Kinetic approach to granular matter in a medium

A kinetic model for a granular gas of particles inelastically scattering between themselves, and interacting simultaneously with a given background medium by conservative binary encounters is analysed in the collision dominated regime (small elastic mean free path). For the hard sphere collision model, the problem of a consistent derivation of macroscopic equations for the fundamental observables (density, mass velocity, and granular temperature) is addressed, for varying mass ratio and inelasticity parameter. Closure is achieved by approximating the distribution function in the appropriate weak forms of the kinetic equation by two suitable expansions around the equilibrium of the dominant operator, one of Grad’s type, the other of local Maxwellian type. In the hydrodynamic limit when the mean free path tends to zero, the same drift–diffusion equation at the Navier–Stokes level is recovered in the two cases for the only hydrodynamic variable of the physical problem.