The nonlinear dynamics of pendent drops on a thin film coating the underside of a ceiling

Abstract

This paper considers the dynamics of a thin film of viscous liquid of density ρ coating the underside of a horizontal rigid boundary under the action of surface tension σ and gravity g, and in the lubrication limit. Gravitational instability for inverse wavenumbers larger than the capillary length ℓ = (σ/ρg)1/2) leads to the formation of quasi-static pendent drops of radius ≈3.83ℓ. If the boundary conditions are such as to pin the positions of the drops then the drops slowly drain fluid from the regions between them through thin annular trenches around each drop. A similarity solution is derived and verified numerically in which the film thickness in the intervening regions scales like t−1/4 and that in the trenches like t−1/2. A single drop placed far from boundaries on an otherwise uniform film, and given an initial perturbation, undergoes self-induced quasi-steady translation during which it grows slowly in amplitude by leaving a wake where the film thickness is reduced by an average of 90. It is driven by release of gravitational potential energy as fluid is collected from the film into the lower lying drop. Analysis of Landau–Levich regions around the perimeter of the translating drop predicts its speed and the profile of the wake. Two translating drops may coalesce if they collide, in contrast with the non-coalescence of colliding collars in the analogous one-dimensional problem (Lister et al., J. Fluid Mech. vol. 552, 2006b, p. 311). Colliding drops may also bounce off each other, the outcome depending on the angle of incidence through complex interactions between their surrounding capillary wave fields.