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# Communications in Mathematical Sciences

## Volume 15 (2017)

### Number 6

### A derivation of the Vlasov–Navier–Stokes model for aerosol flows from kinetic theory

Pages: 1703 – 1741

DOI: https://dx.doi.org/10.4310/CMS.2017.v15.n6.a11

#### Authors

#### Abstract

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. The class of kernels modelling the interaction between the dispersed phase and the gas includes, among others, elastic collisions and inelastic collisions of the type introduced in [F. Charles: in “Proceedings of the 26th International Symposium on Rarefied Gas Dynamics”, *AIP Conf. Proc.* 1084:409–414, 2008].

#### Keywords

Vlasov–Navier–Stokes system, Boltzmann equation, hydrodynamic limit, aerosols, sprays, gas mixture

#### 2010 Mathematics Subject Classification

Primary 35B25, 35Q20. Secondary 76D05, 76T15, 82C40.

Received 6 August 2016

Published 27 June 2017