A geometric theory of swimming: Purcell's swimmer and its symmetrized cousin

TL;DR

This paper introduces a geometric framework for analyzing low Reynolds number swimming, using curvature concepts to understand swimmer behavior without solving differential equations, and applies it to Purcell's swimmer and its symmetric variant.

Contribution

It develops a novel geometric approach based on curvature to analyze swimming at low Reynolds number, avoiding differential equations and providing qualitative insights.

Findings

The approach yields complete information for line swimmers without rotation.

It reveals qualitative features of Purcell's swimmer.

The method is demonstrated on a symmetric version of Purcell's swimmer.

Abstract

We develop a qualitative geometric approach to swimming at low Reynolds number which avoids solving differential equations and uses instead landscape figures of two notions of curvatures: The swimming curvature and the curvature derived from dissipation. This approach gives complete information for swimmers that swim on a line without rotations and gives the main qualitative features for general swimmers that can also rotate. We illustrate this approach for a symmetric version of Purcell's swimmer which we solve by elementary analytical means within slender body theory. We then apply the theory to derive the basic qualitative properties of Purcell's swimmer.