Energy-optimal strokes for multi-link microswimmers: Purcell's loops and Taylor's waves reconciled

TL;DR

This paper derives energy-efficient swimming strokes for multi-link microswimmers, revealing that optimal motions are elliptical and often traveling waves, bridging classical and modern models of microscale propulsion.

Contribution

It introduces a unified optimal control framework for multi-link microswimmers, showing optimal strokes are elliptical and can be high-dimensional loops, extending previous models to larger shape spaces.

Findings

Optimal strokes are elliptical in shape space.

For large N, optimal strokes resemble traveling waves.

Large displacements lead to complex, non-planar loops.

Abstract

Micron-scale swimmers move in the realm of negligible inertia, dominated by viscous drag forces. In this paper, we formulate the leading-order dynamics of a slender multi-link (N-link) microswimmer assuming small-amplitude undulations about its straight configuration. The energy-optimal stroke to achieve a given prescribed displacement in a given time period is obtained as the largest eigenvalue solution of a constrained optimal control problem. Remarkably, the optimal stroke is an ellipse lying within a two-dimensional plane in the (N-1)-dimensional space of joint angles, where N can be arbitrarily large. For large N, the optimal stroke is a traveling wave of bending, modulo edge effects. If the number of shape variables is small, we can consider the same problem when the prescribed displacement in one time period is large, and not attainable with small variations of the joint angles. The fully nonlinear optimal control problem is solved numerically for the cases N=3 (Purcell's three-link swimmer) and N=5 showing that, as the prescribed displacement becomes small, the optimal solutions obtained using the small-amplitude assumption are recovered. We also show that, when the prescribed displacements become large, the picture is different. For N=3 we recover the non-convex planar loops already known from previous studies. For N=5 we obtain non-planar loops, raising the question of characterizing the geometry of complex high-dimensional loops.