  A simplified form of the vorticity equation is derived for arbitrary
coordinate systems. The present work unifies and extends the previous findings
that vorticity is conserved in planar Euler flow, while in axisymmetric Euler
rings it is the ratio of the vorticity to the distance from the symmetry axis
that is conserved. The unifying statement is that in any Euler flow, all
components of the vorticity tensor of a streamline coordinate system that are
normal to the streamline direction are conserved along streamlines. This is
true for both two- and three-dimensional flows, whether the flow is
axisymmetric or not, with or without swirl. What remains of the nonlinear
convective terms in the vorticity equation, after the mathematical
simplification, is the Lie derivative of the vorticity tensor with respect to
fluid velocity. A temporal derivative is defined which, when set to zero,
expresses either the continuity or vorticity equation (excluding the viscous
term), depending upon the argument supplied to it.

Fluid Dynamics