Talk:Vorticity equation

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Derivation

If anyone is interested in creating a wiki page for the derivation of the vorticity equation, I followed the notes given on this page and derived the equation here:

http://earthcubed.wordpress.com/2009/08/25/64/

If you contact me I will supply my Latex code for my wordpress page.

S243a (talk) 03:38, 26 August 2009 (UTC)John Creighton

Wikipedia is meant to be an encyclopedia-like reference, not a textbook; and as such it should not include derivations, exercises, or excessive pedagogical explanations. That sort of material should be most useful and welcome at Wikiversity and Wikibooks. --Jorge Stolfi (talk) 01:16, 13 March 2013 (UTC)

{{vec}}

Any objections? [1] M∧Ŝc²ħεИ_τlk 08:45, 7 March 2013 (UTC)

Equation in the head section

The equation has now been moved back to the head section. The head is supposed to be succint and maximally accessible, but it must contain a definition of the term, not just some general statements about it. Usually mathematical equations should be relegated to the body of the article, but when the article is specifically about an equation, that equation must be given in the head, in order to fulfill that section's only purpose. --Jorge Stolfi (talk) 01:23, 13 March 2013 (UTC)

Incorrect result in introduction

The claim in the introduction that $\left({\frac {d{\vec {w}}}{dt}}\cdot \nabla \right){\vec {v}}$ vanishes is plainly false. In the linked derivation, the author quotes an earlier 'result' of his which relies critically on the following 'fact': If $u(x,y,z)$ is any vector field, then we can write $u(x,y,z)=g(\zeta (x,y,z))$ for some functions $g\colon \mathbb {R} \to \mathbb {R} ^{3},\zeta \colon \mathbb {R} ^{3}\to \mathbb {R}$. This obviously false, since, for example, $\zeta$ would have to be a continuous injection from $\mathbb {R} ^{3}$ into $\mathbb {R}$. A link to the first article is http://www.hrpub.org/download/20131107/UJAM1-12600416.pdf, and the relevant section is titled 'Method of Transformation.' — Preceding unsigned comment added by 140.247.103.144 (talk) 12:22, 30 September 2014 (UTC)

1. The claim “Incorrect result in introduction” is very important but it is false. As we can see this claim contradicts to well-known rule in different textbooks and Wikipedia “…a vector field on a domain in n-dimensional Euclidean space can be represented as a vector-valued function…” http://en.wikipedia.org/wiki/Vector_field . Read more in http://en.wikipedia.org/wiki/Vector_field#Vector_fields_on_subsets_of_Euclidean_space (1.1 Vector fields on subsets of Euclidean space ).

2.This is not a forum for general discussion of the cited article's subject. This is the talk page for discussing improvements to the “Vorticity equation” article.

If you want to discuss the subject of article http://www.hrpub.org/download/20131107/UJAM1-12600416.pdf you should go, for example, here: http://en.wikipedia.org/wiki/Wikipedia_talk:WikiProject_Mathematics. Also, the general discussion of this article's subject you can organize on any journal’s pages of Cambridge Harvard University (as we can see from your IP). --5.45.192.112 (talk) 20:28, 20 November 2014 (UTC)

The entry "This 3-D equation can be transformed to the simpler equation.." has been removed because is false in general; more precisely, the general vorticity equation can be reduced in such a way only in the case of planar velocity fields. For an easy proof, see for example page 3 of:

http://web.mit.edu/2.20/www/lectures/Lecture-2014/lecture9-2014.pdf

To improve the content of this page, the above link is now added to the section "Simplifications". Notice that such results come from basic calculus and are not recent at all; therefore, any link to scientific journals should be avoided, being both unnecessary and misleading. Instead, the bibliography of this page probably still lacks a list of good textbooks for a basic introduction to the subject.

After continuation of transformations in http://web.mit.edu/2.20/www/lectures/Lecture-2014/lecture9-2014.pdf you receive this result http://www.hrpub.org/download/20131107/UJAM1-12600416.pdf . For a refutation of this result it is necessary to deny well-known rule in Wikipedia “…a vector field on a domain in n-dimensional Euclidean space can be represented as a vector-valued function…” http://en.wikipedia.org/wiki/Vector_field . Read more in http://en.wikipedia.org/wiki/Vector_field#Vector_fields_on_subsets_of_Euclidean_space Therefore I do addition in your revision. --Alexandr (talk) 15:29, 27 November 2014 (UTC)

There is also a phenomenological way of seeing that this claim must be wrong. If the simplifaction was holding in 3D as well it would mean that all turbulence as we know it would have been completely different. In particular, if this simplification was holding, it would mean that the vortices could only be moved around and never change direction (exactly as it is the case in 2D). This is obviously not what is measured in real fluids. — Preceding unsigned comment added by Jeanpauldax (talk • contribs) 13:48, 1 December 2014 (UTC)

