On the general impossibility of controllable axi-symmetric Navier-Stokes motions

Steady Navier-Stokes motion of an incompressible linearly viscous fluid of uniform density and viscosity is considered. The motion is called controllable if the velocity field is independent of the kinematic viscosity. A condition for controllable steady Navier-Stokes motion is that the motion be circulation-preserving. For steady isochoric axisymmetric motions, a necessary and sufficient conditions for circulation-preserving flow is that the ratio of the vorticity magnitude to the distance from a representative point to the axis of symmetry be constant on a Lamb surface. In this paper a theorem is proved, stating that the only steady controllable rotational axisymmetric Navier-Stokes motions are motions for which this ratio is spatially constant and motions whose streamlines are parallel straight lines.