Stability and transitions of the second grade Poiseuille flow

• Pipe Poiseuille flow of a second grade fluid is considered.
• Energy and linear stability thresholds for the Reynolds number are found.
• Possible transition scenarios are continuous or catastrophic.
• For small second order viscous effects, the preferred transition is catastrophic.

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 (ϵ=0ϵ=0) case, in the second grade model (ϵ>0ϵ>0 case), the time independent base flow exhibits transitions as the Reynolds number RR exceeds the critical threshold Rc=8.505ϵ−1/2Rc=8.505ϵ−1/2 where ϵϵ is a material constant measuring the relative strength of second order viscous effects compared to inertial effects.At R=RcR=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 RR tends to RcRc. Our numerical calculations suggest that for low ϵϵ values, the system prefers a catastrophic transition where the bifurcation is subcritical.We also show that there is a Reynolds number RERE with RE