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

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. On account of its origins, the CM instability in Dean flow is expected for two-dimensional (azimuthally varying) disturbances with no axial variation, but continues to exist for three-dimensional disturbances. In contrast, the HSM instability is expected primarily in the axisymmetric limit, and again continues to exist for three-dimensional disturbances. We use both the Oldroyd-B and FENE-P models to map out parameter regimes in the italic Wi Wi $\textit{Wi}$ – epsilon ϵ $\epsilon$ – beta β $\beta$ space where the aforementioned instabilities are present. Here, italic Wi Wi $\textit{Wi}$ is a suitably defined Weissenberg number that characterises fluid elasticity, beta β $\beta$ is the ratio of solvent to total solution viscosity and epsilon ϵ $\epsilon$ is the ratio of the gap (channel) width to the radius of curvature. While its origin in rectilinear shearing flows might lead one to expect the CM instability to only be present for small epsilon ϵ $\epsilon$ (the ‘narrow-gap’ limit), we show that it exists even for upper O left parenthesis 1 right parenthesis O ( 1 ) $O(1)$ values of epsilon ϵ $\epsilon$ , and over a larger range of italic Wi Wi $\textit{Wi}$ . Nevertheless, within the Oldroyd-B framework, the HSM determines the stability threshold for experimentally relevant gap-width ratios, corresponding to 0.1 less than or slanted equals epsilon less than or slanted equals 1 0.1 ⩽ ϵ ⩽ 1 $0.1 \leqslant \epsilon \leqslant 1$ , with the CM becoming the most unstable mode only for epsilon less than 0.001 ϵ < 0.001 $\epsilon \lt 0.001$ . For the more accurate FENE-P model, however, decreasing the finite extensibility parameter upper L L $L$ has opposing effects on the HSM and CM instabilities – stabilising the former, but destabilising the latter. In the dilute solution regime ( beta greater than 0.95 β > 0.95 $\beta \gt 0.95$ ) and for realistic values of upper L tilde upper O left parenthesis 100 right parenthesis L ∼ O ( 100 ) $L \sim O(100)$ , corresponding to polymer molecular weights of upper O left parenthesis 10 Superscript 5 minus 6 Baseline right parenthesis O ( 10 5 − 6 ) $O(10^{5-6})$ g mol −1 , the CM remains the most unstable mode for epsilon less than or slanted equals 0.25 ϵ ⩽ 0.25 $\epsilon \leqslant 0.25$ , rendering it potentially relevant to the onset of elastic turbulence in the flow of such polymer solutions through curved channels.