Coating of an inclined plane in the presence of insoluble surfactant

We consider the flow of a thin liquid film coating an inclined plane in the presence of an insoluble surfactant. A fully non-linear two-dimensional system of governing equations is formulated using lubrication theory to describe the dynamics. Numerical simulations of this system highlight a fingering instability present at the main fluid front and elucidate the role of surfactant in the destabilizing mechanism. A full parametric study is undertaken which reveals the dependence of the fingering characteristics on system parameters. Numerical solutions at low angles of inclination are also obtained in order to illustrate the connection between gravitationally driven fingering and the instability induced by surfactant on a flat substrate. The similarities and differences between the destabilizing mechanisms in each case are discussed.