Stability and tip streaming of a surfactant-loaded drop in an extensional flow. Influence of surface viscosity

Abstract

We study numerically the nonlinear stationary states of a droplet covered with an insoluble surfactant in a uniaxial extensional flow. We calculate both the eigenfunctions to reveal the instability mechanism and the time-dependent states resulting from it, which provides 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. Surfactant considerably reduces the interval of stable capillary numbers. 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 transient state resulting from instability is fundamentally different for drops with and without surfactant. Tip streaming occurs only 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. The viscous surface stress does not significantly affect the droplet deformation and the critical capillary number. 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.

Electrically modulated relaxation dynamics of pre-stretched droplets post switched-off uniaxial extensional flow

“…In such flows, small deformation limits are crossed for high Capillary number (Ca) values, implicating strong dominance of viscous stresses over capillary stresses, 22 leading to the possibilities of droplet fragmentation in large deformation limits. [23][24][25][26] In a seminal work, Taylor 21 studied droplet dynamics in an extensional flow using a four-roller apparatus and concurrently developed a theoretical model to calculate the flow-induced deformation. Subsequent numerical and experimental studies investigated the deformation mechanism more rigorously for a wide range of flow parameters.…”

Influence of surface viscosities on the electrodeformation of a prolate viscous drop

“…11,12 The Taylor-Melcher model has been extended in recent years to include the effects of charge relaxation, 13 charge convection, [14][15][16][17] drop shapes, [18][19][20][21] drop instabilities, 10,[22][23][24][25][26][27] and Marangoni stresses. [28][29][30][31] Interfacial rheology is characterized by a shear viscosity and a dilatational viscosity on the drop interface, and may arise for drops covered with colloidal particles and proteins, 32,33 surfactants, 34 and as a property of vesicles' membrane viscosity. 33,35 Studies show a rich range of dynamics over various physical settings as a result of interfacial rheology.…”

“…[36][37][38] Numerical simulations have also been conducted to determine the influence of interfacial rheology in shear 39,40 and extensional flows. For the extensional flow, Herrada et al 34 found that although surface velocity is reduced, the surface viscosities have greater effect on drop breakup, and only minimal effect on the equilibrium deformation. Under the same flow, Singh and Narsimhan 41,42 showed that the dilatational viscosity led to instability, and the shear viscosity acted to stabilize the drop.…”

“…Interfacial rheology is characterized by a shear viscosity and a dilatational viscosity on the drop interface, and may arise for drops covered with colloidal particles and proteins, 32,33 surfactants, 34 and as a property of vesicles’ membrane viscosity. 33,35 Studies show a rich range of dynamics over various physical settings as a result of interfacial rheology.…”

Numerical investigation of the effect of surface viscosity on droplet breakup and relaxation under axisymmetric extensional flow

“…The deformation and breakup of a droplet in shear flow using boundary-element simulations have been explored in Gounley et al (2016) and Luo, Shang & Bai (2019). Herrada et al (2022) also examined the steady-state solutions and stability of a gaseous droplet with interfacial viscosity using linear stability analysis and numerical computations in extensional flow. These previous studies have shown that the surface shear viscosity reduces the droplet deformation and increases the critical capillary number for droplet breakup compared with a clean droplet at the same viscosity ratio.…”