Fluids with Multivalued Internal Energy: The Anisotropic Case

Abstract

This work is concerned with some extensions of the classical compressible Euler model of fluid dynamics in which the fluid internal energy is a measure-valued quantity. A first extension has been derived from the hydrodynamil limit of a kinetic model involving a specific class of collision operators [1,3]. In these papers the collision operator simply describes the isotropization of the kinetic distribution function about some averaging velocity. In the present work we present a new extension of such models in which the relaxed distribution is anisotropic. Similarly to [1] and [3] this model is derived from a kinetic equation with a collision operator that relaxes to anisotropic equilibria. We then investigate diffusive corrections of this fluid dynamical model using Chapman-Enskog techniques and show how the anisotropic character affects the expression of the viscosity and of the heat flux. We argue why such a feature could be used as a tool towards an understanding of fluid turbulence from kinetic theory.