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 flows in an infinite pipe with circular cross section that are independent of the axial coordinate. We show that unlike the Newtonian ($\epsilon=0$) case, in the second grade model ($\epsilon \neq 0$ case), the time independent base flow exhibits transitions as the Reynolds number $R$ exceeds the critical threshold $R_c \approx 4.124 \epsilon^{-1/4}$ where $\epsilon$ is a material constant measuring the relative strength of second order viscous effects compared to inertial effects. At $R=R_c$, we find that generically 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 $R_c$. Our numerical calculations suggest that for low $\epsilon$ values, the system prefers a catastrophic transition where the bifurcation is subcritical. We also find that there is a Reynolds number $R_E$ with $R_E < R_c$ such that for $R<R_E$, the base flow is globally stable and attracts any initial disturbance with at least exponential speed. We show that $R_E \approx 12.87$ at $\epsilon=0$ and $R_E$ approaches $R_c$ quickly as $\epsilon$ increases.