THIN LIQUID LAYERS SUPPORTED BY STEADY AIR-FLOW SURFACE TRACTION

Upward flow of air can support a thin layer of liquid on a plane wall against gravity. Such apparently stationary layers are for example som etimes seen on the windscreen of a car travelling at high speed in rai n. We solve here the two-dimensional case of a layer whose length is f inite, 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 tracti on which is assumed to have a known (small) constant drag coefficient C(D). The air also exerts a non-uniform pressure on the liquid layer, of a magnitude determined by the shape of the layer, and the relations hip between these two quantities can be obtained by thin-airfoil theor y. In the lubrication approximation, the problem can be reduced to a n onlinear singular integro-differential equation to determine the unkno wn shape of the layer boundary. This equation is solved numerically fo r various (small) wall angles, for cases where the effect of surface t ension is confined to a small meniscus region near the layer's leading edge. The numerical results indicate that solutions exist only for wa lls whose inclination is less than 0.70 C(D)1/2, and, for a range of i nclinations below that maximum value, that two distinct steady solutio ns can exist at each inclination.