A modal stability analysis shows that plane Poiseuille flow of an Oldroyd-B fluid becomes unstable to a `center 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 $E = lambda eta/(H^2rho)$, and the ratio of solvent to solution viscosity $eta_s/eta$; 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 predicted critical Reynolds number, $Re_c$, for the center-mode instability is around $200$, with the associated eigenmodes being spread out across the channel. In the asymptotic limit of $E(1 -beta) ll 1$, with $E$ fixed, corresponding to strongly elastic dilute polymer solutions, $Re_c propto (E(1-beta))^{-frac{3}{2}}$ and the critical wavenumber $k_c propto (E(1-beta))^{-frac{1}{2}}$. The unstable eigenmode in this limit is confined in a thin layer near the channel centerline. The above features are largely analogous to the center-mode instability in viscoelastic pipe flow (Garg et al., Phys. Rev. Lett., 121, 024502 (2018)), and suggest a universal linear mechanism underlying the onset of turbulence in both channel and pipe flows of suffciently elastic dilute polymer solutions.
تحميل البحث