Chary et al., private communication. Apart from the evident Ly emission, which implies the presence of young (<10 Myr) stars - i.e., some current''/ongoing star formation - significant nebular emission is also robustly supported by the detection of the N IV] 1486 line. As mentioned above, a similar feature has been identified in the Lynx arc and may indicate a short powerful starburst in which very hot and massive stars (T 80 000 K, Fosbury et al. 2003) or cooler Wolf-Rayet stars are involved (Villar-Martin et al. 2004). A similar blackbody ionizing source may be present in this source. The ongoing star formation activity would also be responsible for the measured outflow, whose spectral signature is in the red tail of the Lyprofile (see Ly modeling in Sect. 4.2). It is beyond the scope of the present work to model the ionizing source; nevertheless, we note that in a pure'' nebular scenario, the continuum is practically flat, and the observed breaks'' are produced by strong nebular emission lines (see Raiter et al. 2010, for a dedicated discussion). Alternatively, a different interpretation suggests a contribution from a relatively evolved stellar population that produces the Balmer break signature (see next section). Given the current spectroscopic and photometric information, the following mixed scenario may be possible: 1) ongoing active star formation in an -like region that produces nebular emission, asprobed by the Ly and N IV] 1486 features; and 2) an already evolved population of stars formed at higher redshift, as probed by the signal detected in the IRAC bands, in particular, redwards of the 4.5 (beyond 7000 Å rest-frame). ## 4 SED and Ly modeling We cannot definitively distinguish between the two scenarios described above, in particular for the explanation of the N IV] 1486 feature. In either case, even though the source reflects an early stage of coevolution of the (circumnuclear starburst) galaxy with its AGN or it is an source, the features of the host galaxy are detected and can be investigated. We therefore need to model the SED allowing for multiple stellar populations. Moreover, valuable information can be derived from the Ly line modeling. This is performed in the following sections. ### 4.1 Modeling the SED The SED modeling was performed adopting the multiwavelength GOODS-MUSIC photometric catalog (Grazian et al. 2006), and the spectral fitting technique was developed in Fontana et al. (2003,2006) (similar to those adopted in other works, e.g., Drory et al. 2004; Dickinson et al. 2003). In the previous section we pointed out that this galaxy likely contains a mixed stellar population, both young stars, as implied by a bright Ly line, the N IV] 1486 line, and old stars that produce the 4000 Å break, clearly observed in broad band photometry. We therefore model the SED of the galaxy both with a single stellar population and with a more plausible double stellar population. More complicated mixes of multiple stellar populations becomes unconstrained by the data given the many degrees of freedom of each population. We actually reduce the number of degree of freedom by imposing the requirement that both populations (old and young) are affected by the same dust extinction (with a Calzetti or a Small Magellanic Cloud extinction curve). Although this might not be true, we feel that it is plausible for very compact objects such as the one we are studying. We fix the rest-frame equivalent width of the Ly line to be EW = 60 Å, as measured from the spectrum and regardless of the star formation rate. As discussed in the previous section, the contribution of nebular lines to the photometry may affect the IRAC magnitudes of the 3.6 and 4.5 bands. In particular, the [O III] emission contributes to the 3.6 channel and the H line to the 4.5 channel. While the H line can be modeled relatively easily and its luminosity can be assumed to be proportional to the global SFR through the well-known Kennicutt relations, disentangling the [O III] contribution is harder. Moustakas et al. (2006) investigated the [O III] nebular emission line as a quantitative SFR diagnostic and conclude that the large dispersion in the [O III]/H ratio among star-forming galaxies precluded its suitability for SFR studies. We therefore treat the [O III] contribution in three different ways: 1) by assuming a mean [O III] flux as inferred in local star-burst galaxies (corresponding to a ratio f([O III])/f([O II]) = 0.32); 2) by assuming a maximum [O III] flux in the 3.6 band 10 times larger than in the previous case, corresponding to the maximum observed [O III]/H in star-burst galaxies; and 3) by neglecting the 3.6 band in the fit. In the SED-fitting computation, the formal errors of the observed magnitudes have a minimum value permitted for each band. This was done to avoid over-fitting in the minimization procedure, and it affects only the 3.6 and 4.5 bands, whose errors are increased to 0.1 (the minimum permitted) during the fit. The results of the various fits are reported in Table 3: for each model, we report the best-fit (bf) total stellar mass, age, (the star formation e-folding timescale), current SFR, and E(B-V) (indicated with EBV) for the single and double populations, as well as the minimum and maximum values allowed by the fit (at 1). For the double stellar populations the best-fit ages of the young and the evolved components are reported (in the last two columns). #### 4.1.1 A single stellar population model Although the single stellar population model with a declining exponential SFR is clearly an oversimplification it can set useful limits. From Table 3, we see that in all cases the best-fit stellar mass is well above and the age more than 700 Myr, implying a formation redshift z > 13. The variation in the [O III] flux of a factor 10 does not have a strong impact on the values of mass, age and SFR. Even neglecting the 3.6 band, the stellar mass is set to 4.5 1010 (with a minimum value of 3.4), but the most notable change is that some dust extinction is allowed, with a best fit = 0.06. It is worth noting that the stellar mass and ages we find are comparable to those derived by Wiklind et al. (2008) and Stark et al. (2007) if the nebular line treatment is not inserted. If we insert the [O III] prescription, our estimates become somewhat smaller, even though still significatively large given the redshift of the source (corresponding to 0.91 Gyr after the Big-Bang). In particular, Stark et al. (2007) derive a mass of 1.4 but without including the 5.8 and 8 IRAC bands in the SED fitting, while Wiklind et al. report a mass of 7 but assume a photometric redshift of 5.2. For comparison we have also fitted the SED with the method of Schaerer & de Barros (2009) allowing for numerous emission lines. The results obtained (see Table 3 and Fig. 5) are compatible with the two other approaches used here (see also next section). In addition, these models predict an intrinsic Ly equivalent width of EW(Ly 49-70 Å (1 interval), in good agreement with the observations. It is worth mentioning that the present source is different from those analyzed by Schaerer & de Barros (2009), extracted from a sample of z 6 star-forming galaxies of Eyles et al. (2007). First, the photometric break between the NIR bands and the first two Spitzer/IRAC channels 3.6 and 4.5 is at least twice for our source, 1.5 mag. The