Stability and transitions of the second grade Poiseuille flow

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.