Bottom reconstruction in thin-film flow over topography: Steady solution and linear stability

Abstract

We consider viscous gravity-driven films flowing over undulated substrates. Instead of the widely studied direct problem of finding the free surface for a given bottom topography, we focus on the inverse problem: Given a specific free surface shape, we seek the corresponding bottom topography which causes this free surface profile. As an asymptotic approach for thin films and moderate Reynolds numbers, we apply the weighted-residual integral boundary-layer method which enables us to derive a set of two evolution equations for the film thickness and the flow rate. We prescribe the free surface as a monofrequent periodic function and discuss the influence of inertia, film thickness, and surface tension on the shape of the corresponding substrate. For small free surface undulations, we can solve the bottom contour analytically and study its parametric dependence. The analytical results are then validated with numerical simulations. Furthermore, we consider the stability of the corresponding direct problem, which reveals that the bottom topography is stabilizing or destabilizing, depending on surface tension.