source is brighter in absolute scale, allowing smaller photometric errors see Table 2. The main nebular contributors at z > 5.6 to the IRAC channels 3.6 and 4.5 are the H, [O III], and H lines, respectively. In this case the H line falls on the blue edge of the 3.6 filter transmission (Fazio et al. 2004), so its contribution is further attenuated. However, more importantly, in our case the source has also been detected where the contribution of nebular lines is less effective (at this redshift), i.e. in the 5.8 and 8.0 bands. Even though the photometric errors slightly increase in these bands, the break is still large (1 mag). For comparison, in the sample of Eyles et al. (2007), the source SBM031 (with z850 = 25.35) has not been detected in the reddest IRAC channels 3 and 4. This is the main reason for this galaxy maintaining a relatively large estimation of the age and stellar mass, even considering the nebular contribution. #### 4.1.2 A mixed stellar population model In this case we allow for two stellar populations with different redshifts of formation and different star formation histories. The only constraint we impose (a part the spectroscopic redshift) is that both stellar populations are affected by the same dust extinction. In Table 3 the best-fit values for total stellar mass and the star formation rate corresponding to the sum of the two contributions are shown, each taken with the relevant normalization factor, while we separately report two best-fit ages for each population. Inall cases, the best-fit solution is made by an older stellar component (of 0.4 Gyr) that contributes to most of the stellar mass, while the young component has high values of SFR. The global stellar mass remains similar to the previous single population case, regardless of the [O III] treatment, with mean values of 5-6 and in all cases greater than 4 . The SFRs increase by a factor of at least two, and solutions with values as high as SFR = 250 are acceptable. In all cases, the e-folding timescale of the star formation rate of the young population is close to 100 Myr. The derived ages of the two stellar populations are 0.01 and 0.4 Gyr, independent of the [O III] treatment. The age of the old component is still quite large (but less than the previous case), implying a formation redshift around z 8. Most important, solutions with small but non negligible dust extinction are always preferred. Formally the best-fit solution among those including all bands is the double population with maximum [O III] contribution ( ). In summary we consider that 5 is a fair estimate of the total stellar mass and that, considering all uncertainties in the data and in the modeling, a solid lower limit can be set at (the contribution of the young component to the total stellar mass is negligible, 1% in all cases). The galaxy contains a stellar population that is at least 400 Myr old, and the average extinction ( = 0.03-0.06) is smaller but not incompatible with the extinction factor that comes from the Ly profile modeling (see below). ### 4.2 Modeling of the Ly line Table 4: Best-fit parameters from the Ly line fitting. Values marked in bold face have been fixed during the fitting procedure. The galaxy shows an Ly emission line with an FWHM of 600 , an evident asymmetric profile, a clear sharp decline in flux on the blue side and a red tail of Ly photons extending up to 40 Å (1500 ) from the peak of the line (see inner box of Figs. 1 and 3). Ly is a resonance line that undergoes a complicated radiation transport, with the line formation under the influence of numerous parameters: not only dust but also the geometry, kinematics, and temperature structure of the neutral ISM (e.g. Ahn et al. 2003; Verhamme et al. 2006). These parameters influence the line profile and, if sufficient care is taken, the line profile itself can be used to provide independent and unique constraints (Verhamme et al. 2008). Using the Monte Carlo Lyman-alpha (MCLya) radiation transfer code of Verhamme et al. (2006), we computed a wide array of possible emergent line profiles. Parameter fitting is performed using a standard least squares fitting engine to minimize the statistic. Details of the software and fitting can be found in Hayes et al. (2009, in prep.). The parameter space is not entirely unconstrained; e.g., it is possible to observationally constrain two of the parameters: the N IV] 1486 line puts the systemic redshift at 5.553; and the velocity shift between N IV] 1486 and Ly constrain the expanding velocity of the gas shell to respect the stellar component, = 457 km s-1 (see Table 1). Since our grid of shell parameters is discrete, we adopted the nearest values of the outflow velocity of 400 and 500 km s-1. We ran six independent fits in total for all combinations of constraints denoted as follows: zsys_V400 and zsys_V500 constraining both z and (400 and 500 km s-1); zsys constraining z; V400 and V500 constraining (400 and 500 km s-1, respectively); and free in which all parameters are fit without constraint. Figure 9: The observed Ly line profile together with the best-fitting synthetic profiles described in the text. Open with DEXTER The results of the fits are presented in Fig. 9 and Table 4. In general, all the fits agree with the case free, in particular V500 and free produce the same values. The Ly modeling favors high column densities ( cm-2), outflow velocities of 400-500 km s-1 (consistent with observations when is allowed to vary), and a 3.0 for all models with error of +1/-2 (in the free case), which provides a rough estimate of the extinction E(B-V 0.3-0.2+0.1. We also explored the possibility that the emerging Ly shape is caused by a static gas ( = 0). In this case the expected double-peaked structure (e.g. Verhamme et al. 2006) would mimic the single peak observed, since the bluer one could be self-absorbed (by the galaxy and IGM). From the modeling it turns out that high values of the Doppler. It is worth mentioning that we do not expect that the extinction undergone by the nebular lines ( from Ly fitting) should match what is undergone by the stars ( , from SED fitting). Indeed Verhamme et al. (2008) find a large scatter in the relation between the extinction determined from the Ly profile fits versus other methods including photometric fit and/or measured spectral slopes (see Fig. 12 of their work). Calzetti et al. (2001) find an empirical relation of = 0.44 , which would make the two results even more consistent (within their uncertainties). ## 5 Discussion ### 5.1 Summary of the modeling The SED fitting and the Ly modeling indicate that: 1. The column density of the nebular neutral gas is high, > 1020.8. We note that this value is comparable to those found for the damped Lyman-alpha systems, e.g., Wolfe et al. (2005). 2. The outflow velocity derived from the Ly modeling is consistent with the observed one, and it is relatively high (greater than 400 km s-1). 