Mathematical modeling and numerical simulation of wave-front flow on a vertical wall with surfactant effects

Abstract

The modeling of thin-layer flow in the presence of surfactant is considered. The classical problem of wave-front flow down a vertical wall, as discussed by Tuck and Schwartz (SIAM Rev 32:453, 1990), is extended to demonstrate the effects of an assumed insoluble surfactant. Results using the thin-layer approximation (“lubrication theory”) are compared with numerical solutions of the full Navier–Stokes problem using a volume-of-fluid method. The surfactant takes the form of either (i) a concentrated clump, or ‘bolus,’ or (ii) an initially uniform distribution. The basic problem is two-dimensional downhill flow from thick to thin wetting layers. If no surfactant is present, or if the surfactant is entirely passive, and if the layers are long, a steadily propagating solution exists. A discrete quantity of surfactant, when deposited on the liquid surface will be transported to the neighbourhood of the wave front. An apparently stable steady-state will ultimately arise. Similar surfactant transport will occur for uniform surfactant, although no steady-state solution is possible then because the quantity of surfactant becomes unbounded. The volume-of-fluid calculations confirm the validity of all the qualitative features revealed by the lubrication analysis. Quantitative discrepancies appear when the evolving free-surface slope or curvature is no longer small, even in the absence of surfactant effect.