Role of finite extensibility on the centre-mode instability in viscoelastic channel flow

A purely elastic linear instability was recently reported for viscoelastic plane Poiseuille flow in the limit of ultra-dilute (solvent to solution viscosity ratio $\beta \gt 0.99$ ), highly elastic (Weissenberg number $W \sim 1000$ ) polymer solutions, within the framework of the Oldroyd-B model (Khalid et al. , Phys. Rev. Lett., vol. 127, 2021, pp. 134–502). This is the first instance of a purely elastic instability in a strictly rectilinear shearing flow, with the phase speed of the unstable ‘centre mode’ being close to the base-state maximum velocity at the channel centreline. Subsequently, Buza, Page and Kerswell ( J. Fluid Mech., vol. 940, 2022, A11) have shown, using the FENE-P model, that the centre-mode instability persists down to moderate elasticities ( $W \sim O (100)$ ), the reduction in threshold evidently due to the finite extensibility of the polymer molecules. In this work, we augment this latter finding and provide a comprehensive account of the effect of finite extensibility on the centre-mode instability in viscoelastic channel flow, using the FENE-P and FENE-CR models, in both the absence and presence of fluid inertia. In both these models, finite extensibility causes a decrease in the polymer relaxation time at high shear rates, and the resulting weakening of elastic stresses would seem to indicate a stabilising effect. The latter trend has been demonstrated by earlier analyses of hoop-stress-driven instabilities in curvilinear flows, and is indeed borne out for the FENE-CR case, where finite extensibility has a largely stabilising influence on the centre-mode instability. In stark contrast, for the FENE-P model, finite extensibility plays a dual role – a stabilising one at lower values of the elasticity number $E$ , but, surprisingly, a destabilising one at higher $E$ values. Further, the centre-mode instability is predicted over a significantly larger domain of the $Re$ – $E$ – $\beta$ parameter space, compared to the Oldroyd-B model, making it more amenable to experimental observations.