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On the large-Weissenberg-number scaling laws in viscoelastic pipe flows

Published online by Cambridge University Press:  28 June 2022

Dongdong Wan ,
Ming Dong Open the ORCID record for Ming Dong [Opens in a new window]  and
Mengqi Zhang Open the ORCID record for Mengqi Zhang [Opens in a new window]
Dongdong Wan*
Affiliation:
Department of Mechanical Engineering, National University of Singapore, 9 Engineering Drive 1, 117575, Singapore
Ming Dong
Affiliation:
State Key Laboratory of Nonlinear Mechanics, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, PR China
Mengqi Zhang
Affiliation:
Department of Mechanical Engineering, National University of Singapore, 9 Engineering Drive 1, 117575, Singapore
*
†Email address for correspondence: [email protected]
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Abstract

This work explains a scaling law of the first Landau coefficient of the derived Ginzburg–Landau equation in the weakly nonlinear analysis of axisymmetric viscoelastic pipe flows in the large-Weissenberg-number ($Wi$) limit, recently reported in Wan et al. (J. Fluid Mech., vol. 929, 2021, A16). Using an asymptotic method, we derive a reduced system, which captures the characteristics of the linear centre-mode instability near the critical condition in the large-$Wi$ limit. Based on the reduced system we then conduct a weakly nonlinear analysis using a multiple-scale expansion method, which readily explains the aforementioned scaling law of the Landau coefficient and some other scaling laws. Particularly, the equilibrium amplitude of disturbance near linear critical conditions is found to scale as $Wi^{-1/2}$, which may be of interest to experimentalists. The current analysis reduces the numbers of parameters and unknowns and exemplifies an approach to studying the viscoelastic flow at large $Wi$, which could shed new light on the understanding of its nonlinear dynamics.

JFM classification

Non-Newtonian Flows: Viscoelasticity Instability: Nonlinear instability
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 944 , 10 August 2022 , A21
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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References

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