A Derivation of the Vlasov-Stokes System for Aerosol Flows from the   Kinetic Theory of Binary Gas Mixtures
  In this short paper, we formally derive the thin spray equation for a steady Stokes gas, i.e. the equation consists in a coupling between a kinetic (Vlasov type) equation for the dispersed phase and a (steady) Stokes equation for the gas. Our starting point is a system of Boltzmann equations for a binary gas mixture. The derivation follows the procedure already outlined in [Bernard-Desvillettes-Golse-Ricci, arXiv:1608.00422 [math.AP]] where the evolution of the gas is governed by the Navier-Stokes equation. 
Etienne Bernard, Laurent Desvillettes, Fran\c{c}ois Golse, and Valeria   Ricci
math.AP