Cuboid drop: A low-dimensional model of drop dynamics on a substrate

Liquid drops placed on substrates may vibrate, slide, take history-dependent shapes, and even detach or break up when subjected to external forces. Although CFD tools can nowadays reproduce these motions, they are affordable for at most a few drops at a time. This work provides a low-dimensional model of such drops, in which the drop shape is approximated by a rectangular cuboid. The model results in three ordinary differential equations per drop. It is sufficiently simple to allow closed-form solutions in a variety of configurations. By systematically comparing the cuboid predictions to experimental, numerical, and theoretical results previously obtained with real drops, we discuss the extent to which the cuboid approximation reproduces the order of magnitude and qualitative dependence on parameters of a series of phenomena. The latter include natural drop vibrations, the detachment of pendant drops, multiple drop shapes allowed by contact angle hysteresis, the nonlinear retraction of drops after spreading, the sliding or climbing of drops on inclined substrates, and drop-induced damping of substrate vibrations. The cuboid model therefore provides a low-cost representation of drop motion on a substrate in response to arbitrarily complex external forcing.