A purely elastic instability in Dean and Taylor–Dean flow

The linear stability of the viscoelastic flow of an Oldroyd‐B fluid between rotating cylinders with an applied, azimuthal pressure gradient is considered. It is found that this Taylor–Dean flow is unstable in certain flow parameter regimes even in the limit of vanishingly small Reynolds number. The critical conditions and the structure of the vortex flow at the onset of instability are presented. These are determined in the limit as the channel width to radius of curvature becomes small. The present results reveal that the instability is a stationary mode when the pressure gradient becomes the dominant flow driving force, while it is an oscillatory instability when the shearing by the cylinder rotation is dominant. In addition, it is found that the direction of the pressure gradient controls the characteristics of the instability: A pressure gradient applied along the cylinder rotation destabilizes the flow, while if applied against the rotation, the flow is substantially stabilized. The mechanism of these instabilities is also investigated through an examination of the disturbance‐energy equation. It is found that the mechanism of the elastic, stationary instability is associated with the coupling of the perturbation velocity field to the polymeric stree gradients in the base flow. To the authors’ knowledge this mechanism has not been reported elsewhere. In contrast, the mechanism for the elastic, oscillatory instability in Taylor–Dean flow involves the coupling between the disturbance polymeric stresses and the base state velocity gradients, as reported by Larson et al. [J. Fluid Mech. 218, 573 (1990)] for the elastic, oscillatory instability in Taylor–Couette flow.