A Theory For Particle Settling and Shear-Induced Migration in Thin Film Flow

Experiments of particle-laden inclined film flow [Zhou, Dupuy, Bertozzi, and Hosoi, Phys. Rev. Lett. 94 (2005)] have displayed different settling behaviors depending on the particle concentration $\phi$ and angle of inclination $\theta$, in which particles accumulate on the substrate or near the advancing contact line, or remain mixed. Zhou et al. presented a lubrication model that captures the qualitative behavior of the high-$\phi$, high-$\theta$ regime, characterized by a particle-rich ridge near the contact line, but cannot explain the other observed settling behaviors. This work presents a model in which $\phi$ varies through the film depth, unlike Zhou et al.'s model. Average velocities for the liquid and particulate phases are computed, and the implications for phase separation are discussed. It is found that the equilibrium depth profile of $\phi$ is more important than gravitational settling in the down-slope direction in determining phase separation. The predicted settling behavior is directly compared with Zhou et al.'s experimental data.