Asymptotics of the centre-mode instability in viscoelastic channel flow: with and without inertia

Motivated by the recent numerical results of Khalid et al. ( Phys. Rev. Lett. , vol. 127, 2021, 134502), we consider the large-Weissenberg-number ( W ) asymptotics of the centre mode instability in inertialess viscoelastic channel flow. The instability is of the critical layer type in the distinguished ultra-dilute limit where W(1-β )=O(1) as W → ∞ ( β is the ratio of solvent-to-total viscosity). In contrast to centre modes in the Orr–Sommerfeld equation, $1-c=O(1)$ as W → ∞ , where c is the phase speed normalised by the centreline speed as a central ‘outer’ region is always needed to adjust the non-zero cross-stream velocity at the critical layer down to zero at the centreline. The critical layer acts as a pair of intense ‘bellows’ which blows the flow streamlines apart locally and then sucks them back together again. This compression/rarefaction amplifies the streamwise-normal polymer stress which in turn drives the streamwise flow through local polymer stresses at the critical layer. The streamwise flow energises the cross-stream flow via continuity which in turn intensifies the critical layer to close the cycle. We also treat the large-Reynolds-number ( $Re$ ) asymptotic structure of the upper (where 1-c=O(Re-2/3) ) and lower branches of the $Re$ – W neutral curve, confirming the inferred scalings from previous numerical computations. Finally, we remark that the viscoelastic centre-mode instability was actually first observed in viscoelastic Kolmogorov flow by Boffetta et al. ( J. Fluid Mech. , vol. 523, 2005, pp. 161–170).