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 `center mode' with phase speed close the maximum base-flow velocity, $U_{max}$. The governing dimensionless groups are Reynolds number $Re = \rho U_{max} H/\eta$, elasticity $E \lambda \eta/(H^2\rho)$, and ratio solvent solution viscosity $\eta_s/\eta$; here, $\lambda$ is polymer relaxation time, $H$ channel half-width, $\rho$ density. For experimentally relevant values (e.g., \sim 0.1$ $\beta 0.9$), predicted critical number, $Re_c$, for center-mode instability around $200$, associated eigenmodes being spread out across channel. In asymptotic limit $E(1 -\beta) \ll 1$, $E$ fixed, corresponding strongly elastic dilute solutions, $Re_c \propto (E(1-\beta))^{-\frac{3}{2}}$ wavenumber $k_c (E(1-\beta))^{-\frac{1}{2}}$. eigenmode in this confined thin layer near centerline. above features largely analogous viscoelastic pipe (Garg et al., Phys. Rev. Lett., 121, 024502 (2018)), suggest universal linear mechanism underlying onset turbulence both flows suffciently solutions.