  In several biologically relevant situations, cell locomotion occurs in
polymeric fluids with Weissenberg {number} larger than one. Here we present
results of three-dimensional numerical simulations for the steady locomotion of
a self-propelled body in a model polymeric (Giesekus) fluid at low Reynolds
number. Locomotion is driven by steady tangential deformation at the surface of
the body (so-called squirming motion). In the case of a spherical squirmer, we
show that the swimming velocity is systematically less than that in a Newtonian
fluid, with a minimum occurring for Weissenberg numbers of order one. The rate
of work done by the swimmer always goes up compared to that occurring in the
Newtonian solvent alone, but is always lower than the power necessary to swim
in a Newtonian fluid with the same viscosity. The swimming efficiency, defined
as the ratio between the rate of work necessary to pull the body at the
swimming speed in the same fluid and the rate of work done by swimming, is
found to always be increased in a polymeric fluid. Further analysis reveals
that polymeric stresses break the Newtonian front-back symmetry in the flow
profile around the body. In particular, a strong negative elastic wake is
present behind the swimmer, which correlates with strong polymer stretching,
and its intensity increases with Weissenberg number and viscosity contrasts.
{The velocity induced by the squirmer is found to decay in space faster than}
in a Newtonian flow, with a strong {dependence} on the polymer relaxation time
and viscosity. Our computational results are also extended to prolate
spheroidal swimmers and smaller polymer stretching are obtained for slender
shapes compared to bluff swimmers. The swimmer with an aspect ratio of two is
found to be the most hydrodynamically efficient.
