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. Phy. Rev. Lett. 121, 024502, 2018) has offered an explanation for the origin of elasto-inertial turbulence (EIT) which occurs at lower Weissenberg (Wi) numbers. In support of this, we show using weakly nonlinear analysis that the subcriticality found in Page et al. (Phys. Rev. Lett. 125, 154501, 2020) is generic across the neutral curve with the instability only becoming supercritical at low Reynolds (Re) numbers and high Wi. We demonstrate that the instability can be viewed as purely elastic in origin even for Re=O(103), rather than `elasto-inertial', as the underlying shear does not energise the instability. It is also found that the introduction of a realistic maximum polymer extension length, Lmax, 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, β. In the dilute limit (β→1) with Lmax=O(100), the linear instability can brought down to more physically-relevant Wi≳110 at β=0.98, compared with the threshold Wi=O(103) at β=0.994 reported recently by Khalid et al. (arXiv: 2103.06794) 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.