Steady solution and spatial stability of gravity-driven thin-film flow: reconstruction of an uneven slippery bottom substrate

Abstract

Steady solutions of an inverse problem relevant to gravity-driven film flows over an undulated slippery bottom are considered. Given a target free surface shape, the goal is to obtain the corresponding bottom topography of a slippery substrate which causes the specified free surface shape for a film flowing over it. The approaches followed by Sellier (Phys Fluids 20:062106, 2008) for creeping films and by Heining and Aksel (Phys Fluids 21:083605, 2009) for inertial films for reconstructing a rigid bottom topography for a target free surface profile are extended to the reconstruction of a slippery bottom topography. The model equations for film thickness above the bottom topography are derived for creeping flows under lubrication approximation and for inertial films using the weighted-residual integral boundary layer method and are solved numerically. The influence of inertia, slip parameter and surface tension on the shape of the reconstructed bottom topography is analyzed for different prescribed free surface shapes (sinusoidal, trench and bell-shaped). It is observed that the nonlinearities that appear in the reconstructed rigid bottom substrate with no slip at the substrate are suppressed by seeking the bottom substrate to be reconstructed as a slippery substrate. A spatial linear stability analysis of the corresponding direct problem is examined using Floquet theory, and the results reveal that the slip parameter and surface tension have a high influence on the critical Reynolds number. The results provide a strategy for controlling surface defects in a gravity-driven film over a substrate; namely, in order to achieve a target free surface profile, one can design the bottom substrate to be an undulated rough/textured/grooved or a superhydrophobic surface which can be modelled as an undulated smooth substrate with velocity slip at the substrate.