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.
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