  We discuss how matrix-free/timestepper algorithms can efficiently be used
with dynamic non-Newtonian fluid mechanics simulators in performing systematic
stability/bifurcation analysis. The timestepper approach to bifurcation
analysis of large scale systems is applied to the plane Poiseuille flow of an
Oldroyd-B fluid with non-monotonic slip at the wall, in order to further
investigate a mechanism of extrusion instability based on the combination of
viscoelasticity and nonmonotonic slip. Due to the nonmonotonicity of the slip
equation the resulting steady-state flow curve is nonmonotonic and unstable
steady-states appear in the negative-slope regime. It has been known that
self-sustained oscillations of the pressure gradient are obtained when an
unstable steady-state is perturbed [Fyrillas et al., Polymer Eng. Sci. 39
(1999) 2498-2504].
  Treating the simulator of a distributed parameter model describing the
dynamics of the above flow as an input-output black-box timestepper of the
state variables, stable and unstable branches of both equilibrium and periodic
oscillating solutions are computed and their stability is examined. It is shown
for the first time how equilibrium solutions lose stability to oscillating ones
through a subcritical Hopf bifurcation point which generates a branch of
unstable limit cycles and how the stable periodic solutions lose their
stability through a critical point which marks the onset of the unstable limit
cycles. This implicates the coexistence of stable equilibria with stable and
unstable periodic solutions in a narrow range of volumetric flow rates.
