Flow of a plane film of viscous incompressible liquid down a dry wall

Abstract

A study is made of the flow regimes of a plane film of viscous liquid whose leading edge moves down a dry vertical wall with constant velocity, The study is based on a nonlinear differential equation obtained by means of integral relations that describes the behavior of the surface of a film of viscous incompressible liquid with allowance for surface tension. It is shown that for fixed physical properties of the liquid flow with a steadily moving leading edge is possible only for a strictly determined flow rate of the liquid reaching the plate at a point infinitely far from the leading edge. The dimensionless flow rate of the liquid is found as a function of the dimensionless parameter γ that characterizes the physical properties of the liquid. It is shown that in addition to monotonic behavior of the free surface of the film along its length it is also possible to have regimes in which the free surface is wavelike near the leading edge, becoming asymptotically straight far from it.