Talk:Navier-Stokes equations

The equation pretending to be the Navier-Stokes equation contains two terms of different physical dimension, rho u and rho d/dt(rho). This cannot be right.

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Formula removed from article, thus:

<math>-\nabla p +

\mu \left( \nabla^2 \mathbf{u} + {1 \over 3} \nabla (\nabla \cdot \mathbf{u} ) \right) + \rho \mathbf{F} = \rho \left( { \partial\mathbf{u} \over \partial t } + \mathbf{u} \cdot \nabla \mathbf{u} \right) <math>

Equations without defined variables and context are useless.

Note that the article in its current form gives the impression that there is "a Navier-Stokes equation", rather than a set of them: we should either give them all, or none of them.

-- The Anome 11:05, 23 Dec 2003 (UTC)

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Come help with Wikipedia:WikiProject Fluid dynamics moink 23:14, 27 Dec 2003 (UTC)

Removed equation again

<math>-\nabla p +

\mu \left( \nabla^2 \mathbf{u} + {1 \over 3} \nabla (\nabla \cdot \mathbf{u} ) \right) + \rho \mathbf{u} = \rho \left( { \partial\mathbf{u} \over \partial t } + \mathbf{u} \cdot \nabla \mathbf{u} \right) <math> where:

p is pressure

<math>\mathbf{u}<math> is fluid velocity

<math>\mu<math> is viscosity

<math>\rho<math> is density

Problems include: no sign of which one this is, no derivation or motivation, and what's with the factor of one-third? -- The Anome 19:28, 19 Jul 2004 (UTC)

It seems a little strange to have an article about a set of equations that doesn't include the equations, but ok. I will fix it when I'm back in the same city as my books. (July 27th or so). moink 19:43, 19 Jul 2004 (UTC)

Why are the equations missing?

The equaitons are correct (yes, the units are correct) except the div(u) term is zero, because constant density is assumed for the "official" Navier-Stokes equations. Also, to address the other concern, this is not one equation, it is a set of equations. u is a vector and has three components. This set of equations should be added to the text with the variables defined.

The problem is that there is no single "official" version of the Navier-Stokes equation or equations. Different ways of posing the problem (streamline/turbulent, incompressible/compressible...) yield different but related sets of equations. Some people prefer to write everything out in components (ugh), others like vectors. Some like the Reynolds-approximation version, some want to do the whole thing right down to thermal terms and shearing. There are lots of different choices for variable names. Just pulling an equation out of a hat and saying "this is the Navier-Stokes equation" is not useful, because it does not provide insight for the reader into what the equation(s) mean(s). What would be useful would be a derivation from scratch, and then some special case versions for simple cases. See some of the references, particularly [1] (http://astron.berkeley.edu/~jrg/ay202/node50.html) for a good example of this.

Just for comparison, the reference above gives the (simplified) Navier-Stokes equations as:

<math>{{\partial \rho} \over {\partial t}} + \nabla \cdot ( \rho \mathbf{u}) = 0<math>

<math>{ \partial \mathbf{u} \over \partial t} + ( {\mathbf{u}}\cdot \nabla ) \mathbf{u} = -{1 \over \rho} \mathbf{ \nabla}p - \mathbf{ \nabla}\phi + {\mu \over \rho } \nabla ^2 {\mathbf{u}},<math>

<math>\rho \left( { \partial \varepsilon \over \partial t} + {\mathbf{u}}\cdot \nabla \varepsilon \right) - \nabla \cdot ( K_H \nabla T) + p \nabla \cdot {\mathbf{u}}= 0.<math>

-- The Anome 07:45, 21 Jul 2004 (UTC)

Those look good. Let's put those in for now, with your caveats and reference, until we can get a full derivation. moink 16:34, 21 Jul 2004 (UTC)

Is a full derivation appropriate here? That seems to be more of a wikibooks thing. However, if we really want one, I'm sure I can cobble one up. -- Kaszeta 21:16, 28 Oct 2004 (UTC)