Efficient swimming of an assembly of rigid spheres at low Reynolds number

TL;DR

This paper develops a mathematical framework to optimize and analyze the swimming efficiency of rigid sphere assemblies at low Reynolds numbers, applicable to small and large amplitude motions.

Contribution

It introduces a method to optimize swimming strokes using eigenvalue problems and extends the analysis to large amplitude motions with harmonic interactions.

Findings

Optimal strokes can be determined through eigenvalue analysis.

The method applies to both small and large amplitude motions.

Performance of a three-sphere collinear swimmer is demonstrated.

Abstract

The swimming of an assembly of rigid spheres immersed in a viscous fluid of infinite extent is studied in low Reynolds number hydrodynamics. The instantaneous swimming velocity and rate of dissipation are expressed in terms of the time-dependent displacements of sphere centers about their collective motion. For small amplitude swimming with periodically oscillating displacements, optimization of the mean swimming speed at given mean power leads to an eigenvalue problem involving a velocity matrix and a power matrix. The corresponding optimal stroke permits generalization to large amplitude motion in a model of spheres with harmonic interactions and corresponding actuating forces. The method allows straightforward calculation of the swimming performance of structures modeled as assemblies of interacting rigid spheres. A model of three collinear spheres with motion along the common axis is studied as an example.