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

Abstract

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 $W(1-\beta )=O(1)$ as $W \rightarrow \infty$ ( $\beta$ is the ratio of solvent-to-total viscosity). In contrast to centre modes in the Orr–Sommerfeld equation, $1-c=O(1)$ as $W \rightarrow \infty$ , 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).