  Simulations of undulatory swimming in viscoelastic fluids with large
amplitude gaits show concentration of polymer elastic stress at the tips of the
swimmers.We use a series of related theoretical investigations to probe the
origin of these concentrated stresses. First the polymer stress is computed
analytically at a given oscillating extensional stagnation point in a
viscoelastic fluid. The analysis identifies a Deborah number (De) dependent
Weissenberg number (Wi) transition below which the stress is linear in Wi, and
above which the stress grows exponentially in Wi. Next, stress and velocity are
found from numerical simulations in an oscillating 4-roll mill geometry. The
stress from these simulations is compared with the theoretical calculation of
stress in the decoupled (given flow) case, and similar stress behavior is
observed. The flow around tips of a time-reversible flexing filament in a
viscoelastic fluid is shown to exhibit an oscillating extension along particle
trajectories, and the stress response exhibits similar transitions. However in
the high amplitude, high De regime the stress feedback on the flow leads to non
time-reversible particle trajectories that experience asymmetric stretching and
compression, and the stress grows more significantly in this regime. These
results help explain past observations of large stress concentration for large
amplitude swimmers and non-monotonic dependence on De of swimming speeds.
