I'm trying to determine the velocity equations, given a known vorticity distribution, and the requirement that the continuity equation be satisfied. In Cartesian, the equations are

w_z = dv/dx - du/dy

0 = du/dx + dv/dy

where (u,v) are the velocities in the (x,y) directions, respectively. I obtained functions for the velocities fairly easily, using a Gaussian distribution for w_z.

I was trying to do the same thing in an axisymmetric cylindrical coordinate system, so that the vorticity in the r-z plane is the same as in the x-y plane as above. Here

w_theta = dv/dz - du/dr

0 = dv/dr + v/r + du/dz

where (u,v) are the velocities in the (z,r) directions, respectively. However, I'm having a lot of trouble trying to determine the (z,r) velocities, given the same distribution as above, due to the extra term in the continuity equation. Using the same velocity fields in the (z,r) plane as the (x,y) plane produces the desired vorticity, but then continuity is not satisfied.

If anyone can give me any pointers, it would be greatly appreciated.