  We investigate the way in which oscillating dumb-bells, a simple microscopic
model of apolar swimmers, move at low Reynold's number. In accordance with
Purcell's Scallop Theorem a single dumb-bell cannot swim because its stroke is
reciprocal in time. However the motion of two or more dumb-bells, with mutual
phase differences, is not time reversal invariant, and hence swimming is
possible. We use analytical and numerical solutions of the Stokes equations to
calculate the hydrodynamic interaction between two dumb-bell swimmers and to
discuss their relative motion. The cooperative effect of interactions between
swimmers is explored by considering first regular, and then random arrays of
dumb-bells. We find that a square array acts as a micropump. The long time
behaviour of suspensions of dumb-bells is investigated and compared to that of
model polar swimmers.
