Effect of axial flow on viscoelastic Taylor–Couette instability

Abstract

Viscoelastic flow instabilities can arise from gradients in elastic stresses in flows with curved streamlines. Circular Couette flow displays the prototypical instability of this type, when the azimuthal Weissenberg number We_θ is O(ε^−1/2), where ε measures the streamline curvature. We consider here the effect of superimposed steady axial Couette or Poiseuille flow on this instability. For inertialess flow of an upper-convected Maxwell or Oldroyd-B fluid in the narrow gap limit (ε[Lt ]1), the analysis predicts that the addition of a relatively weak steady axial Couette flow (axial Weissenberg number We_z=O(1)) can delay the onset of instability until We_θ is significantly higher than without axial flow. Weakly nonlinear analysis shows that these bifurcations are subcritical. The numerical results are consistent with a scaling analysis for We_z[Gt ]1, which shows that the critical azimuthal Weissenberg number for instability increases linearly with We_z. Non-axisymmetric disturbances are very strongly suppressed, becoming unstable only when ε^1/2We_θ= O(We²_z). A similar, but smaller, stabilizing effect occurs if steady axial Poiseuille flow is added. In this case, however, the bifurcations are converted from subcritical to supercritical as We_z increases. The observed stabilization is due to the axial stresses introduced by the axial flow, which overshadow the destabilizing hoop stress. If only a weak (We_z[les ]1) steady axial flow is added, the flow is actually slightly destabilized. The analysis also elucidates new aspects of the stability problems for plane shear flows, including the exact structure of the modes in the continuous spectrum, and illustrates the connection between these problems and the viscoelastic circular Couette flow.