Existence theory for a kinetic-fluid coupling when small droplets are treated as part of the fluid

TL;DR

This paper develops an existence theory for a coupled kinetic-fluid model of spray dynamics, demonstrating that small droplets can be effectively treated as part of the fluid, with proven global weak solutions.

Contribution

It introduces a formal convergence from a kinetic-fluid model to a simplified fluid model when droplets are very small, and proves the existence of global weak solutions for this simplified system.

Findings

Solutions converge to a fluid model when droplets are very small

Global weak solutions exist for the simplified system

The DiPerna-Lions theory is used to establish existence

Abstract

We consider in this paper a spray constituted of an incompressible viscous gas and of small droplets which can breakup. This spray is modeled by the coupling (through a drag force term) of the incom- pressible Navier-Stokes equation and of the Vlasov-Boltzmann equation, together with a fragmentation kernel. We first show at the formal level that if the droplets are very small after the breakup, then the solutions of this system converge towards the solution of a simplified system in which the small droplets produced by the breakup are treated as part of the fluid. Then, existence of global weak solutions for this last system is shown to hold, thanks to the use of the DiPerna-Lions theory for singular transport equations.