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

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