Optimization and small-amplitude analysis of Purcell's three-link microswimmer model

TL;DR

This paper analytically investigates Purcell's three-link microswimmer's motion in viscous flow, deriving formulas for displacement and efficiency, and identifies optimal stroke amplitudes and geometries for enhanced swimming performance.

Contribution

It provides the first analytical perturbation-based analysis of the microswimmer's displacement and efficiency, including correction terms for large strokes.

Findings

Reversal in movement direction at large stroke amplitudes.

Asymptotic expressions for Lighthill's energetic efficiency.

Optimal stroke amplitudes and link ratios for maximum displacement or efficiency.

Abstract

This work studies the motion of Purcell's three-link microswimmer in viscous flow, by using perturbation expansion of its dynamics under small-amplitude strokes. Leading-order expressions and next-order correction terms for the displacement of the swimmer are obtained for the cases of a square or circular gait in the plane of joint angles. The correction terms demonstrate the reversal in movement direction for large stroke amplitudes, which has previously only been shown numerically. In addition, asymptotic expressions for Lighthill's energetic efficiency are obtained for both gaits. These approximations enable calculating optimal stroke amplitudes and swimmer's geometry (i.e. ratio of links' lengths) for maximizing either the displacement or Lighthill's efficiency.