3. A young and an evolved stellar population are both present. The first with an SFR in the range 30-200 and negligible contribution to the total stellar mass (1%). The second with a stellar mass of 1010 and an age of 0.4 Gyr. 4. The extinctions derived from the different methods are compatible within the 1- uncertainties and in general are relatively small (in the range 0 < E(B-V) < 0.3). Although the signal-to-noise ratios are low, the galaxy is strongly detected in both the IRAC 5.8 and 8.0 bands, where no nebular lines or strong nebular continuum should contribute at z = 5.56. Even if emission lines contribute flux to the shorter wavelength IRAC channels, the longer-wavelength IRAC channels indicate a significant increase in flux density relative to the rest-frame UV continuum, suggesting the presence of a Balmer break from an evolved stellar population. Summarizing, this galaxy shows several interesting observed properties: 1) its stellar mass is still relatively high ( 1010 ) with a component of already evolved stellar populations (0.4 Gyr); 2) it contains a star-forming component able to produce nebular emission lines and with an age of 10 Myr; 3) a substantial wind is measured both from the optical spectrum and from the Ly modeling, of 450/500 km s-1; and 4) the source is compact in the rest-frame UV and U-band rest-frame wavelengths. ### 5.2 An already dense galaxy? As described in the previous sections, the SED fitting analysis implies a stellar mass of 1010 , with a significant, evolved component with an age of 0.4 Gyr. If the very compact size measured in the ACS and ISAAC images for the rest-frame ultraviolet light can be assumed to apply to the overall distribution of the evolved stellar population, then it implies a very high stellar mass density: 1. If we assume a constant size over all wavelengths from the 1400 Å to optical bands rest-frame (ACS z850 band, 0.11 kpc physical), the stellar density in a spherical symmetric shape turns out to be = (0.5 )/(4/3 3.5 1012 . 2. Similarly, assuming a constant size over all wavelengths from the 3300 Å rest frame (ISAAC Ks band, 0.9 kpc physical), the stellar density in a spherical symmetric shape turns out to be = (0.5 )/(4/3 8.2 109 . In both cases, given the estimations of the effective radius and the stellar mass of , the source appears to be ultradense if compared with the local mass-size relation. In case (1) the stellar mass density should be considered an upper limit if the derived from the 1400 Å rest-frame is a fair estimate of the smallest size. Locally, on average, the is larger than 2 kpc (3 kpc) for early type (late type) galaxies with the comparable stellar mass (Shen et al. 2003). Interestingly, the Ly modeling suggests (in all cases) a relatively high column density of the neutral gas, the turns out in the range 1020.8-21.4 . We further note that, assuming that the Schmidt law is valid at this redshift (Kennicutt 1998), adopting the observed area of 8.8 and two possible SFRs estimates (see Table 3), 30 and 100 , the mass of the gas turns out to be 8 109 and 2 1010 , respectively, which represents a significant fraction if compared to the stellar mass ( 1010 ). As noted by Buitrago et al. (2008), massive ultradense spheroid observed at intermediate redshift 1.5-3 and the globular clusters have remarkably similar stellar densities (above 1010 ), suggesting a similar origin. A massive ultradense galaxy at z 1.5-3 should form its stars very quickly in earlier epochs and in a high gas-density environment. In this sense the present source may represent a precursor'' of the ultradense spheroids recently discovered at redshift 1.5-3. ### 5.3 Feedback in action? The current burst of star formation may be caused by a previous infall of gas and/or to a merger event (even though the UV morphology is quite regular). A vigorous wind of 450 km s-1 is detected both from the observations (Ly and N IV] 1486 velocity offset) and from the Ly modeling (leaving all parameters free). As discussed above, the Ly emission arise from a compact region with an effective radius not larger than 0.1 kpc (the PSF-deconvolved Re in the i775 band is 0.08 kpc physical), a possible indication that the outflow of gas is in its initial phase of expansion from the central region. This particular phase of the galaxy evolution showing hot and massive stars and/or a low-luminosity AGN may be an efficient mechanism to blow the material out from the potential well of the galaxy, in some way influencing the subsequent star formation activity and/or the surrounding IGM. Wind propagation and escape is quite sensitive to the entrainment fraction and to the velocity of the wind itself. This occurs because the two primary forces limiting wind propagation are the galaxy's potential well and the ram pressure of the gas that must be swept up even if the wind is fast. Moreover, if entrainment is significant, then the mass over which the wind energy and momentum must be shared may be much greater. Therefore it is first useful to compare the escape velocity from the halo with the estimated wind velocity. Following the calculation of Ferrara et al. (2000), the escape velocity can be expressed as (1) with p=1.65. The isothermal halo density profile is assumed ( ), with an extension out to a radius r200 = = , defined as the radius within which the mean dark matter density is 200 times the critical density at redshift z of the galaxy. The turns out to be 30 kpc at this redshift, assuming a halo mass of . Under these assumptions, the escape velocity is , about the same (or a bit larger) as the wind estimated velocity from spectral features, (or from the Ly modeling, ). It is possible that we are observing the transport of material from the galaxy to the halo, which will remain confined. If we assume a slightly lower mass of the halo, e.g. (a factor 10 higher than the stellar mass), then the escape velocity turns out to be and an = 24 kpc. In this case, the velocity of the wind would be sufficient to escape the potential well of the halo. Indeed, from SPH simulations it appears that the main contributors to the metal enrichment of the low-density regions of the IGM are small'' galaxies with stellar masses below (Aguirre et al. 2001b), and similar results have been obtained by other authors (e.g. Oppenheime & Davè 2008; Bertone et al. 2005). In the present case the uncertainty on the halo mass prevents any clear conclusion. If we assume a value lower than , then the expanding material may reach characteristic distances (namely stall radius'') where the outflow ram pressure is balance by the IGM pressure up to few hundred kpc (e.g. Aguirre et al. 2001a). ## 6 Concluding remarks A peculiar galaxy belonging to the GOODS-S field has been discussed. The main observed features are the relatively strong nebular emission in the ultraviolet (Ly and N IV] 1486) and the presence of the Balmer Break detected through the NIR VLT/ISAAC and Spitzer/IRAC data. Indeed, from the SED fitting with single and double stellar populations and the Ly modeling, it turns out that the source seems to have an evolved component with stellar mass of 10 and age 0.4 Gyr, a young component with an age of 0.01 Gyr (contributing to 1% of the total stellar mass), and a star formation rate in the range of 30-200 . At present no evidence of common N IV] emitters'' is observed in surveys of high redshift galaxies or quasars. However, there are rare cases in the literature that show this line emission (together with other atomic transitions), spanning from a pure region source to a subclass of low-luminosity quasars. In the first case, very hot and massive stars with low metallicity are required to produce the N IV] line; however, it is difficult to reproduce the signal measured in the Spitzer/IRAC channels with a pure nebula, in