Influence of surfactant on rounded and pointed bubbles in two-dimensional Stokes flow

A simple plane flow model is used to examine the effects of surfactant on bubbles evolving in slow viscous flow. General properties of the time-dependent evolution as well as exact solutions for the steady state shape of the interface and distribution of surfactant are obtained for a rather general class of far-field extensional flows. The steady solutions include a class for which `stagnant caps' of surfactant partially coat the bubble surface. The governing equations for these stagnant cap bubbles feature boundary data which switches across free boundary points representing the cap edges. These points are shown to correspond to singularities in the surfactant distribution, the location and strength of which are determined as part of the solution. Our steady bubble solutions comprise shapes with rounded as well as pointed ends, depending on the far-field flow conditions. Unlike the clean flow problem, we find in all cases an upper bound on the strain rate for which steady solutions exist. A possible connection with the phenomenon of tip streaming is suggested.