Instability of a liquid film flowing down an inclined wavy plane

A linear stability analysis of a Newtonian liquid film flowing down an inclined wavy plane is carried out. It is studied how wavy bottom variations, which are long compared to the film thickness, modify the stability of the steady film flow with respect to that down a flat inclined plane. By allowing for rather moderate bottom variations, it shows the impact of geometric nonlinearities on the instability. In this case, the spatial growth of disturbances becomes dependent on the phase along the bottom wave. Averaging over the bottom variations, it is found that on a large scale the critical Reynolds number for the onset of surface waves is higher than that for a flat bottom. As in the case of a flat bottom, the instability occurs at long wavelength. Locally, however, at the steep slopes the critical Reynolds number is lower than for a flat incline. In a certain range of waves numbers and Reynolds numbers, shorter waves may be excited at the steep slopes and damped at the flat ones.