Does anyone know how to treat the singularity due to (1/r) in an axis-symmetric problem?

In stream-function vorticity method for 2D axis-symmetric problem:

Ur=(1/r)ds/dz Uz=-(1/r)ds/dr

when r->0 both->infinite. If the mesh size becomes small, it is very hard to get a solution unless you use very very small timestep.

Thanks a lot.

Unless I'm overlooking something (always a possibility), the radial velocity (Ur) has to be zero at r = 0 unless a mass source or sink exists at r = 0.

Also, at r = 0 d(Uz)/dr = 0; the Uz velocity must be symmetric about r = 0. This also says that the stream function is a constant (a streamline!) at r = 0.

These two can be combined to give a boundary condition for the vorticity at r = 0.