A fully coupled mixed finite element method for surfactants spreading on thin liquid films

A model for the spreading of surfactants over thin liquid films on rough surfaces is presented, along with a novel finite element based discretization. An existing framework for surfactants spreading is first augmented by allowing for the influence of a non-trivial surface roughness on the film height and surfactant concentration. In a standard approach, the fourth order equation for the film height is then split into two second order systems. However, distinct from previous approaches, the proposed method also approximates the surface and depth-averaged velocity fields as independent variables, giving rise to a five-field system of coupled nonlinear equations. Benchmark calculations indicate that this approach allows for a Δt∼h² scaling without loss of convergence in the Newton algorithm. Consistent with analytical estimates, simulations of surfactant drops spreading over thin liquid films on smooth substrates exhibit a t^1∕4 temporal scaling for the surfactant leading edge, while constant surfactant sources show faster evolutions and exhibit a t^1∕2 scaling. Finally, simulations of surfactants spreading over extremely thin films demonstrate that fingering instabilities can be triggered by perturbations to either the film height or the substrate roughness.