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

We consider the steady sedimentation under gravity of a viscous drop, suspended in a viscous liquid, along a plane tilted at a small angle . 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.