Reciprocal locomotion of dense swimmers in Stokes flow

TL;DR

This paper demonstrates that dense bodies can swim in Stokes flow despite the scallop theorem, due to body inertia, by deriving new equations and analyzing four simple swimmer models.

Contribution

The study reveals that body inertia enables reciprocal swimming in Stokes flow, extending the understanding beyond fluid inertia effects.

Findings

Dense swimmers can propel themselves without fluid inertia.

Reciprocal swimming is impossible with only tangential surface deformation.

Swimming speed scales linearly with a defined swimmer Reynolds number.

Abstract

Due to the kinematic reversibility of Stokes flow, a body executing a reciprocal motion (a motion in which the sequence of body configurations remains identical under time reversal) cannot propel itself in a viscous fluid in the limit of negligible inertia; this result is known as Purcell's scallop theorem. In this limit, the Reynolds numbers based on the fluid inertia and on the body inertia are all zero. Previous studies characterized the breakdown of the scallop theorem with fluid inertia. In this paper we show that, even in the absence of fluid inertia, certain dense bodies undergoing reciprocal motion are able to swim. Using Lorentz's reciprocal theorem, we first derive the general differential equations that govern the locomotion kinematics of a dense swimmer. We demonstrate that no reciprocal swimming is possible if the body motion consists only of tangential surface deformation (squirming). We then apply our general formulation to compute the locomotion of four simple swimmers, each with a different spatial asymmetry, that perform normal surface deformations. We show that the resulting swimming speeds (or rotation rates) scale as the first power of a properly defined "swimmer Reynolds number'', demonstrating thereby a continuous breakdown of the scallop theorem with body inertia.