Experiments and theory show that hydrodynamic instabilities can arise during flow of viscoelastic liquids in curved geometries. A recent study has found that a relatively weak transverse flow can delay the onset of instability in the circular Couette geometry until the azimuthal Weissenberg number We_theta is significantly higher than without axial flow. In this work we investigate the effect of superposition of a time periodic axial Couette flow on the viscoelastic circular Couette and Dean flow instabilities. The analysis, carried out for UCM and Oldroyd-B fluids generally shows increased stability as compared to when there is no axial flow. However, we also find that the system shows instability - synchronous resonance - for some values of the axial Weissenberg number, We_zand forcing frequency omega. Scaling arguments and numerical results indicate that the high omega, low We_z regime is essentially equivalent to We_z = 0 in the steady case implying no stabilization. At high values of omega and We_z, scaling analysis shows that the flow will always be stable. Numerical results are in agreement with these conclusions. Consistent with previous results on periodically forced systems, we find that the zero frequency limit is singular. In this limit, the disturbances display quiescent intervals punctuated by periods of large transient growth and then decay.

The current study also presents linear and nonlinear results from the addition of steady axial Couette and Poiseuille flows to viscoelastic instabilities in azimuthal Dean flows. It is shown that the qualitative effect of adding a steady axial flow is similar to that of the circular Couette geometry for high We_z with a linear relationship between the critical We_theta and We_z. For low We_z, we find that the flow is stabilized, unlike in the circular Couette flow where the critical value of We_theta decreases at low We_z. Further, weakly nonlinear analysis shows that the criticality of the bifurcation depends on the value of We_z and the solvent viscosity, S. Finally, we also show the presence of a codimension-2 Takens-Bogdanov bifurcation point in the linear stability curve of Dean flow. This point represents a transition from one mechanism of instability to another.