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 steady transverse flow can delay the onset of instability in the circular Couette geometry until the azimuthal Weissenberg number Weθ 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 the upper-convected Maxwell and Oldroyd-B fluids, generally shows increased stability 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, Wez and forcing frequency ω. In particular, instability can be induced not only when ω is of the order of the inverse relaxation time of the fluid but also when it is much smaller. Scaling arguments and numerical results indicate that the high-ω, low-Wez regime is essentially equivalent to Wez=0 in the steady case, implying no stabilization. At high values of ω and Wez, scaling analysis shows that the flow will always be stable. Numerical results are in agreement with these conclusions. Consistent with previous results on parametrically 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 subsequent decay.

This study also presents linear and nonlinear stability results for the addition of steady axial Couette and Poiseuille flows to viscoelastic instabilities in azimuthal Dean flows. It is shown that, for high Wez, the qualitative effect of adding a steady axial flow is similar to that in the circular Couette geometry, with a linear relationship between the critical Weθ and Wez. For low Wez, we find that the flow is stabilized, unlike in the circular Couette flow where the critical value of Weθ decreases at low Wez. Further, weakly nonlinear analysis shows that the criticality of the bifurcation depends on the value of Wez 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.