Viscous flows in rotating pipes with internal cavities and related vortex breakdowns

The problem of predicting vortex breakdown, which is characterized by a drastic change in core size and trajectory, is still basically unsolved. In the present paper, viscous flow in a long rotating pipe is analyzed, as a model for the type of breakdown observed in wake vortices. The full Navier-Stokes equations are shown to simplify to a set of unlinked partial differential equations. These equations allow for two separate solutions, for both full pipe flows (the classic rotating Hagen-Poiseuille flow) and cavity flows, in which a central area, with no net mass flux or viscous dissipation, is produced. Conditions permitting switching from one type of flow to the other, with no change in the mass flux or velocities at the pipe surface, are found. It is then argued that these conditions indicate when vortex breakdown will occur.