  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 $E = \lambda
\eta/(H^2\rho)$, 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.
