RESUMEN

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.

RESUMEN

We study, analytically and numerically, singularity formation in an interface flow driven by a multipole for a two-dimensional Hele--Shaw cell with surface tension. Our analysis proves that singularity formation is inevitable in the case of a dipole. For a multipole of a higher order, we show that the solution does not tend to any stationary solution as time goes to infinity if its initial center of mass is not at the multipole; it is therefore very likely that this solution will develop finite time singularities. Our extensive numerical studies suggest that a solution develops finite time singularities via the interface reaching the multipole while forming a corner at the tip of the finger that touches the multipole. In addition, it is observed that the interface approaches the multipole from directions which can be predicted beforehand. We also estimate as a function of time the distance between the finger tip and multipole, and the results are in excellent agreement with numerical computations.