Effect of surfactants on the instability of a liquid thread: Part II: Extensional flow

The effect of surfactants on the capillary instability of a liquid thread extending under the influence of an ambient flow is studied by linear theory for small-amplitude perturbations, numerical simulations for arbitrary amplitude perturbations based on boundary-integral and finite-volume methods, and numerical simulations based on an approximate model that relies on the long-wave approximation. Theoretical predictions and numerical simulations confirm previous predictions that, in the absence of surfactants, perturbations with a sufficiently small amplitude eventually decay as long as neither the viscosity of the thread nor the viscosity of the ambient fluid is equal to zero. It is shown, however, that for zero or infinite viscosity ratio, disturbances of any wavelength eventually amplify leading to thread breakup at a finite time. Surfactants stabilize the interface during the initial stages of the instability, but the increase in the mean surface tension due to surfactant dilution by stretching leads to higher perturbation amplitudes at long times. An asymptotic flow model is developed to describe the evolution of a viscous thread suspended in an ambient inviscid fluid subject to axisymmetric disturbances with long wavelength. Numerical solutions suggest that the similarity solution developed previously for a thread suspended in a quiescent fluid describes the behavior during the final stages of breakup.