Shock Solutions for Particle-Laden Thin Films

Abstract

We derive a lubrication model describing gravity-driven thin film flow of a suspension of heavy particles in viscous fluid. The main features of this continuum model are an effective mixture viscosity and a particle settling velocity, both depending on particle concentration. The resulting equations form a $2 \times 2$ system of conservation laws in the film thickness $h(x,t)$ and in $\phi h$, where $\phi(x,t)$ is the particle volume fraction. We study flows in one dimension under the constant flux boundary condition, which corresponds to the classical Riemann problem, and we find the system can have either double-shock or singular shock solutions. We present the details of both solutions and examine the effects of the particle settling model and of the microscopic length scale b at the contact line.