A thin drop sliding down an inclined plate

Abstract

We examine two- and three-dimensional drops steadily sliding down an inclined plate. The contact line of the drop is governed by a model based on the Navier-slip boundary condition and a prescribed value for the contact angle. The drop is thin, so the lubrication approximation can be used. In the three-dimensional case, we also assume that the drop is sufficiently small (its size is smaller than the capillary scale). These assumptions enable us to determine the shape of the drop and derive an asymptotic expression for its velocity. For three-dimensional drops, this expression is matched to a qualitative estimate of Kim et al. (J. Colloid Interface Sci., vol. 247, 2002, pp. 372–380) obtained for arbitrary drops, i.e. not necessarily thin and small. The matching fixes an undetermined coefficient in Kim, Lee and Kang’s estimate, turning it into a quantitative result.