Two-dimensional two-layer channel flow near a step

The influence of surface topography on channel flow of immiscible liquids is considered in this paper. Using a diffuse-interface method, we perform two-dimensional numerical simulations of two-layer flow in a channel containing a step. Our simulations indicate that a much simpler model based on lubrication theory predicts well the interface shape provided that the capillary number is small. Inertial effects are also considered. When the inertia of the top layer becomes sufficiently large, the interface shape deviates significantly relative to the case where the inertia of the top layer is negligible. For low to moderate Reynolds numbers, the interface shapes are very similar to those in the creeping-flow limit, indicating that the lubrication model can still be used provided that the capillary number is small. The results presented here provide guidance on when it is appropriate to use lubrication-theory-based models to describe multilayer flow of immiscible liquids in channels containing topographical features.