Continuous breakdown of Purcell's scallop theorem with inertia

TL;DR

This paper demonstrates that inertia causes a continuous breakdown of Purcell's scallop theorem, enabling reciprocal motions to generate net motion even at very low Reynolds numbers, contrary to classical expectations.

Contribution

It introduces elementary examples showing reciprocal motions can induce net velocities at any small Re_f, revealing a continuous breakdown of the scallop theorem due to inertia.

Findings

Reciprocal motions can produce net motion at arbitrarily small Re_f.

Net velocity scales as (Re_f)^α with α > 0.

Inertia causes a continuous breakdown of Purcell's scallop theorem.

Abstract

Purcell's scallop theorem defines the type of motions of a solid body - reciprocal motions - which cannot propel the body in a viscous fluid with zero Reynolds number. For example, the flapping of a wing is reciprocal and, as was recently shown, can lead to directed motion only if its frequency Reynolds number, Re_f, is above a critical value of order one. Using elementary examples, we show the existence of oscillatory reciprocal motions which are effective for all arbitrarily small values of the frequency Reynolds number and induce net velocities scaling as (Re_f)^\alpha (alpha > 0). This demonstrates a continuous breakdown of the scallop theorem with inertia.