Lubrication models for particle-laden thin films.

摘要

We derive 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 x 2 system of conservation laws in the film thickness h(x, t) and in &phis;h, where &phis;( 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. The double-shock solutions correspond to a particle-rich ridge that has been observed in such films. 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. The instability of the contact line is also examined through linear stability analysis, and it is found that the particle-rich ridge makes the contact line somewhat less unstable, while shifting the most unstable mode to a longer wavelength. A separate model is introduced which considers the balance between shear-induced migration and particle settling due to gravity. This model allows the prediction of the qualitative features of the flow, i.e. whether particles accumulate in a ridge or settle to the substrate, in terms of the inclination angle (alpha and bulk concentrations &phis;0, and this prediction is compared with published experimental data.