  Flagella beating in complex fluids are significantly influenced by
viscoelastic stresses. Relevant examples include the ciliary transport of
respiratory airway mucus and the motion of spermatozoa in the mucus-filled
female reproductive tract. We consider the simplest model of such propulsion
and transport in a complex fluid, a waving sheet of small amplitude free to
move in a polymeric fluid with a single relaxation time. We show that, compared
to self-propulsion in a Newtonian fluid occurring at a velocity U_N, the sheet
swims (or transports fluid) with velocity U / U_N = [1+De^2 (eta_s)/(eta)
]/[1+De^2], where eta_s is the viscosity of the Newtonian solvent, eta is the
zero-shear-rate viscosity of the polymeric fluid, and De is the Deborah number
for the wave motion, product of the wave frequency by the fluid relaxation
time. Similar expressions are derived for the rate of work of the sheet and the
mechanical efficiency of the motion. These results are shown to be independent
of the particular nonlinear constitutive equations chosen for the fluid, and
are valid for both waves of tangential and normal motion. The generalization to
more than one relaxation time is also provided. In stark contrast with the
Newtonian case, these calculations suggest that transport and locomotion in a
non-Newtonian fluid can be conveniently tuned without having to modify the
waving gait of the sheet but instead by passively modulating the material
properties of the liquid.
