Kinetic modelling of a heterogeneous dispersed medium

Two-phase dispersed media appear in a wide range of industrial processes and in many problems of pollution. Here, we are interested in the kinetic description for a suspension with two (or more) species of solid particles, the carrier fluid being a viscous gas. Attention is focused on the suspensions of solid spheres of different species. Boltzmann equations for species of hard spheres having binary, inelastic and non-punctual collisions are written. For this purpose, the inelastic collision of two particles is described and the "pseudo-inverse collision" is clearly introduced [1,2]. By considering the asymptotic case where the diameters of the particles are very small in front of the macroscopic length, simple expressions for the collision operators are given. Taylor expansions allow one to separate the collision term contributions in the global balance equations in a flux term and an in source term. In the momentum equation, the source term is zero and the contribution of the collisions may be written as a divergence term like in continuum mechanics.