Pressure driven disturbances on a thin viscous film

Here we investigate the effect of a steady pressure disturbance moving at a constant velocity along the interface of a thin viscous film flowing down an inclined plane. Both steady and unsteady solutions are determined for the interface. We find that it is possible for the steady solutions to have surface waves, either preceding the disturbance or behind it, depending on the values of the Bond and capillary numbers. In addition, for fixed Bond number, disturbance amplitude, and constant film heights both far upstream and downstream, there exists a finite range of capillary numbers for which no steady solutions exist. Within this range of capillary numbers, the transient solution develops a shock like profile which adjusts the heights upstream and downstream of the disturbance and grows in width.