  An analytic, asymptotic approximation of the nonlinear steady-state equations
for viscoelastic creeping flow, modeled by the Oldroyd-B equations with polymer
stress diffusion, is derived. Near the extensional stagnation point the flow
stretches and aligns polymers along the outgoing streamlines of the stagnation
point resulting in a stress-island, or birefringent strand. The polymer stress
diffusion coefficient is used, both, as an asymptotic parameter and a
regularization parameter. The structure of the singular part of polymer stress
tensor is a Gaussian aligned with the incoming streamline of the stagnation
point; a smoothed $\delta$-distribution whose width is proportional to the
square-root of the diffusion coefficient. The amplitude of the stress island
scales with the Wiessenberg number and, although singular in the limit of
vanishing diffusion, it is integrable in the cross stream direction due to its
vanishing width; this yields a convergent secondary flow. The leading order
velocity response to this stress island is constructed and shown to be {\em
independent} of the diffusion coefficient in the limit. The secondary
circulation counteracts the forced flow and has a vorticity jump at the
location of the stress islands, essentially expelling the background vorticity
from the location of the birefringent strands. The analytic solutions are shown
to be in excellent quantitative agreement with full numerical simulations and,
therefore, the analytic solutions elucidate the salient mechanisms of the flow
response to viscoelasticity and the mechanism for instability.
