Modeling draining flow in mobile and immobile soap films

A mathematical model is constructed to describe the two-dimensional how in a vertical soap film that is draining under gravity. An asymptotic analysis is employed that uses the long-wave or "lubrication" approximation. The mo deling results in three coupled partial differential equations that include a number of dimensionless input parameters. The equations are solved numer ically. The three functions calculated, as they vary in space and time, are the film thickness, the surface concentration of an assumed insoluble surf actant, and the slip or surface velocity. The film is assumed to be support ed 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 pi cture of the relevant physics, it appears to capture observed soap film sha pe evolution over a large range of surfactant concentrations. The model pre dicts that, depending on the amount of surfactant that is present, the film profile will pass through several distinct phases. These are (i) rapid ini tial draining with surfactant transport, (ii) slower draining with an almos t immobile interface due to the surface tension gradient effect, and (iii) eventual formation of black spots at various locations on the film. This wo rk is relevant to basic questions concerning surfactant efficacy, as well a s to specific questions concerning film and foam draining due to gravity. P rospects for extension to three-dimensional soap film hows are also conside red. (C) 1999 Academic Press.