Cusp formation for evolving bubbles in 2-D Stokes ow: the e ect of variable surface tension

Analytical and numerical methods are applied to investigate the e ect of variable surface tension, induced by the presence of surfactant, on a bubble evolving in 2-D Stokes ow. The evolution is driven by an extensional ow from a four roller mill. Of particular interest is the possible spontaneous occurrence of a cusp singularity on the bubble surface. Using complex variable methods, exact solutions for the steady interface shape and distribution of surfactant are obtained, in the case when surface diffusion of surfactant is negligible. The steady solutions include those for which the bubble is covered in a nonzero concentration of surfactant, as well as bubbles with `surfactant caps' that collect on the bubble sides. The branch of steady state solutions is shown to terminate at a steady cusped bubble with zero surface tension at the bubble tips. Thus, in contrast to the clean ow (i.e., constant surface tension) problem, there exists an upper bound Q = Qc on the non-dimensional strain rate for which steady bubble solutions exist. Numerical calculations of the transient evolution for Q > Qc suggest that the bubble achieves an unsteady cusped formation in nite time. The numerical computations are greatly simpli ed by exploiting the analytic structure of the governing equations and interfacial shape. This enables the interfacial evolution to be followed nearly up to cusp formation. The role of a nonlinear equation of state and the in uence of surface di usion of surfactant are both considered. Comparison with experiment reveals some interesting similarities, and a possible connection between the observed behavior and the phenomenon of tip streaming is discussed.