Continuous breakdown of Purcell's scallop theorem with inertia

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. (c) 2007 American Institute of Physics.