Reciprocal locomotion of dense swimmers in Stokes flow

Abstract

Due to the kinematic reversibility of Stokes flow, a body executing a reciprocalmotion (a motion in which the sequence of body configurations remains identicalunder time reversal) cannot propel itself in a viscous fluid in the limit of negligibleinertia; this result is known as Purcell’s scallop theorem. In this limit, the Reynoldsnumbers based on the fluid inertia and on the body inertia are all zero. Previousstudies characterized the breakdown of the scallop theorem with fluid inertia.In this paper we show that, even in the absence of fluid inertia, certain densebodies undergoing reciprocal motion are able to swim. Using Lorentz’s reciprocaltheorem, we first derive the general differential equations that govern the locomotionkinematics of a dense swimmer. We demonstrate that no reciprocal swimmingis possible if the body motion consists only of tangential surface deformation(squirming). We then apply our general formulation to compute the locomotion of foursimple swimmers, each with a different spatial asymmetry, that perform normalsurface deformations. We show that the resulting swimming speeds (or rotationrates) scale as the first power of a properly defined ‘swimmer Reynolds number’,demonstrating thereby a continuous breakdown of the scallop theorem with bodyinertia.