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Impact of surface viscosity on the stability of a droplet translating through a stagnant fluid

Published online by Cambridge University Press:  01 October 2021

Natasha Singh  and
Vivek Narsimhan Open the ORCID record for Vivek Narsimhan [Opens in a new window]
Natasha Singh
Affiliation:
Davidson School of Chemical Engineering, Purdue University, 480 Stadium Mall Drive, West Lafayette, IN47907, USA
Vivek Narsimhan*
Affiliation:
Davidson School of Chemical Engineering, Purdue University, 480 Stadium Mall Drive, West Lafayette, IN47907, USA
*
†Email address for correspondence: vnarsim@purdue.edu
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Abstract

This study examines the impact of interfacial viscosity on the stability of an initially deformed droplet translating through an unbounded quiescent fluid. The boundary-integral formulation is employed to investigate the time evolution of a droplet in the Stokes flow limit. The droplet interface is modelled using the Boussinesq–Scriven constitutive relationship having surface shear viscosity $\eta _\mu$ and surface dilatational viscosity $\eta _\kappa$. We observe that, below a critical value of the capillary number, $Ca_C$, the initially perturbed droplet reverts to its spherical shape. Above $Ca_C$, the translating droplet deforms continuously, growing a tail at the rear end for initial prolate perturbations and a cavity for initial oblate perturbations. We find that surface shear viscosity inhibits the tail/cavity growth at the droplet's rear end and increases the $Ca_C$ compared with a clean droplet. In contrast, surface dilatational viscosity increases tail/cavity growth and lowers $Ca_C$ compared with a clean droplet. Surprisingly, both shear and dilatational surface viscosity appear to delay the time at which pinch off occurs, and hence satellite droplets form. Lastly, we explore the combined influence of surface viscosity and surfactant transport on droplet stability by assuming a linear dependence of surface tension on surfactant concentration and exponential dependence of interfacial viscosities on the surface pressure. We find that pressure-thinning/thickening effects significantly affect the droplet dynamics for surface shear viscosity but play a small role for surface dilatational viscosity. We lastly provide phase diagrams for the critical capillary number for different values of the droplet's viscosity ratio and initial Taylor deformation parameter.

JFM classification

Drops and Bubbles: Drops Low-Reynolds-number flows: Boundary integral methods
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 927 , 25 November 2021 , A44
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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