Mobilities of a drop and an encapsulated squirmer

TL;DR

This paper derives exact analytical expressions for the mobility of a spherical squirmer inside a liquid drop, considering external forces and torques, and discusses how these mobilities depend on system parameters.

Contribution

It provides the first exact analytical formulas for the mobility of an encapsulated squirmer in a spherical drop under general conditions.

Findings

Mobility depends on orientation, size, and viscosities.

Analytical expressions for velocities are derived for non-axisymmetric configurations.

Results enable analysis of the trajectories of drop-squirmer systems.

Abstract

We have analyzed the dynamics of a spherical, uni-axial squirmer which is located inside a spherical liquid drop at general position $\bm{r}_s$. The squirmer is subject to an external force and torque in addition to the slip velocity on its surface. We have derived exact analytical expressions for the linear and rotational velocity of the squirmer as well as the linear velocity of the drop for general, non-axisymmetric configurations. The mobilities of both, squirmer and drop, are in general anisotropic, depending on the orientation of $\bm{r}_s$, relative to squirmer axis, external force or torque. We discuss their dependence on the size of the squirmer, its distance from the center of the drop and the viscosities. Our results provide a first step towards a discussion of the trajectories of the composite system of drop and enclosed squirmer.