  This article proposes a derivation of the Vlasov-Navier-Stokes system for
spray/aerosol flows. The distribution function of the dispersed phase is
governed by a Vlasov-equation, while the velocity field of the propellant
satisfies the Navier-Stokes equations for incompressible fluids. The dynamics
of the dispersed phase and of the propellant are coupled through the drag force
exerted by the propellant on the dispersed phase. We present a formal
derivation of this model from a multiphase Boltzmann system for a binary
gaseous mixture, involving the droplets/dust particles in the dispersed phase
as one species, and the gas molecules as the other species. Under suitable
assumptions on the collision kernels, we prove that the sequences of solutions
to the multiphase Boltzmann system converge to distributional solutions to the
Vlasov-Navier-Stokes equation in some appropriate distinguished scaling limit.
Specifically, we assume (a) that the mass ratio of the gas molecules to the
dust particles/droplets is small, (b) that the thermal speed of the dust
particles/droplets is much smaller than that of the gas molecules and (c) that
the mass density of the gas and of the dispersed phase are of the same order of
magnitude.
