Viscous sheets advancing over dry beds

The unsteady creeping motion of a thin sheet of viscous liquid is considered as it advances over a gently sloping dry bed. Attention is focused on the motion of the leading edge under various influences and four problems are discussed. In the first problem the fluid is travelling down an open channel of circular cross-section and the time-dependent shape of the leading-edge is calculated. In the second problem a sheet with a straight leading edge travelling over an inclined plane penetrates a region where the bed is uneven, and the subsequent deformation of the leading edge is followed. The third problem considers the flow down a channel formed by two straight parallel retaining walls placed perpendicular to an inclined plane. When the channel axis is parallel to the fall line, the steady solution corresponds to a straight leading edge. The deviation from this due to inclination of the channel axis is calculated. The fourth problem is that of flow down an inclined plane with a single curved retaining wall. The shape of the leading edge is calculated as it fluctuates due to the wall deformation. These problems are all analysed by assuming that a length characteristic of the geometry is large compared to the fluid depth divided by the bed slope