Linear instability of viscoelastic pipe flow

Abstract

Abstract

Subcritical and supercritical bifurcations in axisymmetric viscoelastic pipe flows

“…Nevertheless, the comparison of the predicted Re c with the transitional Reynolds number Re t observed in an experiment carried out by Chandra, Shankar & Das (2018) showed general consistency. Upon considering the shear-thinning effect by using the FENE-P model (finitely extensible nonlinear elastic model with Peterlin closure) in their scaling analysis, Chaudhary et al (2021) were able to derive Re c ∼ (E(1 − β)) −5/8 , which is consistent with the scaling Re t ∼ (E(1 − β)) −1/2 observed in the experiment. However, in the FENE-P model if the polymer maximum extensibility is small enough (when this parameter approaches infinity, the FENE-P model reduces to the Oldroyd-B model), the linear instability ceases to exist (Zhang 2021).…”

“…As for the bifurcation type of pipe flows, since the transition to turbulence in Newtonian pipe flow is known to be subcritical, a naive speculation would be to argue that the viscoelastic pipe flow will also transition to turbulence only via subcritical routes. However, the finding of the centre-mode instability enriches the picture (Garg et al 2018;Chaudhary et al 2021), implying and confirming the possibility of a supercritical transition route as envisioned by Graham (2014). Samanta et al (2013) first studied experimentally the EIT phenomenon in a viscoelastic pipe flow at a high polymer concentration of 500 p.p.m.…”

“…In the current work, we will perform a weakly nonlinear analysis of axisymmetric viscoelastic pipe flows (of Oldroyd-B fluids) based on multiple-scale expansion around the linear critical point (the damped mode was investigated in Meulenbroek et al 2003). The significance of the current work lies in (1) supplementing the works by Morozov & van Saarloos on the weakly nonlinear stability analysis of viscoelastic plane shear flows; (2) extending the works by Garg et al (2018) and Chaudhary et al (2021) to the study of the weakly nonlinear phase of the linearly unstable centre mode; and (3) providing a theoretical investigation of the bifurcation types in viscoelastic pipe flows in a large parameter space to understand and reconcile the experimental observations and numerical results in the studies reviewed above.…”

“…In this work we will focus on extending the centre-mode instability as discovered by Garg et al (2018). Till now, only the axisymmetric mode in viscoelastic pipe flows have been found to be linearly unstable in the literature (Garg et al 2018;Chaudhary et al 2021;Zhang 2021); therefore, it is legitimate to focus on the weakly nonlinear development of the axisymmetric mode in this work. For the axisymmetric viscoelastic pipe flow, the component-wise governing equations can be obtained from (2.3) as…”

“…The authors concluded that the novel linear instability is a subtle balance among fluid inertia, viscosity and polymer elasticity in viscoelastic pipe flows. Recently, the same research group extended their work to the exploration of instability in a larger parameter space and compared their predictions with direct numerical simulation (DNS) results and experimental measurements (Chaudhary et al 2021). Some of their new observations include that the eigenfunction of unstable modes is not localized near the pipe centreline as β approaches one with E of the order of 0.1; and that the laminar flow remains stable at quite low Re regardless of the values of E and β.…”