Bounded Film Evolution with Nonlinear Surface Properties

Abstract

We study the evolution of a Newtonian free surface of a thin film above a solid wall. We consider the case in which the horizontal solid is covered by a non-wetting fluid and an insoluble monolayer of surfactant is present on the fluid-air interface. We pose a model that incorporates a variety of interfacial effects: van der Waals forces, variable surface tension and surface viscosity. The surface tension and surface viscosity depend nonlinearly on the surfactant concentration. Using lubrication theory we obtain a leading order description of the shape and velocity of the fluid-air interface, and the surfactant concentration, in the form of coupled nonlinear partial differential equations. A linear stability analysis reveals that the wavenumber that characterizes the marginal state is independent of the presence of the surfactant and the nonlinearity of the surface properties. We solve the 1+1-dimensional system numerically to obtain the spatio-temporal evolution of the free surface in the nonlinear regime, and observe the progression to rupture.