Sedimentation of a two-dimensional drop towards a rigid horizontal plane

Abstract

We consider the drainage of fluid trapped beneath a two-dimensional drop that sediments towards a horizontal plane. The governing equation is closely related to that for capillary drainage (without gravity) of an annular film discussed in a companion paper (Lister et al., J. Fluid Mech. vol. 552, 2006, p. 311). When drainage starts, dynamical structures rapidly appear that are usually called dimples in the context of sedimentation. Dimples are constant-pressure regions to which most of the fluid in the film is confined, which are analogous to the collars and lobes that appear in annular capillary drainage.

The process of drainage is controlled by a Bond number, $B$, that measures the relative importance of gravity and surface tension for the sedimenting drop. When $B$ is sufficiently small, all the fluid ultimately drains from a single small dimple and the drop takes a static sessile equilibrium shape. The dimple-drainage process is the same as that of a lobe. When $B$ is sufficiently large, several permanent dimples are formed under the drop, and these exhibit complex dynamics of collision and interaction analogous to that of collars and lobes. No static drop shape is reached, even for long times. For critical values of $B$, fluid may be permanently trapped in one or more stationary dimples (analogous to collars), and families of equilibrium drop shapes are found that depend upon the quantity of trapped fluid.