Kinetic description for a suspension of inelastic spheres—Boltzmann and BGK equations

Abstract

The problem of a two-phase dispersed medium is studied within the kinetic theory. In the case of small and undeformable spheres that have all the same radius, a model for the collision operator of the Boltzmann equation is proposed. The collisions are supposed instantaneous, binary and inelastic. The obtained collision operator allows, to prove the existence of an H theorem in several configurations according to the assumptions made about the particles and particularly in the case of a diluted suspension of weakly inelastic collisions. Because of the complexity of the non linear structure of the collision integral, the Boltzmann equation is very difficult to solve and to analyze. It is therefore interesting to introduce a BGK model equation to study qualitatively its solution. In order to be physically realistic and consistent with the assumptions related to the collisions, a collision frequency depending on the particles velocities is chosen. Moreover, the collision frequency is expected to vary strongly with the particles velocities. Taking this into account, a BGK model is written. All the basic properties of the original operator are retained.