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

A timestepper approach for the systematic bifurcation and stability analysis of polymer extrusion dynamics