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Weakly nonlinear analysis of the viscoelastic instability in channel flow for finite and vanishing Reynolds numbers

Published online by Cambridge University Press:  08 April 2022

Gergely Buza Open the ORCID record for Gergely Buza [Opens in a new window] ,
Jacob Page Open the ORCID record for Jacob Page [Opens in a new window]  and
Rich R. Kerswell
Gergely Buza*
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, CB3 0WA, UK
Jacob Page*
Affiliation:
School of Mathematics, University of Edinburgh, EH9 3FD, UK
Rich R. Kerswell*
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, CB3 0WA, UK
*
†Email addresses for correspondence: gb643@cam.ac.uk, jacob.page@ed.ac.uk, r.r.kerswell@damtp.cam.ac.uk
†Email addresses for correspondence: gb643@cam.ac.uk, jacob.page@ed.ac.uk, r.r.kerswell@damtp.cam.ac.uk
†Email addresses for correspondence: gb643@cam.ac.uk, jacob.page@ed.ac.uk, r.r.kerswell@damtp.cam.ac.uk
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Abstract

The recently discovered centre-mode instability of rectilinear viscoelastic shear flow (Garg et al., Phys. Rev. Lett., vol. 121, 2018, 024502) has offered an explanation for the origin of elasto-inertial turbulence that occurs at lower Weissenberg numbers ($Wi$). In support of this, we show using weakly nonlinear analysis that the subcriticality found in Page et al. (Phys. Rev. Lett., vol. 125, 2020, 154501) is generic across the neutral curve with the instability becoming supercritical only at low Reynolds numbers ($Re$) and high $Wi$. We demonstrate that the instability can be viewed as purely elastic in origin, even for $Re=O(10^3)$, rather than ‘elasto-inertial’, as the underlying shear does not feed the kinetic energy of the instability. It is also found that the introduction of a realistic maximum polymer extension length, $L_{max}$, in the FENE-P model moves the neutral curve closer to the inertialess $Re=0$ limit at a fixed ratio of solvent-to-solution viscosities, $\beta$. At $Re=0$ and in the dilute limit ($\beta \rightarrow 1$) with $L_{max} =O(100)$, the linear instability can be brought down to more physically relevant $Wi\gtrsim 110$ at $\beta =0.98$, compared with the threshold $Wi=O(10^3)$ at $\beta =0.994$ reported recently by Khalid et al. (Phys. Rev. Lett., vol. 127, 2021, 134502) for an Oldroyd-B fluid. Again, the instability is subcritical, implying that inertialess rectilinear viscoelastic shear flow is nonlinearly unstable – i.e. unstable to finite-amplitude disturbances – for even lower $Wi$.

JFM classification

Non-Newtonian Flows: Viscoelasticity Nonlinear Dynamical Systems: Bifurcation
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 940 , 10 June 2022 , A11
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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References

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