  In many biological systems, microorganisms swim through complex polymeric
fluids, and usually deform the medium at a rate faster than the inverse fluid
relaxation time. We address the basic properties of such life at high Deborah
number analytically by considering the small-amplitude swimming of a body in an
arbitrary complex fluid. Using asymptotic analysis and differential geometry,
we show that for a given swimming gait, the time-averaged leading-order
swimming kinematics of the body can be expressed as an integral equation on the
solution to a series of simpler Newtonian problems. We then use our results to
demonstrate that Purcell's scallop theorem, which states that time-reversible
body motion cannot be used for locomotion in a Newtonian fluid, breaks down in
polymeric fluid environments.
