VIII-2

Michael Siegel

New Jersey Institute of Technology

Abstract

Tip streaming refers to a process in which minute droplets or bubbles are formed from a single drop or bubble subjected to an extensional flow. G. I. Taylor first identified this process in a seminal paper describing experiments performed in a four roller mill. Recent numerical simulations have shown that a simple 2-D model of a time evolving bubble in a four roller mill can capture many of the salient features observed in experiments. Among the most interesting results is the observation that 2-D bubbles evolving with non-constant surface tension (such as is induced by surfactant) form regions of very high curvature and can exhibit tip streaming instabilities.

In this talk, complex variable methods are applied to investigate the effect of variable surface tension on the motion of a bubble in 2-D Stokes flow. Analytical results include exact solutions for the steady state interfacial shape and general features of the time dependent evolution for a rather general class of far-field flows. For flow in a four roller mill, we show that true cusp singularities form on the interface during the unsteady evolution, as long as the capillary number exceeds a critical value. The stability of cusped or nearly cusped interfacial profiles is examined. The results are related to the tip streaming observed in experiments and numerical simulations.