Continuum hypothesis of fluid mechanics (& relativistic fluids)

[_sr_]

Continuum approximation of fluid mechanics (& relativistic fluids)

I have a few 'foundational' questions on fluid mechanics which I haven't been able to find quick answers to, any help would be appreciated.

At the start of any course on fluids, one is told of the continuum [STRIKE]hypothesis[/STRIKE] approximation - at large particle numbers and large enough distance scales, the atomic degrees of freedom become irrelevant and the fluid can be accurately modeled by a continuum.

Q1) Can one start from a microscopic theory and recover the Navier-Stokes equations in the appropriate limit? I know there are some attempts at this - but is there a consensus on the correct way to do it?

Q2) If one wishes to describe a relativistic fluid, can one make a continuum hypothesis ( since lengths are not Lorentz invariant) ? I know that relativistic fluid equations are used to describe e.g. neutron stars. Surely some inertial observers would not see a continuum, and would have to use a microscopic theory to describe the fluid... this seems like it could lead to inconsistencies?

Thanks.

[EDIT] Changed title from ...continuum hypothesis... to ...continuum approximation... .

 

[Snicker]

The term "continuum hypothesis" has a very specific and special meaning to mathematicians, so you should not use the term unless you are referring to set theory.

Anyway,

A1) You may view fluid dynamics as the statistical mechanics of fluids; however, it can not be derived as the statistical limiting case of an ensemble of classical particles. The statistics of classical mechanics only allows for the description of gasses. Therefore, unless you are very clever, it might be possible to derive the Navier-Stokes equation from quantum (actually chemical) statistics. However, this is not getting you any closer to a Millenium Prize.

A2) The field of relativistic fluid mechanics, to my knowledge, exists and is actually useful. You can define a continuous fluid in relatavistic space-time just as easily as you are able to describe a field (a fluid may be described by a velocity vector field and the scalar pressure field) produced by source densities.

 

[Andy Resnick]

<snip>
Q1) Can one start from a microscopic theory and recover the Navier-Stokes equations in the appropriate limit? I know there are some attempts at this - but is there a consensus on the correct way to do it?

The typical approach I have seen begins with the fluctuation-dissipation theorem, but then add thermodynamic potentials, which is a continuum field theory. This is treated well in Chaikin and Lubensky's "Principles of condensed matter physics" and Brenner and Edwards "Macrotransport processes".

Q2) If one wishes to describe a relativistic fluid, can one make a continuum hypothesis ( since lengths are not Lorentz invariant) ? I know that relativistic fluid equations are used to describe e.g. neutron stars. Surely some inertial observers would not see a continuum, and would have to use a microscopic theory to describe the fluid... this seems like it could lead to inconsistencies?

I need to get some newer references, all I have that you may like are:

Leaf, B." The continuum in special relativity theory", Phys Rev 84, 345 (1951)
Tolman's "Relativity, thermodynamics, and Cosmology" (Dover)