Asymptotic modelling for the spreading of a thin viscous drop on a chemically heterogeneous surface by gravity and surface tension

The slow motion of a thin droplet on a chemically heterogeneous surface is considered by analytical and numerical means. The long-wave limit of the Stokes regime is invoked, which yields a partial differential equation for the droplet height that takes into account the combined effects of surface tension and gravity. A matched asymptotics analysis is pursued for nearly circular contact lines, by treating separately the dynamics in the bulk of the droplet with that in the vicinity of its contact line. This approach gives rise to a series of boundary value problems, the solution of which is assisted by automatic differentiation techniques, and allows us to deduce a set of evolution equations for the Fourier harmonics of the contact line. Numerical experiments for a number of representative cases highlight the generally favorable agreement between the governing equation and the derived model, which is able to capture the salient features of the dynamics.