  Recently, detailed experiments on visco-elastic channel flow have provided
convincing evidence for a nonlinear instability scenario which we had argued
for based on calculations for visco-elastic Couette flow. Motivated by these
experiments we extend the previous calculations to the case of visco-elastic
Poiseuille flow, using the Oldroyd-B constitutive model. Our results confirm
that the subcritical instability scenario is similar for both types of flow,
and that the nonlinear transition occurs for Weissenberg numbers somewhat
larger than one. We provide detailed results for the convergence of our
expansion and for the spatial structure of the mode that drives the
instability. This also gives insight into possible similarities with the
mechanism of the transition to turbulence in Newtonian pipe flow.
