Specific angular momentum about an axis

The specific angular momentum about the $z$ -axis is

\Gamma=(x\times u)\cdot e_z=x_1u_2-x_2u_1=r u_\theta.

“Specific” means per unit mass. It differs from the angular velocity $u_\theta/r$ by the factor $r^2$ .

Axisymmetric evolution

For a smooth axisymmetric Navier–Stokes solution, axisymmetric pressure and force, and $r>0$ , the azimuthal momentum equation yields

(\partial_t+u_r\partial_r+u_z\partial_z)\Gamma =\nu\left(\partial_r^2-\frac1r\partial_r+\partial_z^2\right)\Gamma+r f_\theta.

Multiply the azimuthal component equation by $r$ and use $D_t r=u_r$ to derive this identity. With $\nu=0$ and no azimuthal forcing, $\Gamma$ is transported by the meridional motion.

Total angular momentum

When the integral converges, total axial angular momentum at density one is $\int\Gamma\,dx$ . Its conservation requires suitable force, boundary, and integrability assumptions; it is not part of the definition of $\Gamma$ .