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

TL;DR

This work reveals the coexistence of two fundamentally different elastic instabilities in inertialess viscoelastic Dean flow: the hoop-stress mode (HSM) driven by streamline curvature and a centre-mode (CM) that does not rely on curvature. Using Oldroyd-B and FENE-P constitutive models, it maps stability in the parameter space defined by the Weissenberg number $Wi$, solvent-relative viscosity $\beta$, and gap ratio $\epsilon$, showing that CM can dominate for realistic finite extensibility $L$ and moderate curvature. Finite extensibility has contrasting effects: it stabilizes axisymmetric HSM but can destabilize CM and non-axisymmetric HSM (HSM2), producing loops in neutral curves and lowering CM thresholds to experimentally accessible values for sufficiently dilute solutions. The results imply two potential routes to elastic turbulence in Dean/Taylor–Dean flows, with significant implications for microfluidic focusing, serpentine channels, and enhanced oil recovery where elastic effects are exploited or mitigated.

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 $W\!i$--$ε$--$β$ space where the aforementioned instabilities are present. Here, $W\!i$ is a suitably defined Weissenberg number that characterizes fluid elasticity, $β$ is the ratio of solvent to total solution viscosity, and $ε$ 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 ($β> 0.95$), 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 $ε\leq 0.25$, rendering it potentially relevant to the onset of elastic turbulence in the flow of such polymer solutions through curved channels.

Figures (34)

• Figure 1: Schematic of the geometry and the coordinate system considered.
• Figure 2: Dean flow velocity profiles for a FENE-P fluid: (a) $\beta = 0.98$ and $\epsilon = 0.1$ for two $(Wi, L)$ pairs, with $W\!i/L = 10$; (b) $\beta = 0.5$ and $\epsilon = 0.1$ with varying $Wi/L$; (c) master curves for $Wi/L = 10$ and $\epsilon = 0.1$ for varying $\beta$; (d) master curves for $Wi/L = 10$ and $\beta = 0.9$ for varying $\epsilon$.
• Figure 3: Benchmarking of neutral curves obtained from our numerical (spectral-cum-shooting) procedure (for $Re = 0$, $n = 0$) with those of joo1992purely, for Dean flow of an Oldroyd-B fluid, for different $\beta$. Continuous lines show results from the present work, while discrete points represent the data of joo1992purely.
• Figure 4: Eigenspectrum showing stationary ($\alpha = 7$) and propagating ($\alpha = 14$) modes of instability via the HSM1 mode in Dean flow of an Oldroyd-B fluid at $Re = 0$, $n = 0$, $\beta = 0.98$, $\epsilon = 0.1$, and $W\!i = 25$.
• Figure 5: Schematic of the continuous spectra branches in Dean flow of a FENE-P fluid with $W\!i/L \sim O(1)$ and $\beta \rightarrow 1$.