Hopf bifurcation in creeping cone-and-plate flow of a viscoelastic fluid
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This paper analyzes the bifurcations which occur in the creeping flow of a viscoelastic fluid subjected to a constant shearing motion in the gap between an inverted cone and a plate. We show that when the Deborah number, a dimensionless relaxation time of the fluid, exceeds a critical value the base `viscometric' flow loses stability and a Hopf bifurcation occurs. The nature of the bifurcation depends on the retardation parameter \( \beta \), defined as the ratio of polymer viscosity to the zero shear rate viscosity of the fluid. Our analysis shows that for \( 0.98 \leq \beta \leq 1\), bifurcation is supercritical and subcritical for \( \beta \leq 0.97 \). The analysis is facilitated by assuming that the gap between the cone and the plate is small. Center manifold theory is then used to derive appropriate amplitude equations in a neighborhood of the critical Deborah number.
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