Cusp Formation and Tip-Streaming Instabilities for Time-Evolving Interfaces in Two-Dimensional Stokes Flow

Abstract

The possible spontaneous occurence of a cusp singularity on the surface of a bubble evolving in an extensional two-dimensional Stokes flow is considered. The role of constant as well as variable surface tension, induced by the presence of surfactant, is considered. For constant surface tension, a previously determined class of exact nearly cusped solutions is found to be linearly stable. However, for large capillary number Q the steady solutions are susceptible to finite amplitude instability, with the dynamics often leading to topological singularities (i.e., tip-streaming). In the case of variable surface tension, a new class of exact steady solutions is determined These include states for which ‘stagnant caps’ of surfactant partially coat the bubble surface. The steady solution branches are found to terminate, suggesting the existence of a critical capillary number above which the bubble achieves an unsteady cusped formation in finite time. Cusp formation is is confirmed through accurate numerical computations.