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The centre-mode instability of viscoelastic plane Poiseuille flow. (English) Zbl 1461.76025
Summary: 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, \(U_{\max}\). 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 \(\beta = \eta_s/\eta\); 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 critical Reynolds number, \(Re_c\), is around \(200\), with the associated eigenmodes being spread out across the channel. For \(E(1-\beta ) \ll 1\), with \(E\) fixed, corresponding to strongly elastic dilute polymer solutions, \(Re_c \propto (E(1-\beta ))^{-3/2}\) and the critical wavenumber \(k_c \propto (E(1-\beta ))^{-1/2}\). 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 [P. Garg et al., “Viscoelastic pipe flow is linearly unstable”, Phys. Rev. Lett. 121, No. 2, Article ID 024502, 6 p. (2018; doi:10.1103/PhysRevLett.121.024502)], 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 \(\beta \sim 10^{-2}\) 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 \(\beta \rightarrow 1\), the centre-mode instability in channel flow continues to exist at \(Re \approx 5\), again in contrast to pipe flow where the instability ceases to exist below \(Re \approx 63\), 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.
MSC:
76A10 Viscoelastic fluids
76F06 Transition to turbulence
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