Viscous flow down a slope in the vicinity of a contact line

Abstract

The two-dimensional base state that ultimately exhibits a contact-line-related instability is investigated by a combination of analysis and numerical solution. Globally, these flows may be characterized as having a down-slope length scale that is much greater than the length scale normal to the slope. This allows a solution by matched expansions, where the outer solution is governed by simple kinematics and describable by lubrication theory. In order to complete the solution, the flow in an inner region near the contact line must be determined. It is shown that it is not possible to model the flow in the inner region using the lubrication approximations while also requiring both that (1) the free surface meets the plane at the contact line and (2) the rate of change of the curvature of the free surface be bounded at the contact line. However, the full two-dimensional Stokes equations allow the inner solution to be determined. A boundary integral technique is used to obtain numerical solutions to the problem in which a contact angle boundary condition is imposed. Solutions are reported for a wide range of the three relevant parameters: the capillary number, the contact angle, and the inclination angle. The computations reveal that the free surface develops a hump near the contact line as a result of a stress field that results from kinematic considerations. A scaling analysis of the numerical results verifies that, except possibly at very small contact angles, the lubrication approximations are inappropriate near the contact line.