Asymptotic study of linear instability in a viscoelastic pipe flow

Abstract

It is recently found that viscoelastic pipe flows can be linearly unstable, leading to the possibility of a supercritical transition route, in contrast to Newtonian pipe flows. Such an instability is referred to as the centre mode, which was studied numerically by Chaudhary et al. (J. Fluid Mech., vol. 908, 2021, p. A11) based on an Oldroyd-B model. In this paper, we are interested in expanding the parameter space investigated and exploring the asymptotic scalings related to this centre instability in the Oldroyd-B viscoelastic pipe flow. It is found from the asymptotic analysis that the centre mode exhibits a three-layered asymptotic structure in the radial direction, a wall layer, a main layer and a central layer, which are driven by pure viscosity, axial and/or radial pressure gradient, and a combined effect of viscosity and elasticity, respectively. Depending on the relations of the control parameters, two regimes, the long-wavelength and short-wavelength centre instabilities, emerge, for which the central-layer thicknesses are of different orders of magnitude. Our large-Reynolds-number asymptotic predictions are compared to the numerical solutions of the original eigenvalue system, and favourable agreement is achieved, especially when the parameters approach their individual limits. In addition to revealing the dominant factors and their balances, the asymptotic analysis describes the instability system by reducing the number of control parameters, and furthermore explaining the collapse of the numerical results for different re-scalings.

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

“…the trend of parametric effects in the experimentally/numerically unavailable regime. Our results extend the linear scaling laws first observed by Garg et al (2018) in viscoelastic pipe flows to the nonlinear regime, utilising the theoretical tools developed in Wan et al (2021) and Dong & Zhang (2022). We hope that our results will evoke more future work along this direction and contribute to the general understanding of the nonlinear dynamics in viscoelastic flows.…”

“…(2018) in viscoelastic pipe flows to the nonlinear regime, utilising the theoretical tools developed in Wan et al. (2021) and Dong & Zhang (2022). We hope that our results will evoke more future work along this direction and contribute to the general understanding of the nonlinear dynamics in viscoelastic flows.…”

“…If they are neglected in the bulk region, then we would obtain a solution which is singular at both r = 1 and r = 0. Thus, a wall layer and a central layer must be taken into account, being similar to the asymptotic structure in Dong & Zhang (2022). Such an unstable mode is found to be possible only when E ∼ σ −1 , corresponding to the long-wavelength centre mode for Re Re c in Dong & Zhang (2022).…”

“…Thus, a wall layer and a central layer must be taken into account, being similar to the asymptotic structure in Dong & Zhang (2022). Such an unstable mode is found to be possible only when , corresponding to the long-wavelength centre mode for in Dong & Zhang (2022). However, when is close to , the three layers merge together, and so the and terms are kept.…”

“…The authors explained these scaling laws using a regular perturbation technique. Instead, by means of an asymptotic technique, Dong & Zhang (2022) analysed the same flow in the large- limit to explain these scaling laws. The authors found a three-layered structure of the centre-mode instability in both a long-wavelength regime and a short-wavelength regime.…”