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, $U_{max}$. The governing dimensionless groups are the Reynolds number $Re = \rho U_{max} H/\eta$, the elasticity number $E =...

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Veröffentlicht in:Journal of fluid mechanics 2021-03, Vol.915, Article A43
Hauptverfasser: Khalid, Mohammad, Chaudhary, Indresh, Garg, Piyush, Shankar, V., Subramanian, Ganesh
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Subramanian, Ganesh
description 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 (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 $\beta \sim 10^{-2}$ 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.
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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 (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 $\beta \sim 10^{-2}$ for pipe flow, it ceases to exist for $\beta &lt; 0.5$ 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 $\beta$. 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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 $\beta \sim 10^{-2}$ for pipe flow, it ceases to exist for $\beta &lt; 0.5$ 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 $\beta$. 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Fluid Mech</addtitle><date>2021-03-12</date><risdate>2021</risdate><volume>915</volume><artnum>A43</artnum><issn>0022-1120</issn><eissn>1469-7645</eissn><abstract>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. 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subjects Base flow
Channel flow
Dilution
Elasticity
Experiments
Flow stability
Flow velocity
Fluid dynamics
Fluid flow
Inertia
Instability
JFM Papers
Laminar flow
Microchannels
Phase velocity
Pipe flow
Polymers
Relaxation time
Reynolds number
Solvents
Stability analysis
Turbulence
Turbulent flow
Viscoelasticity
Viscosity
Wavelengths
title The centre-mode instability of viscoelastic plane Poiseuille flow
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