Rotating flow about a fixed axis

A rotating flow about a specified axis has a swirl component $u_\theta=r\Omega$ . It may also have radial and axial components. The angular velocity $\Omega$ describes its local rotation rate around that axis.

Rigid rotation

The velocity $u=(-\Omega_0 x_2,\Omega_0 x_1,0)$ , with constant $\Omega_0$ , is rigid rotation. It is divergence free, has zero rate of strain, and satisfies both unforced Euler and Navier–Stokes equations with

p=\tfrac12\Omega_0^2(x_1^2+x_2^2),

since the Laplacian of this linear velocity is zero. On the whole space it has infinite kinetic energy.

Differential rotation

For an axisymmetric field, variation of $\Omega(r,z)$ with position is differential rotation. Its radial shear contribution satisfies

2D(u)_{r\theta}=\partial_r u_\theta-\frac{u_\theta}{r} =r\partial_r\Omega.

Consequently a nonzero azimuthal velocity does not by itself imply nonzero shear.