Sliding, slipping and rolling: the sedimentation of a viscous drop down a gently inclined plane

Abstract

We consider the steady sedimentation under gravity of a viscous drop, suspended in a viscous liquid, along a plane tilted at a small angle $\alpha$ to the horizontal. The drop does not wet the wall but is supported by a thin lubricating film of liquid. In the Stokes-flow limit, the problem is parameterized by $\alpha$, the ratio $B$ of buoyancy to capillary forces (a Bond number) and a viscosity ratio $\lambda$. Provided $B$ is not too large ($B\,{\ll}\,\alpha^{-1}$ in two dimensions, $B\,{\ll}\,\alpha^{-4/3}$ in three dimensions), the drop's motion can be described asymptotically by combining a capillary-statics approximation for the drop shape away from the wall, lubrication theory for the thin film and a combination of lubrication theory and a half-plane boundary-integral method for the drop interior.

Systematic scaling arguments for both two- and three-dimensional drops, supported by detailed calculations, are used to survey $(B,\lambda)$-parameter space for fixed $\alpha\,{\ll}\,1$. We find a strong coupling between drop shape (ranging from nearly round to a flattened pancake), kinematics (including slipping, sliding, rolling and tank-treading motions) and the site of dominant viscous dissipation (the edges of the thin film, the bulk of the thin film or the drop interior). Predictions of drop speed and shape are compared with available experimental and computational data.