No abstract

The influence of surfactant on the breakup of a prestretched bubble in a quiescent viscous surrounding is studied by a combination of direct numerical simulation and the solution of a long-wave asymptotic model. The direct numerical simulations describe the evolution toward breakup of an inviscid bubble, while the effects of small but non-zero interior viscosity are readily included in the long-wave model for a fluid thread in the Stokes flow limit.The direct numerical simulations use a specific but realizable and representative initial bubble shape to compare the evolution toward breakup of a clean or surfactantfree bubble and a bubble that is coated with insoluble surfactant. A distinguishing feature of the evolution in the presence of surfactant is the interruption of bubble breakup by formation of a slender quasi-steady thread of the interior fluid. This forms because the decrease in surface area causes a decrease in the surface tension and capillary pressure, until at a small but non-zero radius, equilibrium occurs between the capillary pressure and interior fluid pressure.The long-wave asymptotic model, for a thread with periodic boundary conditions, explains the principal mechanism of the slender thread's formation and confirms, for example, the relatively minor role played by the Marangoni stress. The largetime evolution of the slender thread and the precise location of its breakup are, however, influenced by effects such as the Marangoni stress and surface diffusion of surfactant.

Slender-body theory is used to investigate the steady-state deformation and time-dependent evolution of an inviscid axisymmetric bubble in zero-Reynolds-number extensional flow, when insoluble surfactant is present on the bubble surface. The asymptotic solutions reveal steady ellipsoidal bubbles covered with surfactant, and, at increasing deformation, solutions distinguished by a cylindrical surfactant-free central part, with stagnant surfactant caps at the bubble endpoints. The bubble shapes are rounded near the endpoints, in contrast to the pointed shapes found for clean inviscid bubbles. Simple expressions are derived relating the capillary number $Q$ to the steady bubble slenderness ratio $\epsilon$. These show that there is a critical value $Q_c$ above which steady solutions no longer exist. Equations governing the time-evolution of a slender inviscid bubble with surfactant, valid for large capillary number, are also derived. Numerical solutions of the slender bubble equations for $Q\,{>}\,Q_c$ exhibit spindle shapes with tip-streaming filaments.