Effect of Concentration-dependent Disjoining Pressure on Vertical Draining Flow

Abstract

We construct a mathematical model to describe the flow in a vertical soap film that is draining under gravity, assuming that the film is supported by two wire frame structures at both the top and bottom. A two-term surfactant concentration-dependent disjoining pressure is included in the model allowing the development of thin, stable and rigid film. The modeling results in three coupled partial differential equations including the film thickness, the surfactant concentration, and the slip or surface velocity by lubrication theory, which are solved numerically by using FREEFEM program. The evolution processes of the thin film are investigated numerically under effect of concentration-dependent disjoining pressure as well as concentration-independent disjoining pressure for compared. More over, the effect of two factors, namely the attraction strength coefficient α1 and repulsion strength coefficient α2 in the concentration-dependent disjoining pressure function are investigated thoroughly. Results show that the evolution time varies with α1 and α2. Reducing α1 promotes the draining process of flow, while decreasing α2 impedes the evolution process.