particular at wavelengths beyond 4.5 , i.e. to reconcile the two observed facts: 1) the presence of a relatively evolved stellar population and 2) the low-metallicity environment needed if the N IV] emission arises from stellar photoionization. Alternatively, the low-luminosity quasar/AGN interpetation may explain the N IV] emission, the broad Ly component, and the properties of the host galaxy discussed here, i.e., starforming, massive, and evolved galaxy. The limits on the size derived from the ACS/z850 and VLT/Ks bands indicate that this object is denser than the local ones with similar mass, with a significant mass of the gas still in place (comparable to the stellar one). A relatively high nebular gas column density is also favored from the Ly line modeling, , comparable to those found for the damped Lyman-alpha systems. The region emitting Ly photons is spatially compact, close to that at the continuum emission at 1400 Å, 0.1 kpc, in which a vigorous outflow (450/500 km s-1) has been measured from the spectrum and Ly modeling. The gas is expanding from this region, but given the uncertainty on the halo mass, it is dubious whether it will pollute the IGM to great distances. Such special objects are the key to understanding fundamental passages in the formation and evolution of the galaxy population. Future instruments will shed light on the nature of this interesting object, in particular, the JWST and the ELTs will give better and new constraints on the optical rest-frame morphology and nebular emission. Acknowledgements We would like to thank the anonymous referee for very constructive comments and suggestions. We are grateful to the ESO staff in Paranal and Garching, who greatly helped in the development of this program. We thank J. Retzlaff for the informations about the VLT/Ks images of the GOODS-S field and the useful comments and discussions of P. Tozzi, F. Calura, R. Chary, S. Recchi, P. Monaco, and F. Fontanot about the work. E.V. would like to thank Anna Raiter and R.A.E. Fosbury for precious discussions about the photoionization modeling. We acknowledge financial contributions from contract ASI/COFIN I/016/07/0 and PRIN INAF 2007 A Deep VLT and LBT view of the Early Universe''. ## References 1. Ando, M., Ohta, K., Iwata, I., et al. 2006, ApJ, 645, 9 [NASA ADS] [CrossRef] [Google Scholar] 2. Ando, M., Ohta, K., Iwata, I., et al. 2007, PASJ, 59, 717 [NASA ADS] [CrossRef] [Google Scholar] 3. Aguirre, A., Hernquist, L., Schaye, J., et al. 2001a, ApJ, 561, 521 [NASA ADS] [CrossRef] [Google Scholar] 4. Aguirre, A., Hernquist, L., Schaye, J., & Weinberg, D. H. 2001b, ApJ, 560, 599 [NASA ADS] [CrossRef] [Google Scholar] 5. Ahn, S.-H., Lee, H.-W., & Lee, H. M. 2003, MNRAS, 340, 863 [NASA ADS] [CrossRef] [Google Scholar] 6. Baldwin, J. A., Hamann, F., Korista, K. T., et al. 2003, ApJ, 583, 649 [NASA ADS] [CrossRef] [Google Scholar] 7. Balestra, I., Mainieri, V., Popesso, P., et al. 2010, A&A, 512, A12 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar] 8. Beckwith, S. V. W., Stiavelli, M., Koekemoer, A. M., et al. 2006, AJ, 132, 1729 [NASA ADS] [CrossRef] [Google Scholar] 9. Bentz, M. C., Hall, P. B., & Osmer, P. S. 2004, AJ, 128, 561 [NASA ADS] [CrossRef] [Google Scholar] 10. Bertone, S., Stoehr, F., & White, S. D. M. 2005, MNRAS, 359, 1201 [NASA ADS] [CrossRef] [Google Scholar] 11. Binette, L., Groves, B., Villar-Martin, M., Fosbury, R. A. E., & Axon, D. J. 2003, A&A, 405, 975 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar] 12. Bouwens, R. J., Illingworth, G. D., Blakeslee, J. P., & Franx, M. 2006, ApJ, 653, 53 [NASA ADS] [CrossRef] [Google Scholar] 13. Bouwens, R. J., Illingworth, G. D., Franx, M., & Ford, H. 2007, ApJ, 670, 928 [NASA ADS] [CrossRef] [Google Scholar] 14. Bouwens, R. J., Illingworth, G. D., Franx, M., & Ford, H. 2008, ApJ, 686, 230 [NASA ADS] [CrossRef] [Google Scholar] 15. Buitrago, F., Trujillo, I., Conselice, C. J., et al. 2008, ApJ, 687, L61 [NASA ADS] [CrossRef] [Google Scholar] 16. Castellano, M., Salimbeni, S., Trevese, D., et al. 2007, ApJ, 671, 1497 [NASA ADS] [CrossRef] [Google Scholar] 17. Cimatti, A., Cassata, P., Pozzetti, L., et al. 2008, A&A, 482, 21 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar] 18. Conselice, C. J., Rajgor, S., & Myers, R. 2008, MNRAS, 386, 909 [NASA ADS] [CrossRef] [Google Scholar] 19. Cristiani, S., Alexander, D. M., Bauer, F., et al. 2004, ApJ, 600, 119 [NASA ADS] [CrossRef] [Google Scholar] 20. Chen, Y.-M., Wang, J.-M., Yan, C.-S., Hu, C., & Zhang, S. 2009, ApJ, 695, 130 [NASA ADS] [CrossRef] [Google Scholar] 21. Daddi, E., Renzini, A., Pirzkal, N., et al. 2005, ApJ, 626, 680 [NASA ADS] [CrossRef] [Google Scholar] 22. Davies, R. I., Sanchez, F. M., Genzel, R., et al. 2007, ApJ, 671, 1388 [NASA ADS] [CrossRef] [Google Scholar] 23. Dickinson, M., Papovich, C., Ferguson, H. C., & Budavári, T. 2003, ApJ, 587, 25 [NASA ADS] [CrossRef] [Google Scholar] 24. Dickinson, M., et al. 2003, in the proceedings of the ESO/USM Workshop, The Mass of Galaxies at Low and High Redshift, Venice, Italy, October 2001, ed. R. Bender, & A. Renzini [arXiv:astro-ph/0204213] [Google Scholar] 25. Dickinson, M., Stern, D., Giavalisco, M., et al. 2004, ApJ, 600, 99 [NASA ADS] [CrossRef] [Google Scholar] 26. Drory, N., Bender, R., & Hopp, U. 2004, ApJ, 616, 103 [NASA ADS] [CrossRef] [Google Scholar] 27. Elbaz, D., Daddi, E., Le Borgne, D., et al. 2007, A&A, 468, 33 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar] 28. Eyles, L. P., Bunker, A. J., Stanway, E. R., et al. 2005, MNRAS, 364, 443 [NASA ADS] [CrossRef] [Google Scholar] 29. Eyles, L. P., Bunker, A. J., Ellis, R. S., et al. 2007, MNRAS, 374, 910 [NASA ADS] [CrossRef] [Google Scholar] 30. Fan, X., Strauss, M. A., Schneider, D. P., & Becker, R. H. 2003, AJ, 125, 1649 [NASA ADS] [CrossRef] [Google Scholar] 31. Fazio, G. G., Hora, J. L., Allen, L. E., et al. 2004, ApJS, 154, 10 [NASA ADS] [CrossRef] [Google Scholar] 32. Ferguson, H. C., Dickinson, M., Giavalisco, M., et al. 2004, ApJ, 600, 107 [NASA ADS] [CrossRef] [Google Scholar] 33. Ferrara, A., Pettini, M., & Shchekinov, Y. 2000, MNRAS, 319, 539 [NASA ADS] [CrossRef] [Google Scholar] 34. Fontana, A., Donnarumma, I., Vanzella, E., et al. 2003, ApJ, 594, 9 [NASA ADS] [CrossRef] [Google Scholar] 35. Fontana, A., Salimbeni, S., Grazian, A., et al. 2006, A&A, 459, 745 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar] 36. Fontana, A., Santini, P., Grazian, A., et al. 2009, A&A, 501, 15 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar] 37. Fontanot, F., Cristiani, S., Monaco, P., et al. 2007, A&A, 461, 39 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar] 38. Fosbury, R. A. E., Villar-Martín, M., Humphrey, A., et al. 2003, ApJ, 596, 797 [NASA ADS] [CrossRef] [Google Scholar] 39. Giavalisco, M., Ferguson, H. C., Koekemoer, A. M., et al. 2004a, ApJ, 600, L93 [NASA ADS] [CrossRef] [Google Scholar] 40. Giavalisco, M., Dickinson, M., Ferguson, H. C., et al. 2004b, ApJ, 600, 103 [NASA ADS] [CrossRef] [Google Scholar] 41. Glikman, E., Djorgovski, S. G., Stern, D., Bogosavljevic, M., & Mahabal, A. 2007, ApJ, 663, 73 [NASA ADS] [CrossRef] [Google Scholar] 42. Gonzalez Delgado, R. M., Cid Fernandes, R., Perez, E., et al. 2004, ApJ, 605, 127 [NASA ADS] [CrossRef] [Google Scholar] 43. Grazian, A., Fontana, A., de Santis, C., et al. 2006, A&A, 449, 951 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar] 