Thin liquid layers supported by steady air-flow surface traction

Abstract

Upward flow of air can support a thin layer of liquid on a plane wall against gravity. Such apparently stationary layers are for example sometimes seen on the windscreen of a car travelling at high speed in rain. We solve here the two-dimensional case of a layer whose length is finite, but significantly greater than the meniscus length. The flow is steady, with a fixed layer boundary, inside which there is a steadily circulating viscous liquid, and outside which the air exerts a traction which is assumed to have a known (small) constant drag coefficient CD. The air also exerts a non-uniform pressure on the liquid layer, of a magnitude determined by the shape of the layer, and the relationship between these two quantities can be obtained by thin-airfoil theory. In the lubrication approximation, the problem can be reduced to a nonlinear singular integro-differential equation to determine the unknown shape of the layer boundary. This equation is solved numerically for various (small) wall angles, for cases where the effect of surface tension is confined to a small meniscus region near the layer's leading edge. The numerical results indicate that solutions exist only for walls whose inclination is less than 0.70 C½D, and, for a range of inclinations below that maximum value, that two distinct steady solutions can exist at each inclination.