Stokes streamfunction for an axisymmetric meridional field

For an axisymmetric field with radial and axial components depending only on $(r,z)$ , a Stokes streamfunction $S$ uses the convention

u_r=-\frac1r\partial_z S,\qquad u_z=\frac1r\partial_r S,\qquad r>0.

Substitution in the cylindrical divergence formula shows that the meridional field $u_re_r+u_ze_z$ is divergence-free. The swirl component $u_\theta e_\theta$ is separate and, if axisymmetric, also has zero divergence.

Potential and axis

The azimuthal potential $A=(S/r)e_\theta$ has curl $u_re_r+u_ze_z$ . If $S=r^2a(r^2,z)$ with $a$ smooth on a neighborhood of the relevant half-plane, then

A=(-y\,a(x^2+y^2,z),\ x\,a(x^2+y^2,z),\ 0)

is smooth at the axis, so its curl is smooth there too. An arbitrary smooth function of $(r,z)$ need not have this property. Authors sometimes choose the opposite sign for $S$ ; the component formulas fix the convention.

Meridional motion

In an axisymmetric fluid, the streamfunction represents the meridional velocity. The swirl component is additional velocity data.