Transition in elastic Dean flow: the centre-mode versus hoop-stress pathways

Abstract

We analyse the stability of viscoelastic Dean flow (flow of an elastic fluid through a curved two-dimensional channel, driven by an azimuthal pressure gradient) in the absence of fluid inertia. This configuration is well known to exhibit a hoop-stress-driven `purely elastic' instability (referred to henceforth as the hoop-stress mode -- `HSM') on account of the base-flow streamline curvature. The objective of this study is to demonstrate the existence and importance of a distinct elastic instability in this flow configuration, which is not driven by hoop-stresses, but instead is a continuation of a novel `centre-mode' (CM) instability recently identified in rectilinear shear flows. We use both the Oldroyd-B and FENE-P models to map out parameter regimes in the $W\!i$--$\epsilon$--$\beta$ space where the aforementioned instabilities are present. Here, $W\!i$ is a suitably defined Weissenberg number that characterizes fluid elasticity, $\beta$ is the ratio of solvent to total solution viscosity, and $\epsilon$ is the ratio of the gap (channel) width to the radius of curvature. For FENE-P model, decreasing the finite extensibility parameter $L$ has opposing effects on the HSM and CM instabilities -- stabilising the former, but destabilising the latter. In the dilute solution regime ($\beta > 0.95$), and for realistic values of $L \sim O(100)$, corresponding to polymer molecular weights of $O(10^{5-6})$g/mol, the CM remains the most unstable mode for $\epsilon \leq 0.25$, rendering it potentially relevant to the onset of elastic turbulence in the flow of such polymer solutions through curved channels.