The limiting vortex in the similarity solution for a swirling flow

摘要

It is known that the similarity solution for a viscous swirling flow over a stationary disk does not exist if the driving vortex far away from the disk is a potential vortex, while the solution exists for a rigid body vortex. Previously, the breakdown has been determined to occur if the azimuthal velocity of the driving vortex decreases faster than a certain power of the radial distance from the axis of symmetry. The decay parameter at which the similarity solution ceases to exist is computed here by a more direct method, and the reason for the breakdown becomes apparent. The analysis confirms (and slightly improves) the known value of the parameter. The case where the fluid, now assumed to be conducting, is subject to an axial magnetic field and the asymptotic behavior of the solution far away from the axis are also briefly discussed.