A rigorous proof of the scallop theorem and a finite mass effect of a microswimmer

TL;DR

This paper provides a rigorous proof of Purcell's scallop theorem in Stokes flow, discusses its breakdown at finite Stokes numbers, and explores conditions under which the theorem remains valid.

Contribution

It offers a mathematically rigorous proof of the scallop theorem including body rotation and analyzes the effects of finite Stokes number on the theorem's validity.

Findings

The scallop theorem is proven rigorously including body rotation.

Breakdown of the theorem occurs at first order of the Stokes number.

Symmetric strokes can preserve the theorem at higher orders.

Abstract

We reconsider fluid dynamics for a self-propulsive swimmer in Stokes flow. With an exact definition of deformation of a swimmer, a proof is given to Purcell's scallop theorem including the body rotation. The breakdown of the theorem due to a finite Stokes number is discussed by using a perturbation expansion method and it is found that the breakdown generally occurs at the first order of the Stokes number. In addition, employing the Purcell's "scallop" model, we show that the theorem holds up to a higher order if the strokes of the swimmer has some symmetry.