Your doubts demand only visible mathematical proofs or simple mathematical counterexample because we discuss a correctness of the Navier-Stokes exact transformation. --Alexandr (talk) 20:13, 2 December 2014 (UTC)

I haven't spotted yet the mathematical error in your transformation. It is clear that there must be one since the Navier-Stokes equations (and therefore the vorticity equations) reproduce correctly the fluid motion in 2 and 3 dimensions. This great difference is represented by the term you removed by your transformation (as I explained in my previous comment). Until you can explain the physical difference between 2 and 3 dimensional turbulence even without this term I suggest you remove your remark about this term cancelling also in 3D. There are no other option. Either you made a mathematical mistake, either you just claimed that the Navier-Stokes equations cannot reproduce the motion of turbulent flows accurately. I think that the scientific community that has worked on the topic for the last couple of centuries does not believe that the latter is true. — Preceding unsigned comment added by Jeanpauldax (talk • contribs) 16:06, 3 December 2014 (UTC)

Jeanpauldax wrote:

“I haven't spotted yet the mathematical error in your transformation.”

This is very important claim! Mathematicians have not found an error in these transformations too.

“Until you can explain the physical difference between 2 and 3 dimensional turbulence even without this term I suggest you remove your remark about this term cancelling also in 3D.”

In this case explanations without concrete formulas have no sense. While look formulas (6) and (7) in article

“Either you made a mathematical mistake, either you just claimed that the Navier-Stokes equations cannot reproduce the motion of turbulent flows accurately. I think that the scientific community that has worked on the topic for the last couple of centuries does not believe that the latter is true.”

The answer to your comment you can read here: "It is believed, though not known with certainty, that the Navier–Stokes equations describe turbulence properly."[5] http://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations#Turbulence --93.74.76.101 (talk) 14:11, 4 December 2014 (UTC)

OK leave aside turbulence (since the Navier-Stokes equations are describing turbulent flows so poorly). We do not need turbulent motion to still need a way to stretch and rotate vortices in 3D. These vortices are also present in laminar flows. Are we also not sure that the Navier-Stokes describe correctly laminar Newtonian fluids? Do you have any hint? I think they have proven to be quite robust to testing for the last 200 years or so. — Preceding unsigned comment added by Jeanpauldax (talk • contribs) 22:27, 4 December 2014 (UTC)

Alexandr: Please stop this edit war. Your previous attempts at including your clearly erroneous paper on another wikipedia page resulted in the Navier-Stokes_existence_and_smoothness page being semi-protected. I'm certain that if you continue to redo your edits here that this page will be protected soon too. See the discussion at Talk:Navier-Stokes_existence_and_smoothness#Yet_another_solution_proposed.3F, and the edit history of the main article. If you are still convinced of your correctness, you should have your paper reviewed by a reputable journal with non-anonymous reviewers. AndyBloch 07:29, 5 December 2014 (UTC) — Preceding unsigned comment added by AndyBloch (talk • contribs)

AndyBloch! Your refer to another's not proved claim. The mathematical proofs are needed. Show your own proof or at least the proof of another mathematics. Have courage to rely on own mind (Habe Mut, dich deines eigenen Verstandes zu bedienen! https://ru.wikipedia.org/wiki/Sapere_aude) I suggest please undo your unreasonable revision if you have no the convincing arguments.--Alexandr (talk) 12:50, 5 December 2014 (UTC)

To Jeanpauldax! Probably, in these citations you will find some answers to your important remarks:

www.hrpub.org/download/20131107/UJAM1-12600416.pdf Important note (extraordinary). "Formulas (5*) means that we can not receive true solutions of any vector equations without these additional equalities. Therefore all solutions of the NSE and other vector PDE should satisfy these requirements."

http://www.hrpub.org/download/20140205/UJAM3-12601996.pdf "Equations (3**) and (11) in [1] can be used for the checking of so called NSE exact solutions. All true exact solutions of the NSE should satisfy these equations. Otherwise such “NSE exact solutions” are false." --Alexandr (talk) 19:54, 5 December 2014 (UTC)

Alexandr: First, even if your paper had no errors, it would be wrong for you to be including it in a wikipedia page for several reasons, some of which were mentioned at Talk:Navier-Stokes_existence_and_smoothness#Yet_another_solution_proposed.3F. Second, as mentioned there also, you misapplied the chain rule in your UJAM1-12600416.pdf paper. Equation (3) is not generally valid. The chain rule only applies to functions which are differentiable at the point(s) where you are taking the differentiation. You have not proven that U(x,y,z) can be be decomposed into the form F(g(x,y,z)) where both F (a vector-valued function) and g (a scalar function) are differentiable. See Chain_rule#Statement. Am I wrong about this? AndyBloch 06:41, 6 December 2014 (UTC) — Preceding unsigned comment added by AndyBloch (talk • contribs)

