Governing equations for the stream function

Now we would like to get the identical answer by using the stream function.  The stream function can always be defined in two-dimensions from the continuity equation and for axisymmetric 3-D flows.  For spherical coordinates, the velocity components are (in terms of ψ)

=     = -

To get  the governing equation for ψ, we use our definition obtained from the continuity equation and substitute into the vorticity components.  Inspection of the components reveals that the and ==0 identically, the only interesting component is   (i. e. coordinate φ).  This component is   

We can simplify this to be

which = 0 if there is no vorticity.  Thus with the restriction of no vorticity, which should not be produced by an inviscid flow, then a simple PDE can be obtained for the stream function.

Note that this is NOT the Laplace equation (It would be for a planar flow!!), even though it looks similar.  Compare to the Laplace equation from above.