Weakly nonlinear analysis of the viscoelastic instability in channel flow for finite and vanishing Reynolds numbers

Abstract

The recently discovered centre-mode instability of rectilinear viscoelastic shear flow (Garg et al. , Phys. Rev. Lett. , vol. 121, 2018, 024502) has offered an explanation for the origin of elasto-inertial turbulence that occurs at lower Weissenberg numbers ( $Wi$ ). In support of this, we show using weakly nonlinear analysis that the subcriticality found in Page et al. ( Phys. Rev. Lett. , vol. 125, 2020, 154501) is generic across the neutral curve with the instability becoming supercritical only at low Reynolds numbers ( $Re$ ) and high $Wi$ . We demonstrate that the instability can be viewed as purely elastic in origin, even for $Re=O(10^3)$ , rather than ‘elasto-inertial’, as the underlying shear does not feed the kinetic energy of the instability. It is also found that the introduction of a realistic maximum polymer extension length, $L_{max}$ , in the FENE-P model moves the neutral curve closer to the inertialess $Re=0$ limit at a fixed ratio of solvent-to-solution viscosities, $\beta$ . At $Re=0$ and in the dilute limit ( $\beta \rightarrow 1$ ) with $L_{max} =O(100)$ , the linear instability can be brought down to more physically relevant $Wi\gtrsim 110$ at $\beta =0.98$ , compared with the threshold $Wi=O(10^3)$ at $\beta =0.994$ reported recently by Khalid et al. ( Phys. Rev. Lett. , vol. 127, 2021, 134502) for an Oldroyd-B fluid. Again, the instability is subcritical, implying that inertialess rectilinear viscoelastic shear flow is nonlinearly unstable – i.e. unstable to finite-amplitude disturbances – for even lower $Wi$ .