We are interested in coupled microscopic/macroscopic models describing the evolution of particles dispersed in a fluid. The system consists in a Vlasov-Fokker-Planck equation to describe the microscopic motion of the particles coupled to the Euler equations for a compressible fluid. We investigate dissipative quantities, equilibria and their stability properties and the role of external forces. We also study some asymptotic problems, their equilibria and stability and the derivation of macroscopic two-phase models.

The paper is devoted to the analysis of a hydrodynamic limit for the Vlasov-Navier-Stokes equations. This system is intended to model the evolution of particles interacting with a fluid. The coupling arises from the force terms. The limit problem consists of an advectiondiffusion equation for the macroscopic density of the particles, the drift velocity being solution of the incompressible Navier-Stokes equation.

The paper is devoted to the analysis of a hydrodynamic limit for the Vlasov-Navier-Stokes equations.This system is intended to model the evolution of particles interacting with a fluid. The coupling arises from the force terms. The limit problem is the Navier-Stokes system with non constant density. The density which is involved in this system is the sum of the (constant) density of the fluid and of the macroscopic density of the particles. The proof relies on a relative entropy method.