Stability and transitions of the second grade Poiseuille flow

Saadet Özer, Taylan Şengül*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

9 Citations (Scopus)


In this study we consider the stability and transitions for the Poiseuille flow of a second grade fluid which is a model for non-Newtonian fluids. We restrict our attention to perturbation flows in an infinite pipe with circular cross section that are independent of the axial coordinate. We show that unlike the Newtonian (∈=0) case, in the second grade model (∈>0 case), the time independent base flow exhibits transitions as the Reynolds number R exceeds the critical threshold Rc=8.505∈-1/2 where ∈ is a material constant measuring the relative strength of second order viscous effects compared to inertial effects. At R=Rc, we find that the transition is either continuous or catastrophic and a small amplitude, time periodic flow with 3-fold azimuthal symmetry bifurcates. The time period of the bifurcated solution tends to infinity as R tends to Rc. Our numerical calculations suggest that for low ∈ values, the system prefers a catastrophic transition where the bifurcation is subcritical. We also show that there is a Reynolds number RE with RE<Rc such that for R<RE, the base flow is globally stable and attracts any initial disturbance with at least exponential speed. We show that the gap between RE and Rc vanishes quickly as ∈ increases.

Original languageEnglish
Pages (from-to)71-80
Number of pages10
JournalPhysica D: Nonlinear Phenomena
Publication statusPublished - 15 Sept 2016

Bibliographical note

Publisher Copyright:
© 2016 Elsevier B.V.


  • Energy stability
  • Linear stability
  • Poiseuille flow
  • Principal of exchange of stabilities
  • Second grade fluids
  • Transitions


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