Fluid accumulation in thin-film flows driven by surface tension and gravity

In this note we derive a model describing the two-dimensional viscous flow driven by surface tension and gravity of a thin liquid film near a stagnation point. In the thin-film approximation of such a flow, accumulation takes place where the combined effects of gravity and surface tension stop the flow. These stagnation points are characterised to leading order by the geometry of the substrate. We first derive the thin-film approximation that describes the flow away from such accumulation regions. Then, assuming the existence of isolated stagnation points, we derive the boundary layer equation describing the inner structure of solutions describing accumulation. The existence of these solutions has been proved by the authors elsewhere. Finally, in order to justify the model we prove the existence of curves that give a substrate with an isolated stagnation point.