A reduced model for droplet dynamics with interfacial viscosity

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.