44. Grazian, A., Salimbeni, S., Pentericci, L., et al. 2007, A&A, 465, 393 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar] 45. Jiang, L., Fan, X., & Vestergaard, M. 2008, ApJ, 679, 962 [NASA ADS] [CrossRef] [Google Scholar] 46. Keenan, F. P., Ramsbottom, C. A., Bell, K. L., et al. 1995, ApJ, 438, 500 [NASA ADS] [CrossRef] [Google Scholar] 47. Kurk, J., Cimatti, A., Zamorani, G., et al. 2009, A&A, 504, 331 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar] 48. Labbé, I., Bouwens, R., Illingworth, G. D., & Franx, M. 2006, ApJ, 649, 67 [NASA ADS] [CrossRef] [Google Scholar] 49. Le Fèvre, O., Vettolani, G., Garilli, B., et al. 2005, A&A, 439, 845 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar] 50. Luo, B., Bauer, F. E., Brandt, W. N., et al. 2008, ApJS, 179, 19 [NASA ADS] [CrossRef] [Google Scholar] 51. Miller, N. A., Fomalont, E. B., Kellermann, K. I., et al. 2008, ApJS, 179, 114 [NASA ADS] [CrossRef] [Google Scholar] 52. Moustakas, J., Kennicutt, R. C., Jr., & Tremonti, C. A. 2006, ApJ, 642, 775 [NASA ADS] [CrossRef] [Google Scholar] 53. Nagao, T., Sasaki, S. S., Maiolino, R., & Grady, C. 2008, ApJ, 680, 100 [NASA ADS] [CrossRef] [Google Scholar] 54. Nonino, M., Dickinson, M., Rosati, P., et al. 2009, ApJ, 183, 244 [NASA ADS] [CrossRef] [Google Scholar] 55. Oppenheimer, B. D., & Davè, R. 2008, MNRAS, 387, 577 [NASA ADS] [CrossRef] [Google Scholar] 56. Ouchi, M., Shimasaku, K., Akiyama, M., et al. 2008, ApJ, 176, 301 [NASA ADS] [CrossRef] [Google Scholar] 57. Panessa, F., Bassani, L., Cappi, M., et al. 2006, A&A, 455, 173 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar] 58. Peng, C. Y., Ho, L. C., Impey, C. D., & Rix, H.-W. 2002, AJ, 124, 266 [NASA ADS] [CrossRef] [Google Scholar] 59. Pentericci, L., Grazian, A., Fontana, A., et al. 2007, A&A, 471, 433 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar] 60. Pentericci, L., Grazian, G., Fontana, A., et al. 2009, A&A, 494, 553 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar] 61. Polletta, M. d. C., Wilkes, B. J., Siana, B., et al. 2006, ApJ, 642, 673 [NASA ADS] [CrossRef] [Google Scholar] 62. Polletta, M., Omont, A., Berta, S., et al. 2008, A&A, 492, 81 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar] 63. Popesso, P., Dickinson, M., Nonino, M., et al. 2009, A&A, 494, 443 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar] 64. Raiter, A., Fosbury, R. A. E., & Teimoorinia, H. 2010, A&A, 510, A109 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar] 65. Ravindranath, S., Giavalisco, M., Ferguson, H. C., et al. 2006, ApJ, 652, 963 [NASA ADS] [CrossRef] [Google Scholar] 66. Santini, P., Fontana, A., Grazian, A., et al. 2009, A&A, 504, 751 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar] 67. Schaerer, D., & Verhamme, A. 2008, A&A, 480, 369 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar] 68. Schaerer, D., & de Barros, S. 2009, A&A, 502, 423 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar] 69. Shapley, A. E., Steidel, C. C., Pettini, M., & Adelberger, K. L. 2003, ApJ, 588, 65 [NASA ADS] [CrossRef] [Google Scholar] 70. Shen, S., Mo, H. J., White, S. D. M., & Blanton, M. R. 2003, MNRAS, 343, 978 [NASA ADS] [CrossRef] [Google Scholar] 71. Silverman, J. D., Lamareille, F., Maier, C., et al. 2008, ApJ, 696, 396 [NASA ADS] [CrossRef] [Google Scholar] 72. Songaila, A. 2004, AJ, 127, 2598 [NASA ADS] [CrossRef] [Google Scholar] 73. Stark, D. P., Bunker, A. J., Ellis, R. S., Eyles, L. P., & Lacy, M. 2007, ApJ, 659, 84 [NASA ADS] [CrossRef] [Google Scholar] 74. Stark, D. P., Ellis, R. S., Bunker, A., et al. 2009, ApJ, 697, 1493 [NASA ADS] [CrossRef] [Google Scholar] 75. Steidel, C. C., Adelberger, K. L., Giavalisco, M., Dickinson, M., & Pettini, M. 1999, ApJ, 519, 1 [NASA ADS] [CrossRef] [Google Scholar] 76. Stern, D., Bunker, A., Spinrad, H., & Dey, A. 2000, ApJ, 537, 73 [NASA ADS] [CrossRef] [Google Scholar] 77. Szokoly, G. P., Bergeron, J., Hasinger, G., et al. 2004, ApJS, 155, 271 [NASA ADS] [CrossRef] [Google Scholar] 78. Taniguchi, Y., Ajiki, M., Nagao, T. K., et al. 2005, PASJ, 57, 165 [NASA ADS] [Google Scholar] 79. Taniguchi, Y., Murayama, T., Scoville, N. Z., et al. 2009, ApJ, accepted [arXiv:0906.1873] [Google Scholar] 80. Tapken, C., Appenzeller, I., Noll, S., et al. 2007, A&A, 467, 63 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar] 81. Taylor-Mager, V. A., Conselice, C. J., Windhorst, R. A., & Jansen, R. A. 2007, ApJ, 659, 162 [NASA ADS] [CrossRef] [Google Scholar] 82. Tozzi, P., Mainieri, V., Rosati, P., et al. 2009, ApJ, 698, 740 [NASA ADS] [CrossRef] [Google Scholar] 83. Trujillo, I., Conselice, C. J., Bundy, K., & Cooper, M. C. 2007, MNRAS, 382, 109 [NASA ADS] [CrossRef] [Google Scholar] 84. Vanden Berk, D. E., Richards, G. T., Bauer, A., et al. 2001, AJ, 122, 549 [NASA ADS] [CrossRef] [Google Scholar] 85. van Dokkum, P. G., Franx, M., Kriek, M., et al. 2008, ApJ, 677, 5 [NASA ADS] [CrossRef] [Google Scholar] 86. Vanzella, E., Cristiani, S., Dickinson, M., et al. 2005, A&A, 434, 53 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar] 87. Vanzella, E., Cristiani, S., Dickinson, M., et al. 2006, A&A, 454, 423 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar] 88. Vanzella, E., Cristiani, S., Dickinson, M., et al. 2008, A&A, 478, 83 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar] 89. Vanzella, E., Giavalisco, M., Dickinson, M., et al. 2009, ApJ, 695, 1163 [NASA ADS] [CrossRef] [Google Scholar] 90. Verhamme, A., Schaerer, D., & Maselli, A. 2006, A&A, 460, 397 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar] 91. Verhamme, A., Schaerer, D., Atek, A., & Tapken, C. 2008, A&A, 491, 89 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar] 92. Villar-Martin, M., Cerviño, M., & González Delgado, R. M. 2004, MNRAS, 355, 1132 [NASA ADS] [CrossRef] [Google Scholar] 93. Wiklind, T., Dickinson, M., Ferguson, H. C., et al. 2008, ApJ, 676, 781 [NASA ADS] [CrossRef] [Google Scholar] 94. Wolfe, A. M., Gawiser, E., & Prochaska, J. X. 2005, ARA&A, 43, 861 [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar] 95. Wuyts, S., Labbè, I., Schreiber, N. M. F., et al. 2008, ApJ, 682, 985 [NASA ADS] [CrossRef] [Google Scholar] 96. Yan, H., Dickinson, M., Stern, D., et al. 2005, ApJ, 634, 109 [NASA ADS] [CrossRef] [Google Scholar] ## Footnotes ... 