Stagnation-point flow against a liquid film on a plane wall

Abstract

Stagnation-point flow of a semi-infinite viscous fluid against a liquid film resting on a plane wall is considered under conditions of Stokes flow and for arbitrary Reynolds numbers. Assuming that the interface remains flat and parallel to the wall at all times, the lateral spatial coordinate is scaled out to yield a simplified system of governing equations in the transverse coordinate normal to the wall, describing the evolution of the flow and displacement of the interface. In this inherently unsteady flow the film keeps thinning in time, while the rate of thinning decreases as the interface approaches the wall. Orthogonal two-dimensional, axisymmetric, three-dimensional, and oblique two-dimensional flow are individually considered. In each case, exact solutions of a similarity form are constructed, and an evolution equation describing the film thickness is formulated and solved by numerical methods. Quasi-steady solutions compare favourably with full calculations of the unsteady flow, suggesting that the unsteady terms have only a minor effect on the rate of film thinning. Dual solutions are uncovered in the case of three-dimensional flow.