Surface tank-treading: propulsion of Purcell's toroidal swimmer

TL;DR

This paper investigates the propulsion mechanisms of a doughnut-shaped swimmer using surface tank-treading, analyzing different motion modes, efficiencies, and proposing a practical design with numerical and analytical solutions.

Contribution

It provides a detailed analysis of surface tank-treading propulsion for toroidal swimmers, including efficiency optimization and explicit solutions for simplified models.

Findings

Maximum propulsion speed is 66% of surface tank-treading velocity.

Swimming efficiency can reach up to 13% with simple modes.

Numerical demonstration of a practical swimmer design.

Abstract

In this work we address the "smoking ring" propulsion technique, originally proposed by E. M. Purcell. We first consider self-locomotion of a doughnut-shaped swimmer powered by surface tank-treading. Different modes of surface motion are assumed and propulsion velocity and swimming efficiency are determined. The swimmer is propelled against the direction of its outer surface motion, the inner surface having very little affect. The simplest swimming mode corresponding to constant angular velocity, can achieve propulsion speeds of up to 66% of the surface tank-treading velocity and swimming efficiency of up to 13%. Higher efficiency is possible for more complicated modes powered by twirling of extensible surface. A potential practical design of a swimmer motivated by Purcell's idea is proposed and demonstrated numerically. Lastly, the explicit solution is found for a two-dimensional swimmer composed of two counter-rotating disks, using complex variable techniques.