The centre-mode instability of viscoelastic plane Poiseuille flow

A modal stability analysis shows that plane Poiseuille flow of an Oldroyd-B fluid becomes unstable to a ‘centre mode’ with phase speed close to the maximum base-flow velocity, $U_{max}$. The governing dimensionless groups are the Reynolds number $Re = \rho U_{max} H/\eta$, the elasticity number $E = \lambda \eta /(H^2 \rho )$ and the ratio of solvent to solution viscosity $\beta = \eta _s/\eta$; here, $\lambda$ is the polymer relaxation time, $H$ is the channel half-width and $\rho$ is the fluid density. For experimentally relevant values (e.g. $E \sim 0.1$ and $\beta \sim 0.9$), the critical Reynolds number, $Re_c$, is around $200$, with the associated eigenmodes being spread out across the channel. For $E(1-\beta ) \ll 1$, with $E$ fixed, corresponding to strongly elastic dilute polymer solutions, $Re_c \propto (E(1-\beta ))^{-3/2}$ and the critical wavenumber $k_c \propto (E(1-\beta ))^{-1/2}$. The unstable eigenmode in this limit is confined in a thin layer near the channel centreline. These features are largely analogous to the centre-mode instability in viscoelastic pipe flow (Garg et al., Phys. Rev. Lett., vol. 121, 2018, 024502), and suggest a universal linear mechanism underlying the onset of turbulence in both channel and pipe flows of sufficiently elastic dilute polymer solutions. Although the centre-mode instability continues down to $\beta \sim 10^{-2}$ for pipe flow, it ceases to exist for $\beta < 0.5$ in channels. Whereas inertia, elasticity and solvent viscous effects are simultaneously required for this instability, a higher viscous threshold is required for channel flow. Further, in the opposite limit of $\beta \rightarrow 1$, the centre-mode instability in channel flow continues to exist at $Re \approx 5$, again in contrast to pipe flow where the instability ceases to exist below $Re \approx 63$, regardless of $E$ or $\beta$. Our predictions are in reasonable agreement with experimental observations for the onset of turbulence in the flow of polymer solutions through microchannels.