On the dependence of the Navier Stokes equations on the distribution of moleular velocities

TL;DR

This paper develops a general Chapman Enskog method to derive hydrodynamic equations from arbitrary local velocity distributions, revealing that the incompressible limit is distribution-independent while the compressible case includes an extra heat flux term.

Contribution

It introduces a recursive Chapman Enskog procedure for arbitrary distributions, generalizing the derivation of Navier-Stokes equations and identifying an additional heat flux term in the compressible limit.

Findings

Incompressible equations are distribution-independent.

Compressible energy equation includes an extra heat flux term.

The extra term aligns with previous findings in Tsallis thermostatistics.

Abstract

In this work we introduce a completely general Chapman Enskog procedure in which we divide the local distribution into an isotropic distribution with anisotropic corrections. We obtain a recursion relation on all integrals of the distribution function required in the derivation of the moment equations. We obtain the hydrodynamic equations in terms only of the first few moments of the isotropic part of an arbitrary local distribution function. The incompressible limit of the equations is completely independent of the form of the isotropic part of the distribution, whereas the energy equation in the compressible case contains an additional contribution to the heat flux. This additional term was also found by Boghosian and by Potiguar and Costa in the derivation of the Navier Stokes equations for Tsallis thermostatistics, and is the only additional term allowed by the Curie principle.