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.
Bibliographical noteFunding Information:
The authors would like to thank Dirk Peschka and Roman Taranets for fruitful discussions and Sarah Burnett, Jesse Kreger, Hanna Kristensen, and Andrew Stocker for their experimental work. This work is funded by NSF grants DMS-1312543 and DMS-10455536, and in part by Simons Math +X award #510776.
- lubrication model
- surface tension
- thin films