In this thesis we develop theory for an experiment done by Snow and coworkers at Dow Corning that involves a vertically-oriented, thinned polyurethane film with silicone surfactant, draining under gravity.

We present the mathematical formulation for a 1+1- and 2+1-dimensional model to study the evolution of a vertically-oriented thin liquid film draining under gravity when there is an insoluble surfactant with finite surface viscosity on its free surface. This formulation has all the ingredients that include: surface tension, gravity, surface viscosity, the Marangoni effect, convective and diffusive surfactant transport; essential to describe the behavior of a vertical draining film with surfactant.

We study a hierarchy of mathematical models with increasing complexity starting with the flat film model where gravity balances viscous shear and surface tension is neglected, this is generalized to include surface tension. We further generalize to incorporate variable surface viscosity and more complicated constitutive laws for surface tension as a function of surfactant concentration. Lubrication theory is employed to derive three coupled nonlinear partial differential equations (PDEs) describing the free surface shape, a component of surface velocity and the surfactant transport at leading order. A large surface viscosity limit recovers the tangentially-immobile model; for small surface viscosity, the film is mobile. Transition from a mobile to an immobile film is observed for intermediate values of surface viscosity and Marangoni number. The above models reproduce a number of features observed in experiments, these include film shapes and thinning rates which can be correlated to experiment.

The 2+1-dimensional model for simplified surface properties has also been studied. Numerical experiments were performed to understand the stability of the system to perturbations across the film. An instability was seen in the mobile case; this was caused by a competition between gravity and the Marangoni effect. The behavior observed from this model qualitatively matches the structures observed in Dow Corning experiments; more work is needed to compare our numerics with experiment quantitatively.