Free-surface viscous flow over a depression

Abstract

The flow of viscous fluid from a line source on an inclined plane containing a topographic depression is studied using a lubrication model. The steady flow is perturbed by the depression. We focus on the regime in which the flow thickness is small relative to the lengthscale of the depression. For shallow depressions, the flow is only slightly perturbed. However, for deeper depressions a large pond of fluid with a horizontal free-surface is observed. This ponding of fluid is associated with the topography having uphill regions, which in the absence of inertia can only be surmounted by developing a deep fluid layer. The accumulation of fluid in the depression leads to a focusing of the flow, which is observed even far downstream of the depression.