A 2+1 dimensional insoluble surfactant model for a vertical draining free film

A 2+1-dimensional mathematical model is constructed to study the evolution of a vertically-oriented thin, free liquid film draining under gravity when there is an insoluble surfactant, with finite surface viscosity, on its free surface. Lubrication theory for this free film results in four coupled nonlinear partial differential equations (PDEs) describing the free surface shape, the surface velocities and the surfactant transport, at leading order. Numerical experiments are performed to understand the stability of the system to perturbations across the film. In the limit of large surface viscosities, the evolution of the free surface is that of a rigid film. In addition, these large surface viscosities act as stabilizing factors due to their energy dissipating effect. An instability is seen for the mobile case; this is caused by a competition between gravity and the Marangoni effect. The behavior observed from this model qualitatively matches some structures observed in draining film experiments.