Steady deformation and tip-streaming of a slender bubble with surfactant in an extensional flow

Abstract

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.

Selective withdrawal of polymer solutions: Experiments

“…In some cases, a thin filament is drawn out from the cusp [27,28]. Another related phenomenon is tip-streaming [30][31][32][33][34]. Subject to extensional flows, surfactant-laden bubbles and drops deform into a spindle shape with pointed ends, and then emit a train of tiny bubbles or droplets from these points.…”

Microscale tipstreaming in a microfluidic flow focusing device

“…Other theoretical work investigating tipstreaming has focused on using slender body theory to predict the pointed shapes observed in tipstreaming. 30 Numerical and theoretical work on tipstreaming to date has primarily utilized insoluble surfactants. The important variables for the process established through this work, in addition to the capillary number and viscosity ratio, are the initial dimensionless surface coverage X and an elasticity number El, reflecting the sensitivity of surface tension to the local surface concentration.…”

A hybrid numerical method for interfacial fluid flow with soluble surfactant

“…A striking example is the tipstreaming of thin threads or small droplets from a drop or bubble that is stretched in an imposed extensional flow. Although first observed by Taylor [1] in his seminal four-roller mill experiments, tipstreaming has only recently been ascribed to the presence of surfactant [2][3][4], and surfactant mediated tipstreaming has now been utilized to synthesize micron-sized and smaller droplets in a flow focusing device [5]. Other applications of surfactants include their addition to two-phase mixtures or emulsions to facilitate breakup of large droplets and prevent coalescence of smaller ones, thereby stabilizing the emulsion [6].…”