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 $σ$, 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.

💡 Research Summary

The paper presents an extension of the phenomenological Maffettone‑Minale (MM) model to account for interfacial shear and dilatational viscosities, thereby creating an “Extended MM” (EMM) framework capable of describing droplet deformation in simple shear flow when the interface possesses finite viscous resistance. The authors introduce two dimensionless Boussinesq numbers, Bq_s = μ_s/(μR) and Bq_d = μ_d/(μR), which quantify the relative magnitude of interfacial shear (μ_s) and dilatational (μ_d) viscosities with respect to bulk viscosity μ and droplet radius R. By incorporating the Boussinesq‑Scriven surface stress law into the MM tensorial evolution equation for the morphological tensor S, they derive modified coefficients f₁ and f₂ that now depend on the viscosity ratio λ, the capillary number Ca = μRG/σ, and the two Boussinesq numbers. The resulting expressions (Eq. 8) reduce to the original MM coefficients when Bq_s = Bq_d = 0, ensuring consistency with the clean‑droplet limit.

To evaluate the predictive capability of the EMM model, the authors perform fully resolved numerical simulations (FRS) using an immersed‑boundary lattice‑Boltzmann (IB‑LB) method. The computational domain is a 128³ lattice with a droplet of radius 20 LU discretized by ~20 k triangular elements. Simple shear flow is imposed via periodic boundaries in x and z and no‑slip walls in y. Simulations span a range of capillary numbers (Ca = 0.1, 0.2, 0.3, 0.5) and Boussinesq numbers (0–50), while keeping the viscosity ratio λ = 1. The droplet shape is quantified by the deformation parameter D(t) = (L−B)/(L+B), where L and B are the major and minor axes in the shear plane, extracted from the eigenvalues of the inertia tensor (identical to S in the EMM).

The authors compare time‑dependent deformation curves from the EMM (obtained by integrating Eq. 2 with the new f₁, f₂) with those from the FRS. They introduce a mean relative error ⟨ε⟩_T (Eq. 10) that averages the absolute deviation between the two predictions over a time window covering both transient and steady‑state regimes. Results show that for Bq_s = 0, Bq_d > 0 (pure dilatational viscosity) the steady‑state deformation slightly increases with Bq_d, whereas for Bq_s = Bq_d > 0 the deformation monotonically decreases as interfacial viscosity grows. When only shear viscosity is present (Bq_d = 0), the deformation exhibits a mixed behavior reflecting the combined influence of the surface‑divergence term and the surface rate‑of‑deformation tensor. Across all cases, the EMM reproduces the steady‑state deformation D_s predicted by the analytical expression (Eq. 4) and matches the FRS data within a few percent for Ca ≤ 0.2 and moderate Boussinesq numbers (product Bq_s·Bq_d ≤ 25). As Ca rises above 0.3 or the Boussinesq numbers become large (≥30), the mean error grows beyond 10 %, indicating the breakdown of the small‑deformation assumption inherent in the MM formulation.

A systematic error map (Fig. 5) across the (Bq_s, Bq_d) plane for various Ca values highlights the parameter region where the EMM remains quantitatively reliable. The map reveals that high interfacial viscosities tend to suppress deformation, extending the range of Ca over which the model stays accurate, because viscous resistance dominates over surface tension. Conversely, low interfacial viscosities allow larger deformations at modest Ca, narrowing the validity window.

In conclusion, the paper demonstrates that the EMM provides a compact, tensor‑based reduced‑order model that captures the essential physics of interfacial viscous effects on droplet deformation. It offers a computationally cheap alternative to full three‑dimensional simulations while maintaining quantitative agreement (≤5 % error) for a broad but bounded set of capillary and Boussinesq numbers. The authors acknowledge that the current formulation, which retains linear dependence of f₁ and f₂ on Ca, cannot capture strongly nonlinear deformation or breakup phenomena observed at higher Ca. Future work is suggested to incorporate nonlinear corrections, explore confined geometries, and extend the framework to multiphase flows with surfactant transport or elastic membranes. This work thus bridges the gap between detailed numerical simulations and practical engineering models for droplets with complex interfacial rheology.