5.56 Based on observations made at the European Southern Observatory, Paranal, Chile (ESO program 170.A-0788 The Great Observatories Origins Deep Survey: ESO Public Observations of the SIRTF Legacy/HST Treasury/Chandra Deep Field South). ... limit The ratio of those components is related to the electron density, Raiter et al. (2010), Keenan et al. (1995). ...) We performed separate fits with enlarged errors in the 5.8 and 8 channels to explore their possibile underestimation, and the consequence is even higher stellar masses, since more weight is given to the 3.6 and 4.5 bands. ... Doppler Where the Doppler parameter is as the contribution of thermal and turbulent motions. parameter b and intrinsic FWHM are favored, 160 km s-1 and 1000 km s-1, respectively. This is not surprising since it is a way to drive photons away from line center in the absence of a wind. However, in all cases the resulting fit worsens in general and, in particular, the extended red and wavy tail of the line is no longer reproduced. Conversely, this feature favor the above interpretation of backscattered photons from an expanding shell (the presence of a wind would agree also with the ongoing star formation activity). ## All Tables Table 1: Summary of the physical quantities derived from the spectral features and morphological analysis. Table 2: Summary of the photometric information (magnitudes and 1- errors) for our source. Table 3: SED modeling: results. Table 4: Best-fit parameters from the Ly line fitting. Values marked in bold face have been fixed during the fitting procedure. ## All Figures Figure 1: 1-dimensional spectrum of the galaxy discussed in the present work. The Ly and N IV] 1486 emission lines are evident, together with the detection of the continuum and the IGM attenuation. The inner box shows a zoom of the Ly line, the asymmetry and the red tail are visible (the latter is marked with a solid segment, 40 Å long, or 1500 ). The transmissions of the ACS V606, i775, and z850 are also shown. The exposure time was 14 400 s. Open with DEXTER In the text Figure 2:Three zoomed regions of the 1D spectrum (red dotted line is the solid black line smoothed over two pixels). Dotted plots show the rescaled sky spectrum. Filled circles mark the position of the lines at the redshift 5.553 (case in which only the N IV] 1486.5 component is detected). Dotted open circles denote the other (but less probable) case in which the N IV] 1483.3 component is detected (z=5.568). Red filled triangles mark the positions at the observed Ly redshift, z=5.563. Top: the positions of the N IV] 1486 and C IV] features are shown. In the sky spectrum two sky emission lines are marked: (1) 10082.46 Å and (2) 10124.01 Å (also shown in the 2-dimensional spectrum of Fig. 8). Middle: the region where N V is expected to lie is shown. The open blue triangle indicates the estimated intrinsic Ly redshift from the Ly modeling. Bottom: the region around the SiIV feature is shown. In the middle ( 8150 Å) and bottom ( 9170 Å) cases, the sky-lines contamination is minimal. Open with DEXTER In the text Figure 3: Contour plot of the 2-dimensional Ly region, spatial direction (pixels, 1 pixel = 1.5 kpc) versus wavelength (Å, 20 Å corresponds to 751 ). The spectral interval of 7960-8280 Å and spatial extension of 22.5 kpc are shown. Thin dotted and continuous black lines mark 0, 5, 10, 20 above the mean signal. Thick lines mark 1 (red) and 3 (blue). Horizontal dotted lines indicate the FWHM of the profiles (spatial extension) by collapsing columns in the Ly region and on the continuum region. Open with DEXTER In the text Figure 4: Cutouts of the source GDS J033218.92-275302.7. From left to right: the B435, V606, i775, and z850 HST/ACS bands. The box side is 1.0 arcsec (6 kpc proper at the redshift of the source). Drawn from the V2.0 ACS catalog http://archive.stsci.edu/prepds/goods/. The last right box is the ISAAC/Ks band. The box side is 3 arcsec. Open with DEXTER In the text Figure 5: The resulted template fitting over the MUSIC multi-band catalog of the GDS J033218.92-275302.7. Left: single stellar population modeling. Blue solid line is the fit adopting the maximum ratio [O III]/[O II] (prescription Single/[O III] max'' in Table 3), and red dashed line is the fit with the Schaerer & de Barros (2009) method (Single SB09'' in the same table), see text for details. Right: double stellar population modeling. Green dashed line shows the evolved component (age 0.4 Gyr), the thin cyan line the young contribution (age 0.01 Gyr), and black thick line the best fit composition of the two. Open with DEXTER In the text Figure 6: The effective radius measured with the SExtractor algorithm (pixel units) as a function of the z850 magnitude for a sample of stellar-like sources selected having S/G classifier larger than 0.97 (crosses). Solid line is the median value and the dotted lines the 1 percentiles. The open red circles are stars confirmed spectroscopically. The filled blue circle indicates the source described in the present work. Open with DEXTER In the text Figure 7: The ACS z850 image of the source ( left, 100 100 pixels) and the residual image after subtraction of the model derived from GALFIT ( middle). In the right panel the behavior of the residuals (standard deviation calculated on a 20 20 pixels area centered on the source) as a function of the Re (fixed during GALFIT runs) is shown. The minimum value corresponds to Re 0.65 pixels, i.e. 0.02 arcsec or 0.12 kpc proper. The dotted line indicates the median value of the background residuals (standard deviation calculated in the blank regions). Open with DEXTER In the text Figure 8: Extracted VLT/FORS2 2-dimensional spectrum of the galaxy discussed in the present work (the spectral interval 9403-10 164 Å is shown). As a check of the wavelength calibration the skyline position derived from the FORS2 spectrum are reported (the sky line measurement performed by VLT/UVES and Keck/HIRES are indicated within parenthesis www.eso.org/observing/dfo/quality/UVES/pipeline/sky_spectrum.html). The expected positions of the C IV] feature are also shown with thick arrows: case (A) at the redshift of the observed N IV] 1486 (z=5.553) and case (B) at the redshift of the Ly line (z=5.563); see text for details. Open with DEXTER In the text Figure 9: The observed Ly line profile together with the best-fitting synthetic profiles described in the text. Open with DEXTER In the text Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform. Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9018454551696777, "perplexity": 4302.911372668486}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623488253106.51/warc/CC-MAIN-20210620175043-20210620205043-00168.warc.gz"}
https://stats.stackexchange.com/questions/331803/what-are-the-properties-of-the-unfolded-gamma-distribution-generalization-of-a	# What are the properties of the “unfolded” gamma distribution generalization of a normal distribution? In a prior post, I developed an "unfolded" gamma distribution generalization of a normal distribution as an example of how to relate a gamma distribution to a normal distribution. This yielded $$\text{ND}(x;\mu,\sigma^2,a) = \dfrac{a e^{-2^{-\frac{a}{2}} \left(\frac{1}{\sigma }\right)^a \left\| x-\mu\right\| ^a}}{2 \sqrt{2} \sigma \Gamma \left(\frac{1}{a}\right)} \,,$$ where the mean is $\mu$, the variance is $\sigma^2$, and the shape is $a>0$, where $a=2$ for an ordinary normal distribution. This appears to be a different distribution from the generalized error distribution $$\text{GED}(x;\mu,\alpha,\beta)=\frac{\beta}{2\alpha\Gamma(1/\beta)} \; e^{-(\|x-\mu\|/\alpha)^\beta}\,,$$ where $\mu$ is the location, $\alpha>0$ is the scale, and $\beta>0$ is the shape and where $\beta=2$ yields a normal distribution. It includes the Laplace distribution when $\beta=1$. As $\beta\rightarrow\infty$, the density converges pointwise to a uniform density on $(\mu-\alpha,\mu+\alpha)$. Questions Are these distributions the same? If so, how does one convert between them? If not, and $\text{ND}(x;\mu,\sigma^2,a)$ is a different distribution, what are its other properties, for example, does it reduce to a distribution other than a normal distribution? • Assuming the domain is $x\in\mathbb R$, comparing the parts of the two densities that depend on $x$ shows immediately that the role played by $\mu$ is the same in both (it's the location parameter) and $a=\beta$ (the shape parameter) and $\sigma=\alpha$ (the scale parameter). – whuber Mar 5 '18 at 13:53 • @whuber Close, but when one lets $\beta=a=1$, then $\sqrt{2} \sigma=\alpha$. There is a bit more to this, will explore further. – Carl Mar 5 '18 at 18:47 • I'm having trouble reading the "ND" expression. If it means the argument of the exponential is $$-2^{-a/2} \sigma^{-a}\|x-\mu\|^a=-\left(\frac{\|x-\mu\|}{\sigma\sqrt{2}}\right)^a,$$ then the scale factor must be $\sigma\sqrt{2}$ and everything works out. – whuber Mar 5 '18 at 19:32 ## 2 Answers Indeed, my notation was bad. Better notation is now used on the post this came from, The generalized gamma distribution $$\text{GD}\left(x;\alpha ,\beta ,\gamma ,\mu \right)=\begin{array}{cc} & \begin{cases} \dfrac{\gamma e^{-\left(\dfrac{x-\mu }{\beta }\right)^{\gamma }} \left(\dfrac{x-\mu }{\beta }\right)^{\alpha \gamma -1}}{\beta \,\Gamma (\alpha )} & x>\mu \\ 0 & \text{other} \\ \end{cases} \\ \end{array}\,,$$ implies that \begin{align} \text{GND}(x;\mu,\alpha,\beta) &= \frac{1}{2}\text{GD}\left(x;\frac{1}{\beta},\alpha,\beta,\mu \right)+\frac{1}{2}\text{GD}\left(-x;\frac{1}{\beta},\alpha,\beta,\mu \right)\\ &= \frac{\beta e^{-\left(\dfrac{\left\|x-\mu\right\|}{\alpha }\right)^{\mathrm{\Large{\beta}}}}}{2 \alpha \Gamma \left(\dfrac{1}{\beta }\right)}\\ \end{align} \,, is a generalization of the normal distribution, where $\mu$ is the location, $\alpha>0$ is the scale, and $\beta>0$ is the shape and where $\beta=2$ yields a normal distribution. It includes the Laplace distribution when $\beta=1$. As $\beta\rightarrow\infty$, the density converges pointwise to a uniform density on $(\mu-\alpha,\mu+\alpha)$. See the original post for the logic behind this. The result from Whuber's comment in an answer: It seems like both expressions can be related to the function below $$f(x;c_1,c_2,c_3) = c_1 e^{-c_2 \vert x - \mu \vert^{c_3}}$$ ### GED: $\frac{\beta}{2\alpha\Gamma(1/\beta)} \; e^{-(\|x-\mu\|/\alpha)^\beta}$ $$\begin{array}{rcl} c_1 &=& \frac{\beta}{2\alpha\Gamma(1/\beta)} \\ c_2 &=& \left(\frac{1}{\alpha} \right)^\beta \\ c_3 &=& \beta \end{array}$$ ### ND: $\dfrac{a e^{\left(-2^{-\frac{a}{2}} \left(\frac{1}{\sigma }\right)^a \left\| x-\mu\right\| ^a\right)}}{2 \sqrt{2} \sigma \Gamma \left(\frac{1}{a}\right)}$ $$\begin{array}{rcl} c_1 &=& \frac{a}{2 \sqrt{2} \sigma\Gamma(1/a)} \\ c_2 &=& 2^{-\frac{a}{2}}\left( \frac{1}{\sigma} \right)^a \\ c_3 &=& a \end{array}$$ and you can say $$\begin{array}{rcl} \mu &=& \mu \\ {\alpha} &=& \sqrt{2}{\sigma} \\ \beta &=& a \end{array}$$	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9874038696289062, "perplexity": 380.05702223120613}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243991650.73/warc/CC-MAIN-20210518002309-20210518032309-00106.warc.gz"}
http://scienceblogs.com/evolutionblog/2010/08/02/monday-math-the-infinitude-of/	# Monday Math: The Infinitude of the Primes Last week we saw that every positive integer greater than one can be factored into primes in an essentially unique way. This week we ask a different question: Just how many primes are there? Euclid solved this problem a little over two thousand years ago by showing there are infinitely many primes. His proof was by contradiction. If there are only finitely many primes then we can list them all: $p_1, \ p_2, \ p_3, \ p_4, \ \dots, \ p_k$ We can now define a new number, which we shall call $N$, by the formula $N=p_1p_2p_3 \dots p_k +1$ That is, $N$ is obtained by multiplying all the primes together and adding one. The fundamental theorem of arithmetic tells us that $N$ must have at least one prime factor. (Of course, $N$ might be prime itself, but that is OK since every number is a factor of itself.) But $N$ is clearly not divisible by any of the primes on the list, since it leaves a remainder of one upon division by any of our primes. That shows there is a prime that is not on our list, which is a contradiction. An ingenious argument! G. H. Hardy used it as one of two examples of mathematical elegance in his famous essay “A Mathematician’s Apology.” I doubt many mathematicians would disagree with the choice. (Hardy’ other selection, for the record, was the classic proof of the irrationality of the square root of two.) There are many other proofs, of course. Kummer devised a clever variation on Euclid’s argument. He defined $N$ by the formula $N=p_1p_2p_3 \dots p_k.$ No adding one this time. But then he observed that $N-1$ must have a prime factor $p_i$. Since $N$ itself is also a multiple of $p_i$ we see that $p_i \mid N-(N-1)=1,$ which is plainly impossible. Actually, that last little trick shows that any two consecutive numbers must be relatively prime. (That means they have no prime factors in common.) Filip Saidak used that to produce a direct proof of the infinitude of the primes. Let $n$ be a positive integer greater than one. Define the numbers $A=n(n+1) \phantom{xxx} \textrm{and} \phantom{xxx} B=A(A+1)$ Then $A$ has at least two prime factors, since it is the product of consecutive numbers. But then $B$ must have at least three prime factors. Since we can continue this sequence indefinitely we see that the number of primes must be greater than any finite number. It follows that there are infinitely many primes. Euclid’s trick of multiplying all the primes and adding one is so good that it would be a shame if the infinitude of the primes was the only thing we could prove with it. So let us generalize things. By an arithmetic progression we mean a sequence of positive integers in which each element is obtained from the one before by adding the same number. An example of an arithmetic progression is $1, \ 2, \ 3, \ 4, \ 5, \ 6, \ \dots$ That is the progression where we begin with one, and keep adding one to produce the subsequent numbers. We have shown there are infinitely many primes in that sequence. So let us ask the question: If we pick an arbitrary arithmetic progression, how many prime numbers will it contain? Let’s experiment! Here are two progressions: \begin{align} 4k+1 &: \phantom{x} 1, \ 5, \ 9, \ 13, \ 17, \ 21, \ 25, \ 29, \ \dots \\ 4k+3 &: \phantom{x} 3, \ 7, \ 11, \ 15, \ 19, \ 23, \ 27, \ 31, \ \dots \end{align} In the first sequence we begin with one and keep adding four. In the second we began with three and kept adding four. A quick scan reveals that some elements of the sequence are prime and some are not. Let us try to use Euclid’s method to prove there are infinitely many primes in the progression $4k+3$. Suppose there were only finitely many such primes. Then list them all. For reasons that will become clear momentarily, remove the number three itself from the list. Now construct the number $N=4 \left( p_1p_2 \dots p_k)+3,$ Plainly $N$ has a prime factor. But it is not divisible by any of the primes on our list. It also is not divisible by three, since we deliberately left that number off our list. (Had we not done so, the number $N$ would be a multiple of three.) It follows that there are primes not on our list. So far, all is essentially identical to Euclid. The trouble is, in this case we already knew there were primes not on our list. Specifically, all of the primes that are one more than a multiple four were omitted. We have not yet shown there are any primes of the form $4k+3$ that have been left off our list. But wait! It cannot be that all of $N$’s factors are of the from $4k+1$. Note that $N$ itself is