The center-mode instability of viscoelastic plane Poiseuille flow

Abstract

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 $Re = ρU_{max} H/η$, the elasticity number $E = λη/(H^2ρ)$, and the ratio of solvent to solution viscosity $η_s/η$; here, $λ$ is the polymer relaxation time, $H$ is the channel half-width, and $ρ$ is the fluid density. For experimentally relevant values (e.g., $E \sim 0.1$ and $β\sim 0.9$), 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 $E(1 -β) \ll 1$, with $E$ fixed, corresponding to strongly elastic dilute polymer solutions, $Re_c \propto (E(1-β))^{-\frac{3}{2}}$ and the critical wavenumber $k_c \propto (E(1-β))^{-\frac{1}{2}}$. 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.