A reduced model for droplet dynamics with interfacial viscosity

TL;DR

This paper extends the Maffettone-Minale model to include interfacial viscosities, enabling more accurate predictions of droplet deformation in shear flows by comparing with detailed numerical simulations.

Contribution

The paper introduces an extended MM model that accounts for interfacial shear and dilatational viscosities, enhancing the modeling of droplet deformation under shear flow.

Findings

The extended model accurately predicts droplet deformation within certain parameter ranges.

Interfacial viscosities significantly influence droplet shape dynamics.

Model validation against numerical simulations confirms its applicability.

Abstract

We propose an extension of the phenomenological Maffettone-Minale (MM) model (P.L. Maffettone and M. Minale, J. Non-Newton. Fluid Mech. 78, 227-241 (1998)) to describe the time-dependent deformation of a droplet with interfacial viscosity in a shear flow. The droplet, characterised by surface tension $\sigma$, is spherical at rest with radius $R$ and deforms into an ellipsoidal shape under a shear flow of rate $G$, described by a symmetric second-order morphological tensor $\boldsymbol{S}$. In addition to surface tension, the extended MM (EMM) model incorporates interfacial shear and dilatational viscosities, $\mu_s$ and $\mu_d$, through the corresponding Boussinesq numbers $\mbox{Bq}_s=\mu_s/\mu R$ and $\mbox{Bq}_d=\mu_d/\mu R$, where $\mu$ is the bulk viscosity. A central goal of this work is to quantify the parameter range over which the EMM model provides a realistic description of droplet deformation, as a function of the capillary number Ca$=\mu R G/\sigma$ and the Boussinesq numbers. To this end, model predictions are systematically compared with fully resolved numerical simulations.