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 × 2 system of conservation laws in the film thickness h(x, t) and in φh, where φ(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.

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Particles suspended in a film flow can either settle out of the flow, remain well mixed, or even advance faster than the fluid, accumulating at the moving contact line. Recent experiments by Zhou et al. [Phys. Rev. Lett. 94, 117803 (2005)] have demonstrated that these three settling behaviors can be achieved by control of the average particle concentration phi and inclination angle theta . This work presents a theory for determining the settling behavior in the Stokes regime by calculating the depth profile of phi and the depth-averaged velocities of the liquid and particle phases. It is found that shear-induced particle fluxes can lead to an inversely stratified flow, in which the particles move on average faster than the liquid. The theory is directly compared to Zhou et al.'s experimental data, and the implications of stratification for lubrication-type models are also discussed.

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