A similarity solution for slow viscous flow down an inclined plane

Abstract

A similarity solution is found for the distribution of layer thickness in viscous source flow down an inclined plane. With this soldion a t lowest order, an asymptotic expansion in inverse powers of the downstream co-ordinate x allows a small correction for upstream influence. A simple experiment confirms the major features of the similarity solution: (i) a parabolic cross-stream variation in the layer thickness; (ii) spreading of the flow according to an ; and (iv) surface velocities which vary as the square of the layer thickness. Deviations of the layer-thickness measurements from the parabolic profile follow the trend predicted by the first-order corrections, whereas systematically high measured values are explained qualitatively in terms of waves at the free surface.