Effect of long undulated bottoms on thin gravity-driven films

We carry out a perturbation analysis for steady gravity-driven film flow over undulations of moderate steepness that are long compared to the film thickness and study the linear stability of the flow in the framework of Floquet analysis. The effect of geometric nonlinearities on the instability becomes relevant for moderate bottom variations. We find that the critical Reynolds number for the onset of surface waves is higher than that for a flat bottom. At higher inclination angles, the theoretical results are in good quantitative agreement with experiment. At inclination angles where the flat part of the undulation is close to being horizontal, the basic solution for the steady flow fails to describe the flow in the flat part, and the linear stability analysis overestimates the critical Reynolds number.