Finite element approximation of a two-layered liquid film in the presence of insoluble surfactants

摘要

We consider a system of degenerate parabolic equations modelling a thin film, consisting of two layers of immiscible Newtonian liquids, on a solid horizontal substrate. In addition, the model includes the presence of insoluble surfactants on both the free liquid-liquid and liquid-air interfaces, and the presence of both attractive and repulsive van der Waals forces in terms of the heights of the two layers. We show that this system formally satisfies a Lyapunov structure, and a second energy inequality controlling the Laplacian of the liquid heights. We introduce a fully practical finite element approximation of this nonlinear degenerate parabolic system, that satisfies discrete analogues of these energy inequalities. Finally, we prove convergence of this approximation, and hence existence of a solution to this nonlinear degenerate parabolic system.