Talk:Vorticity equation

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)