A Derivation of the Vlasov-Navier-Stokes Model for Aerosol Flows from Kinetic Theory

TL;DR

This paper derives a coupled Vlasov-Navier-Stokes model for aerosol flows from kinetic theory, showing convergence from a multiphase Boltzmann system under specific assumptions about particle and gas properties.

Contribution

It provides a formal derivation and rigorous convergence proof of the Vlasov-Navier-Stokes system from a multiphase Boltzmann model for aerosol flows.

Findings

Convergence of Boltzmann solutions to Vlasov-Navier-Stokes solutions under scaling limits.

Derivation of a coupled kinetic-fluid model for spray/aerosol flows.

Conditions on particle-to-gas mass ratio and thermal speeds for model validity.

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.