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

Two different physical problems are considered: the magnetic shaping of a liquid metal column and the distortion of a bubble in a corner vortex flow. It is shown that the two problems can be modeled with a virtually identical set of equations. These equations are solved numerically using a conformal mapping and a series truncation method, which permits fast and efficient computation of the bubble or column shapes. It is found that the two problems exhibit different limiting configurations. For the bubble problem, the deformation becomes more severe as the vortex moves further into the corner until eventually the free surface makes contact with the walls. For the magnetic shaping problem, columns approach a limiting configuration featuring either a finite number of cusps or a fixed number of trapped bubbles along the perimeter. The division between these two different behaviors is explained by means of an exact solution for zero surface tension.