Mathematical Modeling of Fluids from the Kinetic Theory

A synthesis of mathematical fluid models derived from the kinetic theory of gases is presented. Kinetic theory yields well structured set of partial equations with entropies compatible with convection, diffusion and sources. It also yields properly structured nonlinear source terms as well as the transport coefficients. We first address multicomponent flow models including detailed transport and chemical reactions. Symmetrization, hyperbolic-parabolic structure and asymptotic stability of constant equilibrium states are established. We next investigate relaxation issues for a two temperature model that leads to symmetrized hyperbolic-parabolic systems with stiff source terms. Local existence of solutions is established and the apparition of the bulk viscosity coefficient is justified in the fast relaxation limit. We finally consider nonideal fluids and their cohesive properties that lead to diffuse interfaces models with capillary effects. Using an augmented formulation—with density gradient added as an extra unknown—a new type of hyperbolic-parabolic-dispersive structure is obtained with antisymmetric second order coupling terms arising from capillarity. Local existence of solutions is established as well as asymptotic stability of constant states.