Two-dimensional nonlinear advection-diffusion in a model of surfactant spreading on a thin liquid film

The spreading of a localized monolayer of dilute, insoluble surfactant, discharged from a point source that moves at constant speed over a thin liquid film coating a planar substrate, is described according to lubrication theory by a pair of coupled nonlinear evolution equations for the monolayer concentration Γ and the film depth h. Numerical and asymptotic techniques are here used to show that the extent and structure of such a spreading asymmetric monolayer can be well approximated by a single nonlinear advection-diffusion equation involving Γ alone. At large times the solution is composed of three, spatially distinct, asymptotic regions: (i) a quasi-steady `nose' region (containing the source), in which there is a dominant balance between two-dimensional nonlinear diffusion and advection; (ii) an `advective' region, in which longitudinal advection balances transverse diffusion; and (iii) a `tail' region, in which unsteady diffusion is dominant. In each region, local similarity solutions are obtained either exactly (in the advective region) or approximately (elsewhere) by rescaling numerical solutions of the initial-value problem. If the source concentration decreases with time, it is demonstrated that the monolayer's width is greatest in the tail region, whereas for a source of increasing concentration the monolayer is widest in the advective region. For the simpler one-dimensional problem of a monolayer spreading from a line source, the same balances hold but with transverse diffusion eliminated; here self-similar solutions are found in all three regions that agree closely with numerical solutions of the initial-value problem.