# The autophoretic torus

## TL;DR

This paper derives an exact analytical solution for the flow around an axisymmetric phoretic torus, revealing its potential as a micro-pump and optimizing its swimming capabilities based on surface chemistry and geometry.

## Contribution

It provides the first analytical solution for phoretic flow around a torus and explores its applications as a pump and swimmer, extending beyond spheroidal shapes.

## Key findings

- A torus can act as a micro-pump due to confinement effects.
- Optimal swimming tori do not occur when the central hole vanishes.
- Analytical solutions agree with boundary element computations.

## Abstract

Phoretic swimmers provide new avenues to study non-equilibrium statistical physics and are also hailed as a promising technology for bioengineering at the cellular scale. Exact solutions for the locomotion of such swimmers have been restricted so far to spheroidal shapes. In this paper we solve for the flow induced by the canonical non-simply connected shape, namely an axisymmetric phoretic torus. The analytical solution takes the form of an infinite series solution, which we validate against boundary element computations. For a torus of uniform chemical activity, confinement effects in the hole allow the torus to act as a pump, which we optimize subject to fixed particle surface area. Under the same constraint, we next characterize the fastest swimming Janus torus for a variety of assumptions on the surface chemistry. Perhaps surprisingly, none of the optimal tori occur in the limit where the central hole vanishes.