Kinetic Theory of Inelastic Hard Spheres

Abstract

The present chapter provides a concise introduction to the kinetic theory of granular gases (namely, gases of hard spheres with inelastic collisions) at low and moderate densities. We briefly review first the dynamics of binary collisions for some of the models most widely used in the literature, and then we outline heuristically the derivation of the Boltzmann and Enskog kinetic equations for monocomponent granular gases. A connection with hydrodynamics is established where the corresponding macroscopic balance equations for the densities of mass, momentum and energy are exactly derived from the above kinetic equations with expressions for the momentum and heat fluxes and the cooling rate as functionals of the one-particle velocity distribution function. These kinetic equations are then extended to the interesting case of multicomponent granular mixtures. The complexity of the kernel of the Enskog-Boltzmann collision operator, however, prevents the possibility of obtaining exact results, and for this reason there is often a preference for kinetic models that are mathematically simpler than the original equations but capture their most relevant physical properties. Thus, the chapter ends with the construction of some kinetic models proposed in the literature of granular gases based on the popular BGK model for ordinary (elastic) gases.