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

TL;DR

This paper derives a scaling law for the Landau coefficient in viscoelastic pipe flows at large Weissenberg numbers, revealing how the flow's nonlinear behavior scales with $Wi$ and simplifying the analysis of such flows.

Contribution

It introduces an asymptotic and weakly nonlinear analysis approach to explain the large-$Wi$ scaling laws in viscoelastic pipe flows, reducing complexity and providing new insights.

Findings

The equilibrium amplitude scales as $Wi^{-1/2}$.

The reduced system captures the linear centre-mode instability.

The approach simplifies understanding of nonlinear dynamics at large $Wi$.

Abstract

This work explains a scaling law of the first Landau coefficient of the derived Ginzburg-Landau equation (GLE) 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. (2021), vol. 929, 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.