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 Weθ
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 Wez=O(1))
can delay the onset of instability until
Weθ 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 Wez[Gt ]1, which shows
that the critical azimuthal Weissenberg
number for instability increases linearly with Wez.
Non-axisymmetric disturbances are very strongly suppressed, becoming unstable
only when
ε1/2Weθ=
O(We2z). 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 Wez 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
(Wez[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.
Axial Flow Effect on the Stability of Circular Couette Flow
