Self-similar, surfactant-driven flows

Abstract

Consider a dilute, insoluble surfactant monolayer on the free surface of a thin viscous film. A gradient in surfactant concentration generates a gradient in surface tension, driving a flow that redistributes the surfactant so that these gradients decay. The nonlinear evolution equations governing such flows, derived using lubrication theory, have previously been shown to admit a set of simple similarity solutions representing the spreading of a monolayer over an uncontaminated interface. Here, a much more general class of similarity solutions is considered, and a transformation is identified reducing the governing partial differential equations to a set of nonlinear ordinary differential equations, the solutions of which correspond to integral curves in a two-dimensional phase plane. This allows the construction of solutions to a wide range of problems. Many new solutions are revealed, including one that cannot be determined by simpler techniques, namely the closing of an axisymmetric hole in a monolayer, the radius of which is shown to be proportional to (−t)δ as t→0−, where δ≊0.807 41; this solution corresponds to a heteroclinic orbit in the phase plane.