We are interested in models describing the evolution of particles (such as solid dust particles or droplets) in a rarefied gas. We present here a derivation of a fluid-kinetic system in which the gas and the disperse phase are coupled not only by the drag force, but also by the exchange of temperature between the gas and the droplets/dust specks.
We start from a kinetic collisional model, based on the hypothesis that collisions between dust particles and gas molecules are inelastic and are given by a diffuse reflection mechanism on the surface of dust particles. We propose a model which preserves the total energy by introducing a new variable in the density function of macroscopic particles. We then derive an asymptotics to a Vlasov-Euler equation of compressible fluid when two small parameters tend to zero, namely the ratio of masses between gas molecules and dust particles on the one hand, and the Knudsen number of the gas on the other hand. We thus obtain an explicit expression of the transfer of energy between the phases.