The Free-boundary Problem for Gravity-driven Unidirectional Viscous Flows

If a layer of viscous fluid drains steadily under gravity, whilein contact with an upward-moving vertical cylinder of a generalcross section with boundary C_B its velocity w(x, y) satisfiesa Poisson equation w_xx+w_yy = g/v = constant. There are two boundaries,the given curve C_B on which w takes a prescribed constant value,and a free boundary C_p marking the edge of the layer, whoseshape we wish to determine. In order to do this, we need twoboundary conditions on C_F; one is the zero-stress conditionw/n = 0, and we assume that the other is that w is constantaround C_F. The latter condition is shown to be equivalent toa requirement that the volume flux in the layer is maximal.The properties of the resulting free-boundary problem are discussed,and numerical solutions obtained for some special cases. Forexample, a solution for the case when C_B is a 270? corner showsthat the layer thickness at the corner is reduced to 76% ofthe thickness on the flat part of the boundary. The presentanalysis is relevant to industrial coating problems, such asedge effects in continuous galvanizing of sheet steel.