European Journal of Applied Mathematics



On a slender dry patch in a liquid film draining under gravity down an inclined plane


S. K. WILSON a1, B. R. DUFFY a1 and S. H. DAVIS a2
a1 Department of Mathematics, University of Strathclyde, Livingstone Tower, 26 Richmond Street, Glasgow G1 1XH, UK (email: s.k.wilson@strath.ac.uk)
a2 Department of Engineering Sciences and Applied Mathematics, Robert R. McCormick School of Engineering and Applied Science, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208, USA

Abstract

In this paper two similarity solutions describing a steady, slender, symmetric dry patch in an infinitely wide liquid film draining under gravity down an inclined plane are obtained. The first solution, which predicts that the dry patch has a parabolic shape and that the transverse profile of the free surface always has a monotonically increasing shape, is appropriate for weak surface-tension effects and far from the apex of the dry patch. The second solution, which predicts that the dry patch has a quartic shape and that the transverse profile of the free surface has a capillary ridge near the contact line and decays in an oscillatory manner far from it, is appropriate for strong surface-tension effects (in particular, when the plane is nearly vertical) and near (but not too close) to the apex of the dry patch. With the average volume flux per unit width (or equivalently with the uniform height of the layer far from the dry patch) prescribed, both solutions contain a free parameter. For each value of this parameter there is a unique solution in the first case and either no solution or a one-parameter family of solutions in the second case. The solutions capture some of the qualitative features observed in experiments.

(Received October 9 1998)
(Revised February 21 2000)