We study numerically the nonlinear stationary states of a droplet covered with an insoluble surfactant in a uniaxial extensional flow. We calculate the eigenfunctions to reveal the instability mechanism and determine the nonlinear states resulting from it to obtain a coherent picture of the phenomenon. The transition is of the saddle-node type, both with and without surfactant. The flow becomes unstable under stationary linear perturbations. We find that the soluto-capillarity effect is dominant in determining the stability. Surfactant considerably reduces the interval of stable capillary numbers. The nonlinear state resulting from instability is fundamentally different for drops with and without surfactant. Tip streaming only occurs in the presence of surfactants. The critical eigenmode leading to tip streaming is qualitatively the same as that yielding the central pinching mode for a clean interface, which indicates that the small local scale characterizing tip streaming is set during the nonlinear droplet deformation. Inertia increases the droplet deformation and decreases the critical capillary number. In the presence of the surfactant monolayer, neither the droplet deformation nor the stability is significantly affected by the droplet viscosity. The viscous surface stress does not significantly affect the droplet deformation. However, the damping rate of the dominant mode considerably decreases for viscous surfactants. Interestingly, shear viscous surface stress considerably alters the tip streaming arising in the supercritical regime, even for very small surface viscosities. The viscous surface stresses alter the balance of normal interfacial stresses and affect the surfactant transport over the stretched interface.
|Journal||Journal of Fluid Mechanics|
|Publication status||Submitted - 14 Jul 2021|