On the shapes of droplets that are sliding on a vertical wall

This paper presents a straightforward theoretical model for the slow motion of a viscous liquid drop on an inclined or vertical wall. The model simulates three-dimensional unsteady motion using the long-wave or lubrication approximation. The static contact angle, or wetting angle, between the liquid and the substrate is a given finite value. By using a “disjoining pressure” model, the time-dependent dynamic changes in the contact angle are predicted without the need for further modeling. For a small droplet on a vertical wall, all its properties, including its size, weight, surface tension, and contact angle, can be collapsed into a single dimensionless control parameter. A wide variety of drop shapes can be calculated by varying this parameter. Apparently all experimentally observed regimes can be reproduced. These include single sliding drops of various forms, an almost-periodic shedding of droplets from the main drop, and virtually chaotic motions. The detailed agreement with experiment provides strong evidence for the validity and usefulness of the present theory.