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, Umax. The governing dimensionless groups are the Reynolds number Re = rho UmaxH/eta, the elasticity number E = lambda eta/(H-2 rho) and the ratio of solvent to solution viscosity beta = eta(s)/eta; here,. is the polymer relaxation time, H is the channel half-width and. is the fluid density. For experimentally relevant values (e.g. E similar to 0.1 and ss similar to 0.9), the critical Reynolds number, Rec, is around 200, with the associated eigenmodes being spread out across the channel. For E(1 - beta) << 1, with E fixed, corresponding to strongly elastic dilute polymer solutions, Rec. (E(1 - beta))(-3/2) and the critical wavenumber k(c) proportional to (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 similar to 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 -> 1, the centre-mode instability in channel flow continues to exist at Re approximate to 5, again in contrast to pipe flow where the instability ceases to exist below Re approximate to 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.

Automated summary

A modal stability analysis reveals that plane Poiseuille flow of an Oldroyd-B fluid is prone to instability towards a 'centre mode' with a phase speed close to the maximum base-flow velocity, Umax. The critical Reynolds number, Rec, is around 200 for experimentally relevant values, and the unstable eigenmodes are spread out across the channel. In the limit where E(1 - beta) << 1, the critical Reynolds number, Rec, decreases with (E(1 - beta))(-3/2) and the critical wavenumber, kc, is proportional to (E(1 - beta))(-1/2).