Effects of surface viscosities on the motion of a droplet enclosing a translating particle

TL;DR

This study analyzes how surface viscosities affect the motion of a droplet containing a translating particle, using analytical and numerical methods to reveal the roles of interfacial rheology and geometry.

Contribution

It provides the first exact analytical solution for concentric configurations and extends analysis to eccentric geometries, highlighting the effects of surface viscosities on droplet dynamics.

Findings

Surface shear viscosity does not affect droplet velocity in concentric configurations.

Surface dilatational viscosity can either enhance or suppress droplet motion depending on conditions.

Eccentric positioning introduces a dependence on surface shear viscosity, generally increasing droplet motion.

Abstract

We investigate the influence of interfacial rheology on the motion of a compound particle consisting of a viscous droplet enclosing a translating rigid particle in the Stokes flow regime. The droplet interface is modeled using the Boussinesq-Scriven constitutive law, incorporating both surface shear and dilatational viscosities. An exact analytical solution is derived for the concentric configuration, and the analysis is extended to eccentric geometries using a spectral boundary integral method, enabling a systematic examination of confinement, viscosity contrast, and interfacial properties. For concentric configurations, we show that the induced droplet velocity is independent of surface shear viscosity, while surface dilatational viscosity can either enhance or suppress the droplet motion depending on the interplay between confinement and viscosity ratio. This behavior is rationalized in terms of competing effects between reduced interfacial mobility and increased driving force required to maintain the prescribed particle speed. In contrast, when the particle is eccentrically positioned within the droplet, a dependence on surface shear viscosity emerges, leading to a consistent enhancement of droplet motion that becomes more pronounced with increasing eccentricity. The analytical and numerical results are in excellent agreement and reveal how interfacial rheology, confinement, and symmetry breaking jointly govern the dynamics of compound particle systems. These findings provide mechanistic insight and establish a quantitative benchmark for future studies of active compound particles with complex interfaces.