Surface tension effects for particle settling and resuspension in viscous thin films

We consider flow of a thin film on an incline with negatively buoyant particles. We derive a one-dimensional lubrication model, including the effect of surface tension, which is a nontrivial extension of a previous model (Murisic et al 2013 J. Fluid Mech. 717 203-31). We show that the surface tension, in the form of high order derivatives, not only regularizes the previous model as a high order diffusion, but also modifies the fluxes. As a result, it leads to a different stratification in the particle concentration along the direction perpendicular to the motion of the fluid mixture. The resulting equations are of mixed hyperbolic-parabolic type and different from the well-known lubrication theory for a clear fluid or fluid with surfactant. To study the system numerically, we formulate a semi-implicit scheme that is able to preserve the particle maximum packing fraction. We show extensive numerical results for this model including a qualitative comparison with two-dimensional laboratory experiments.