A reduced model for droplet dynamics with interfacial viscosity

TL;DR

The paper addresses how interfacial viscosity alters droplet deformation in shear flows and asks whether a reduced tensorial model can capture the dynamics. It extends the Maffettone–Minale framework to include interfacial shear and dilatational viscosities via $Bq_{\mathrm{s}}$ and $Bq_{\mathrm{d}}$, deriving $f_1^{(\mathrm{EMM})}$ and $f_2^{(\mathrm{EMM})}$ that govern the stationary deformation $D_{\mathrm{s}}$ and validating the model against fully resolved simulations across $Ca$, $Bq_{\mathrm{s}}$, and $Bq_{\mathrm{d}}$. Key contributions include explicit expressions for the EMM coefficients, quantitative maps of the model’s validity, and insights into how interfacial viscosity slows transient deformation and shifts breakup thresholds. The work provides a fast, accurate tool for predicting droplet behavior with viscous interfaces in engineering contexts and outlines avenues for extending the framework to larger deformations and surfactant-laden interfaces.

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 $σ$, 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, $μ_s$ and $μ_d$, through the corresponding Boussinesq numbers $\mbox{Bq}_s=μ_s/μR$ and $\mbox{Bq}_d=μ_d/μR$, where $μ$ 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$=μR G/σ$ and the Boussinesq numbers. To this end, model predictions are systematically compared with fully resolved numerical simulations.

Figures (5)

• Figure 1: Shear plane section of a droplet with viscous interface deforming due to a simple shear flow with intensity $G$. The drop with radius at rest $R$, surface tension $\sigma$ and inner viscosity $\lambda\mu$ is immersed in a fluid with viscosity $\mu$. The shear interfacial viscosity $\mu_{\hbox{\scriptsize s}}$ weights the in-plane interface friction, while the dilatational interfacial viscosity $\mu_{\hbox{\scriptsize d}}$ weights the friction in the normal direction of $\boldsymbol{\bm{\hat{n}}}$ (green arrows). The drop shape is described via a second order tensor $\boldsymbol{\bm{S}}(t)$, while $L(t)$ and $B(t)$ represent the major and minor axes in the shear plane.
• Figure 2: EMM model parameters $1/f_1^{(\mathrm{EMM})}$ (panel (a)) and $f_2^{(\mathrm{EMM})}$ (panel (b)) as functions of $\text{Bq}_{\hbox{\scriptsize s}}$ and $\text{Bq}_{\hbox{\scriptsize d}}$ (see Eq. \ref{['ourf1f2']}). The thick red and blue curves highlight the two boundary sections of the surface corresponding to $\text{Bq}_{\hbox{\scriptsize s}}=0$ and $\text{Bq}_{\hbox{\scriptsize d}}=0$, respectively.
• Figure 3: Stationary deformation $D_{\hbox{s}}$ predicted by Eq. \ref{['eq:Dsteady']} as a function of $\text{Bq}_{\hbox{\scriptsize s}}$ and $\text{Bq}_{\hbox{\scriptsize d}}$, evaluated with $f_1=f_1^{(\mathrm{EMM})}$ and $f_2=f_2^{(\mathrm{EMM})}$ as given in Eq.\ref{['ourf1f2']}, for $\text{Ca}=0.1$ and $\lambda=1$.
• Figure 4: Time evolution of the deformation $D(t)$ in shear flow (see Eq. \ref{['eq:D']} and Fig. \ref{['fig:drop']}) for different values of Boussinesq numbers $\text{Bq}_{\hbox{\scriptsize s}}=\mu_{\hbox{\scriptsize s}}/\mu R$ and $\text{Bq}_{\hbox{\scriptsize d}}=\mu_{\hbox{\scriptsize d}}/\mu R$, where $\mu_{\hbox{\scriptsize s}}$ and $\mu_{\hbox{\scriptsize d}}$ denote the shear and dilatational interfacial viscosities, $\mu$ is the fluid viscosity and $R$ is the radius of the droplet at rest. Time is made dimensionless using the characteristic droplet time $\tau_\sigma=\mu R/\sigma$, where $\sigma$ is the surface tension; the capillary number $\text{Ca}=\mu R G/\sigma$ is $\text{Ca}=0.1$, where $G$ is the intensity of the shear rate; the viscosity ratio is $\lambda=1$. Fully resolved simulations (FRS) results are shown with symbols, while predictions from the numerical integration of Eq. \ref{['eq:MMmodel']} with $f_1$ and $f_2$ given by Eq. \ref{['ourf1f2']} (EMM) are reported as lines. The horizontal ticks shown outside the plotting area indicate the stationary deformation values predicted by Eq.\ref{['eq:Dsteady']}. Different combinations of Boussinesq numbers $\text{Bq}_{\hbox{\scriptsize s}}$ and $\text{Bq}_{\hbox{\scriptsize d}}$ have been reported: panel (a), $\text{Bq}_{\hbox{\scriptsize s}}=0$ and $\text{Bq}_{\hbox{\scriptsize d}}=0,5,25,50$; panel (b), $\text{Bq}_{\hbox{\scriptsize s}}=\text{Bq}_{\hbox{\scriptsize d}}=0,5,25,50$; panel (c), $\text{Bq}_{\hbox{\scriptsize s}}=0,5,25,50$ and $\text{Bq}_{\hbox{\scriptsize d}}=0$.
• Figure 5: Mean relative error $\langle \epsilon \rangle_T$ (see Eq. \ref{['eq:averr']}) between FRS and EMM predictions for the deformation $D(t)$, shown across the ($\text{Bq}_{\hbox{\scriptsize s}},\text{Bq}_{\hbox{\scriptsize d}}$)-plane for different capillary numbers $\text{Ca}$.