Angular velocity about an axis

For $r>0$ , the angular velocity of a fluid about the $z$ -axis is the swirl velocity divided by radius:

\Omega(t,r,\theta,z)=\frac{u_\theta(t,r,\theta,z)}r.

Along a particle trajectory away from the axis, $d\theta/dt=\Omega(t,X(t))$ . Its units are inverse time when the angle is measured in radians.

Rigid and differential rotation

Rigid rotation with rate $\Omega_0$ has $u_\theta=r\Omega_0$ . When $\Omega$ varies spatially, different locations rotate at different rates. The angular velocity is not generally half the axial vorticity: for axisymmetric swirl,

\omega_z=\frac1r\partial_r(ru_\theta) =2\Omega+r\partial_r\Omega.

The quotient at $r=0$ is interpreted through smooth extension when the Cartesian field has the required axis regularity.