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 Gamma 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 Gamma 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. [References: 33]
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