Meridional velocity - Knowlpedia

The meridional velocity relative to a chosen axis is

u_{\mathrm{mer}}=u_r e_r+u_z e_z.

It is often represented by the ordered pair $(u_r,u_z)$ on the $(r,z)$ half-plane. A meridional plane contains the symmetry axis; its radial and axial directions span this part of the velocity.

Axisymmetric incompressible representation

For an axisymmetric divergence-free velocity, a Stokes streamfunction $S$ locally represents these components by

u_r=-\frac1r\partial_z S,\qquad u_z=\frac1r\partial_r S.

The swirl component can be prescribed separately, subject to the fluid equations and axis regularity. The pair $(u_r,u_z)$ satisfies the weighted divergence relation $\partial_r(ru_r)+\partial_z(ru_z)=0$ .