# Swimming with a cage: Low-Reynolds-number locomotion inside a droplet

## TL;DR

This study investigates how a spherical microswimmer inside a droplet can propel both itself and the droplet through hydrodynamic interactions, revealing stable co-swimming states and flow reversals relevant for targeted drug delivery.

## Contribution

It provides an analytical and numerical analysis of droplet-encaged microswimmers, demonstrating stable co-movement, flow reversal effects, and stability scenarios based on swimmer gait.

## Key findings

- Droplet can be propelled by the encaged swimmer.
- Stable co-swimming states are achievable with specific actuation patterns.
- Flow field around the swimmer is reversed by the droplet, affecting its pushers and pullers.

## Abstract

Inspired by recent experiments using synthetic microswimmers to manipulate droplets, we investigate the low-Reynolds-number locomotion of a model swimmer (a spherical squirmer) encapsulated inside a droplet of comparable size in another viscous fluid. Meditated solely by hydrodynamic interactions, the encaged swimmer is seen to be able to propel the droplet, and in some situations both remain in a stable co-swimming state. The problem is tackled using both an exact analytical theory and a numerical implementation based on boundary element method, with a particular focus on the kinematics of the co-moving swimmer and droplet in a concentric configuration, and we obtain excellent quantitative agreement between the two. The droplet always moves slower than a swimmer which uses purely tangential surface actuation but when it uses a particular combination of tangential and normal actuations, the squirmer and droplet are able to attain a same velocity and stay concentric for all times. We next employ numerical simulations to examine the stability of their concentric co-movement, and highlight several stability scenarios depending on the particular gait adopted by the swimmer. Furthermore, we show that the droplet reverses the nature of the far-field flow induced by the swimmer: a droplet cage turns a pusher swimmer into a puller, and vice versa. Our work sheds light on the potential development of droplets as self-contained carriers of both chemical content and self-propelled devices for controllable and precise drug deliveries.