Non-Axisymmetric Subcritical Bifurcations in Viscoelastic Taylor-Couette Flow (original) (raw)

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 1994

Abstract

In this work, we calculate the bifurcating families corresponding to each one of the two possible non-axisymmetric patterns emerging at the point of criticality, namely the spirals and ribbons and determine their stability. It is shown that for a narrow gap size, for upper convected Maxwell and Oldroyd-B fluids, at least one of the non-axisymmetric families bifurcates subcritically. This result, in conjunction with the theoretical analysis of Hopf bifurcation in presence of symmetries (Golubitsky et al. 1988), implies tha neither of the bifurcating families is stable. Consequently, there is a finite transition corresponding to infinitesimal changes of the flow parameters in the vicinity of the Hopf bifurcation point. Although a change in the ratio of the Deborah and Reynolds numbers has not been found to qualitatively affect this behaviour, calculations with a wider gap size have shown that both bifurcating families become supercritical. There, a Ginzburg-Landau analysis shows that the ribbons are the stable pattern. This behavior is qualitatively similar to that seen for the newtonian fluid, but for counterrotating cylinders, albeit there, spirals have been found to be stable (Golubitsky & Langford 1988).

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