Alexandr: I do not see how these references answer my concern. Your aim here is to show that the vorticity equation can be simplified a lot in 3D. I am no mathematician and I think many people more capable than me in mathematics on wikipedia pointed errors in your papers. Here my argument is plain physics. The NS equations are known for describing correctly 3D fluid motion (do you agree? i'm not even talking about turbulent motion here). The term you are removing from the equation is the one responsible for the deformation of vortices due to velocity gradients which is present in nature. If you remove it fluid motion would be completely different and cannot be observed except in 2D limits. Your argument is therefore equivalent than to claim that the NS equations are not reproducing correctly fluid motion (I am not talking about a small difference here, but a fundamental difference). Do you agree with this statement? Is that what you truly believe? — Preceding unsigned comment added by Jeanpauldax (talk • contribs) 23:33, 6 December 2014 (UTC)

To AndyBloch.

1. As you can see Talk:Navier-Stokes_existence_and_smoothness#Yet_another_solution_proposed.3F the main reason (secondary sources) already is eliminated. Formulate please your position about these secondary sources and edit proposed section’s revision (Attempt at solution).

There are other reasons. For example, the "journal" you "published" in is not a reputable journal. They are listed as a predatory publisher. I'm sorry but you've been mislead as to their reputation and you may have been scammed by them. --AndyBloch 09:08, 12 December 2014 (UTC)

2. In first my message I already made comments on this remark 140.247.103.144: “ …well-known rule in different textbooks and Wikipedia “…a vector field on a domain in n-dimensional Euclidean space can be represented as a vector-valued function…” http://en.wikipedia.org/wiki/Vector_field . Read more in http://en.wikipedia.org/wiki/Vector_field#Vector_fields_on_subsets_of_Euclidean_space. — Preceding unsigned comment added by 5.45.192.109 (talk) 12:33, 9 December 2014 (UTC)

You keep repeating that, but that does not have anything to do with the error I've been pointing out. That error is with applying the Chain Rule to functions which are not differentiable. Yes, it is possible to write any vector function U(x,y,z) as a vector function U'(ζ(x,y,z)), where ζ is a scalar function. It's almost trivial to find a function ζ which will always work. For example, ζ(x,y,z) could be composed of the alternating digits of x, y, and z. As an illustration, ζ(1.23, 4.56, 7.89) = 147.258369. (U'(ζ) would then be U(inverse function of ζ).) But this function is not differentiable. It is not always possible to find such functions U' and ζ where both are differentiable. For example, suppose U(x,y,z) = (x,y,0). How would you write that as U'(ζ(x,y,z)), where both functions are differentiable? (Hint: it's not possible. If it was, then R^m and Rⁿ would be homeomorphic even when m ≠ n.) --AndyBloch 09:08, 12 December 2014 (UTC)

To Jeanpauldax! Thanks for your persistence to get to the truth.

I hope, in these brief notes you will find some answers to your important remarks:

https://www.idmarch.org/document/Navier–Stokes+equations/6LnF-show/Vector+PDE+Paradoxes+Alexandr+Kozachok+Kiev%2C+Ukraine+a-kozachok1%40narod.ru+http%3A++continuum-paradoxes.narod.ru++In+this+paper+the+additional+equalities+%28differ

http://analysis3.com/Helmholtz-decomposition-contradictions-download-w117376.html

--5.45.192.112 (talk) 12:26, 11 December 2014 (UTC)

AndyBloch, you wrote:

1.“…the "journal" you "published" in is not a reputable journal. They are listed as a predatory publisher. I'm sorry but you've been mislead as to their reputation and you may have been scammed by them”

The journal’s reputation is important only for an estimation of difficult mathematical works which are impossible to understand without the experts help. In our case we have two brief articles (3 and 5 pages) which even the student can understand. Do you agree?

2. “…For example, suppose U(x,y,z) = (x,y,0). How would you write that as U'(ζ(x,y,z)), where both functions are differentiable?”

You have paid attention to very important but independent problem. However your example U (x, y, z) = (x, y, 0) is erroneous. Read more https://www.idmarch.org/document/Navier–Stokes+equations/6LnF-show/Vector+PDE+Paradoxes+Alexandr+Kozachok+Kiev%2C+Ukraine+a-kozachok1%40narod.ru+http%3A++continuum-paradoxes.narod.ru++In+this+paper+the+additional+equalities+%28differ --93.74.76.101 (talk) 21:14, 14 December 2014 (UTC)