Modeling Draining Flow in Mobile and Immobile Soap Films

Abstract

A mathematical model is constructed to describe the two-dimensional flow in a vertical soap film that is draining under gravity. An asymptotic analysis is employed that uses the long-wave or “lubrication” approximation. The modeling results in three coupled partial differential equations that include a number of dimensionless input parameters. The equations are solved numerically. The three functions calculated, as they vary in space and time, are the film thickness, the surface concentration of an assumed insoluble surfactant, and the slip or surface velocity. The film is assumed to be supported by “wire frame” elements at both the top and the bottom; thus the liquid area and the total surfactant are conserved in the simulation. A two-term “disjoining” pressure is included in the model that allows the development of thin, stable, i.e., “black,” films. While the model uses a simplified picture of the relevant physics, it appears to capture observed soap film shape evolution over a large range of surfactant concentrations. The model predicts that, depending on the amount of surfactant that is present, the film profile will pass through several distinct phases. These are (i) rapid initial draining with surfactant transport, (ii) slower draining with an almost immobile interface due to the surface tension gradient effect, and (iii) eventual formation of black spots at various locations on the film. This work is relevant to basic questions concerning surfactant efficacy, as well as to specific questions concerning film and foam draining due to gravity. Prospects for extension to three-dimensional soap film flows are also considered.