Gradient flow perspective of thin-film bilayer flows

Abstract

We study gradient flow formulations of thin-film bilayer flows with triple-junctions between liquid/liquid/air. First we highlight the gradient structure in the Stokes free-boundary flow and identify its solutions with the well-known PDE with boundary conditions. Next we propose a similar gradient formulation for the corresponding thin-film model and formally identify solutions with those of the corresponding free-boundary problem. A robust numerical algorithm for the thin-film gradient flow structure is then provided. Using this algorithm we compare the sharp triple-junction model with precursor models. For their stationary solutions a rigorous connection is established using [Gamma]-convergence. For time-dependentsolutions the comparison of numerical solutions shows a good agreement for small and moderate times. Finally we study spreading in the zero-contact angle case, where we compare numerical solutions with asymptotically exact source-type solutions.