  In a world without inertia, Purcell's scallop theorem states that in a
Newtonian fluid a time-reversible motion cannot produce any net force or net
flow. Here we consider the extent to which the nonlinear rheological behavior
of viscoelastic fluids can be exploited to break the constraints of the scallop
theorem in the context of fluid pumping. By building on previous work focusing
on force generation, we consider a simple, biologically-inspired geometrical
example of a flapper in a polymeric (Oldroyd-B) fluid, and calculate
asymptotically the time-average net fluid flow produced by the reciprocal
flapping motion. The net flow occurs at fourth order in the flapping amplitude,
and suggests the possibility of transporting polymeric fluids using reciprocal
motion in simple geometries even in the absence of inertia. The induced flow
field and pumping performance are characterized and optimized analytically. Our
results may be useful in the design of micro-pumps handling complex fluids.
