| Abstrakt: |
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 = ρUmaxH/η, the elasticity number E = λη/(H²ρ) and the ratio of solvent to solution viscosity β = ηs/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 ∼ 0.1 and β ∼ 0.9), the critical Reynolds number, Rec, is around 200, with the associated eigenmodes being spread out across the channel. For E(1 - β) 1, with E fixed, corresponding to strongly elastic dilute polymer solutions, Rec ∝ (E(1 - β))-3/2 and the critical wavenumber kc ∝ (E(1 - β))-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 β ∼ 10-2 for pipe flow, it ceases to exist for β < 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 β → 1, the centre-mode instability in channel flow continues to exist at Re ≈ 5, again in contrast to pipe flow where the instability ceases to exist below Re ≈ 63, regardless of E or β. Our predictions are in reasonable agreement with experimental observations for the onset of turbulence in the flow of polymer solutions through microchannels. [ABSTRACT FROM AUTHOR] |
