The free vortex.

This is for a flow pattern of concentric circles that is not driven mechanically but left to find its own relationships between radius and pressure and radius and velocity and still be stable. It is a flow pattern that cannot have a steady state because at every circle there is a velocity gradiant and therefore a shearing force owing to the effects of viscosity with no way to drive it.

 

We need some condition comparable to that for the forced vortex that must be fulfilled as well as the condition for stable swirling flow.