  The current work studies the dynamics of a microswimmer in pressure-driven
flow of a weakly viscoelastic fluid. Employing the second-order fluid model, we
show that the self-propelling swimmer experiences a viscoelastic swimming lift
in addition to the well-known passive lift that arises from its resistance to
shear flow. Using the reciprocal theorem, we evaluate analytical expressions
for the swimming lift experienced by neutral and pusher/puller-type swimmers
and show that they depend on the hydrodynamic signature associated with the
swimming mechanism. We find that for neutral swimmers focusing towards the
centerline is accelerated by two orders of magnitude, while for force-dipole
swimmers no net modification in cross-streamline migration occurs.
