Viscous Centre Modes in the Stability of Swirling Poiseuille Flow

Centre modes in the neighbourhoods of both branches of the neutral curve are identified for viscous rotating flow in a pipe when the Reynolds number is sufficiently large. Limit equations satisfied by these modes are established, and solutions are computed as functions of the azimuthal wavenumber and one additional parameter, μ say, representing the distance from a neutral curve; these compare favourably with existing calculations of the full equations at large but finite values of R. The question of the attainment of an inviscid limit as |μ|--> ∞ is addressed, and it is shown that the solution on the unstable side of the neutral curve is dominantly viscous. The resulting highly oscillatory viscous modes are examined and are shown to be present throughout the region bounded by the neutral curve. It is anticipated that the results may have application in the study of vortex breakdown.