New results in rotating Hagen–Poiseuille flow

Abstract

New three-dimensional finite-amplitude travelling-wave solutions are found in rotating Hagen–Poiseuille flow (RHPF[Ω a , Ω p ]) where fluid is driven by a constant pressure gradient along a pipe rotating axially at rate Ω a and at Ω p about a perpendicular diameter. For purely axial rotation (RHPF[Ω a , 0]), the two-dimensional helical waves found by Toplosky & Akylas (1988) are found to become unstable to three-dimensional travelling waves in a supercritical Hopf bifurcation. The addition of a perpendicular rotation at low axial rotation rates is found only to stabilize the system. In the absence of axial rotation, the two-dimensional steady flow solution in RHPF[0, Ω p ] which connects smoothly to Hagen–Poiseuille flow as Ω p → 0 is found to be stable at all Reynolds numbers below 10 4 . At high axial rotation rates, the superposition of a perpendicular rotation produces a ‘precessional’ instability which here is found to be a supercritical Hopf bifurcation leading directly to three-dimensional travelling waves. Owing to the supercritical nature of this primary bifurcation and the secondary bifurcation found in RHPF[Ω a , 0], no opportunity therefore exists to continue these three-dimensional finite-amplitude solutions in RHPF back to Hagen–Poiseuille flow. This then contrasts with the situation in narrow-gap Taylor–Couette flow where just such a connection exists to plane Couette flow.