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 was derived from the hydrodynamic limit of a kinetic model involving a specific class of collision operators typical from quasi-linear plasma theory (see Eur. J. Mech. B Fluids 20 (2001) 303-327, and Contin. Mech. Thermodyn. 10 (1998) 153-178). In these papers the collision operator simply describes the isotropization of the kinetic distribution function about some averaging velocity. In the present work we introduce a new extension of such models in which the relaxed distribution is anisotropic. Similarly to (Eur. J. Mech. B Fluids 20 (2001) 303-327) and (Contin. Mech. Thermodyn. 10 (1998) 153-178) 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. © 2003 Éditions scientifiques et médicales Elsevier SAS. All rights reserved.
Degond, P., Lemou, M., & López, J. L. (2003). A kinetic description of anisotropic fluids with multivalued internal energy. European Journal of Mechanics, B/Fluids, 22(5), 487–509. https://doi.org/10.1016/S0997-7546(03)00060-8