Stability of Poiseuille flow in pipes, annuli, and channels

The value of $R = 180$ which has been given by Orr [1] as a limit for sure stability of Hagen-Poiseuille flow is incorrect. A lower value, $R = 82.88$, can be associated with an eigenfunction possessing a first mode azimuthal variation $\left ( {N = 1} \right )$ and no streamwise variation. This eigenfunction is obtained as an exact solution of the appropriate Euler equation. A yet lower value, $R = 81.49$, is associated with a spiral mode with $N = 1$ and wave number $\alpha \approx 1$. Corresponding results are obtained for Poiseuille flow between cylinders. For all but the very smallest radius ratios the smallest eigenvalue of Euler’s equation is assumed for the purely azimuthal disturbance. The mode shape for the pipe flow is consistent with the experimental situation as it is now understood [2], though the stability limit is much smaller than the experimental value $\left ( {R \approx 2100} \right )$. For the annulus, the variation of the energy limit with the radius ratio (from pipe flow to channel flow) is consistent with the experimentally observed stability limit. All the results of the energy analysis hold equally if the pipe is in rigid rotation about its axis. If the rotation is “fast” linear and energy results nearly coincide.