The Motion of a Thin Liquid Film Driven by Surfactant and Gravity

We investigate wave solutions of a lubrication model for surfactant-driven flow of a thin liquid film down an inclined plane. We model the flow in one space dimension with a system of nonlinear PDEs of mixed hyperbolic-parabolic type in which the effects of capillarity and surface diffusion are neglected. Numerical solutions reveal distinct patterns of waves that are described analytically by combinations of traveling waves, some with jumps in height and surfactant concentration gradient. The various waves and combinations are strikingly different from what is observed in the case of flow on a horizontal plane. Jump conditions admit new shock waves sustained by a linear surfactant wave traveling upstream. The stability of these waves is investigated analytically and numerically. For initial value problems, a critical ratio of upstream to downstream height separates two distinct long-time wave patterns. Below the critical ratio, there is also an exact solution in which the height is piecewise constant and the surfactant concentration is piecewise linear and has compact support.