three more than a multiple of four. But if we multiply two numbers together, each of which is one more than a multiple of four, we get this: $\left( 4k+1 \right) \left( 4 \ell + 1 \right)=\left( 16k \ell + 4k + 4 \ell \right) +1$ The number in the parentheses is clearly a multiple of four. It follows that our product is one more than a multiple of four. This shows that it cannot be that all of $N$’s prime factors are one more than a multiple of four. It must have at least one prime factor that is three more than a multiple of four. And since none of the primes on our list fits the bill, the proof is complete. Well, that’s one sequence down! Alas, we cannot apply the same method to the sequence $4k+1$. You see, it is possible for a number that is one more than a multiple of four to have all of its prime factors be three more than a multiple of four. For example: $21=7 \times 3$ In fact, while Euclid’s method will allow us to pick off certain progressions, it cannot be used to solve the general question we asked. So I repeat, if $ak+b$ is an arithmetic progression (in which we start with $b$ and keep adding $a$ over and over again), how many prime numbers will it contain? Obviously we need to require that $a$ and $b$ be relatively prime. If they had a common factor other than one, then every number in the sequence would be divisible by that common factor. That means every number after the first would definitely fail to be prime. So let us make this one assumption. It turns out that with that one simple assumption, the progression $ak+b$ contains infinitely primes. Sadly, there is no elementary proof of that fact. Dirichlet proved that in 1837, but he since he needed to invent the field of analytic number theory to do it we shall definitely save that for a different post. 1. #1 Owlmirror August 2, 2010 Perhaps an overly-pedantic point — given that 3 is the first prime in the “4k+3” sequence, and you’re specifically leaving it out, shouldn’t that be: $N=4 \left( p_2p_3 \dots p_k)+3,$ </hyperpedant> 2. #2 Jason Rosenhouse August 2, 2010 Pedantic indeed! Good point though. 3. #3 David August 3, 2010 Reminds me of this lovely song: 4. #4 Darlingtonia August 5, 2010 So, do mathematicians who are looking for the next largest known prime utilize the fact that it is between Pk and N? 5. #5 g August 10, 2010 Darlingtonia: No, for several reasons. 1. N is much, much larger than the next prime in all reasonable cases. 2. The largest known primes all have very special forms (e.g., 2^n-1) because there are very efficient algorithms for testing primality for numbers of these forms. If you want to find a larger prime number than anyone has yet found, what you do is go looking more or less at random for primes of the form 2^n-1 for n somewhat bigger than the current record. These numbers would, again, be much smaller than N. 3. In particular, no one goes looking for “the next largest prime”. It’s probably not at all feasible to determine the next prime after the currently-biggest-known prime. 4. There are other provable bounds on “the next prime” that are much, much smaller than N. For instance (though this is still far from the best one can do) there is a theorem called “Bertrand’s postulate” that says that the next prime after n is always less than 2n. (Unless n<2.)	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9184190630912781, "perplexity": 213.4100564099995}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-30/segments/1469257832939.48/warc/CC-MAIN-20160723071032-00059-ip-10-185-27-174.ec2.internal.warc.gz"}
http://math.stackexchange.com/questions/60010/equivalence-of-certain-comma-categories	# Equivalence of certain comma categories Let $\mathcal{C}$ be some category, $A \in \mathcal{C}$, $\mathbf{Grpd}_A$ be a groupoid consisting of all objects isomorphic to $A$ and all isomorphisms between them, and $i: \mathbf{Grpd}_A \to \mathcal{C}$ be the embedding. Are categories $\mathcal{C} \downarrow A$ and $\mathcal{C} \downarrow i$ necessarily equivalent? I tried to construct the equivalence for a special case of bundles but there was an obstruction (one triangle that failed to be commutative), and I don't know how to approach the proof of inequivalence in general. Should I seek to prove inequivalence directly for some toy category (which I'd prefer not to do, see below) or is there an actual equivalence? My motivation lies with bundles (so I'm mostly concerned with the special case $\mathcal{C} = \mathbf{Top}$): it seems 'evil' to fix one space as the base because we don't generally distinguish between isomorphic spaces, so a natural question arises: does it matter if we fix just one or the whole isomorphism class? A hint would be nice, but if the question is actually too answer with only little knowledge of category theory (up to limits), a full answer is welcome too. - Is the issue that you can't pick a coherent system of isomorphisms from objects in the groupoid to $A$? – Dylan Moreland Aug 26 '11 at 23:39 @Dylan: arrange $p': E' \to B'$, $p'': E'' \to B''$ together with $u: E' \to E''$ and $f: B' \to B''$ as a convex trapezoid. Then add $\varphi': B' \to B$ and $\varphi'': B'' \to B$ so that arrows form a large triangle. That's how I tried to construct one functor of the equivalence (the other being, of course, the embedding). However, there is no guarantee that the small triangle in the diagram commutes, and this later came back to bite me when I was constructing a natural transformation to $\mathrm{id}_{\mathcal{C} \downarrow i}$. – Alexei Averchenko Aug 26 '11 at 23:40 @Dylan: yes, and I'm not sure if picking such a system even fits the spirit of what I'm pondering :) – Alexei Averchenko Aug 26 '11 at 23:41 These two categories are not equivalent in general, because $\mathcal{C}\downarrow A$ always has a terminal object $id: A \to A$ whereas $\mathcal{C}\downarrow i$ may fail to have one. Assume that your original category $\mathcal{C}$ has an initial object $0$ (this will in particular apply to the case $\mathcal{C}=Top$), and consider any object $p: E \to B$ in $\mathcal{C}\downarrow i$. Suppose that $A$ has some automorphism different from the identity. Then the same holds for $B$. This nonidentity automorphism $u: B \to B$ together with $id: B \to B$ will provide two different maps from $(0 \to B)$ to $p$. Consequently, $p$ cannot be a terminal object in $\mathcal{C}\downarrow i$. In fact, if $A$ does not have any automorphisms beside the identity then $Grpd_A$ has exactly one isomorphism between any two of its objects and your attempt will then provide an equivalence. As to your general question, I do not think that the choice of a particular object $B$ out of $Grpd_A$ runs against the categorical spirit. The important thing is that the result, the category $\mathcal{C}\downarrow B$, does not depend on this choice: any map $f: B \to B'$ induces a functor $F: \mathcal{C}\downarrow B \to \mathcal{C}\downarrow B'$ and if $f$ is an isomorphism then $F$ will be an equivalence. Therefore one may as well use $\mathcal{C}\downarrow A$ right from the beginning.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9375101923942566, "perplexity": 159.2464159236496}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-07/segments/1454701157075.54/warc/CC-MAIN-20160205193917-00165-ip-10-236-182-209.ec2.internal.warc.gz"}

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