On Instabilities in Thin-Film Flows

Abstract

Thin-film instabilities can appear in several applications, e.g. the formation of drops on a non-wetting substrate, the bursting of bubbles, and the formation of liquid fingers in a film spreading under the effect of gravity are a few examples. In this work, we investigate the dynamics of a thin film undergoing an instability in three different settings.

First, we study the unsteady flow of an inviscid bubble displacing a non-wetting viscous fluid in a tube. The film deposited in the annular region between the bubble interface and the tube walls is considered thin enough, such that intermolecular forces in the form of van der Waals attractions are significant, and thin-film rupture is possible. We obtain, both numerically and analytically, the time-scale over which rupture occurs. This leads to a critical capillary number below which the film is expected to rupture.

Our second study focuses on the effects of curvature on the classical Rayleigh-Taylor instability. We analyze the dynamics of a thin liquid film on the underside of a curved substrate, namely a cylinder and a sphere. We classify the dynamics into three different flow regimes: a stable film, a film that undergoes an initial growth of perturbations, followed by decay, and an unstable film. We show that below a critical value of a modified Bond number, all small initial perturbations decay. However, once this critical value is exceeded, linear stability predicts an initial growth before the eventual decay. Our asymptotic analysis is confirmed with both numerics and experiments for the case of a cylinder.

Lastly, we investigate the dynamics of a thin film on a curved substrate using an embedding numerical scheme, called the Closest Point Method. This technique allows for arbitrary geometries, without any restrictions on topology. We validate the method against analytical and numerical results for a film under a cylinder. We then use it to study the spreading of a thin film undergoing a fingering instability on a sphere. Our numerical results are found to be in agreement with experiments.