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

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 -> 1) with L-max = 0(100), the linear instability can be brought down to more physically relevant Wi greater than or similar to 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.