Q:

Fluid Mechanics: Stream Function for Axisymmetric flow

I have problem in understanding the result of stream function in Axisymmetric 3D flow:
I know that the result is (for spherical coordinates):
$$u_r=\frac{1}{r^2sin\theta}\frac{\partial\psi}{\partial\theta},$$  and
$$u_\theta=-\frac{1}{rsin\theta}\frac{\partial\psi}{\partial r}.$$
But I cannot see how this comes from the continuity equation.
What is derivation for this result?

A:

The continuity equation is described by:
$$ \frac{\partial \rho}{\partial t} + \nabla\cdot(\rho{\vec u}) = 0 $$
For incompressible and steady flow, it reduces to:
$$ \nabla\cdot{\vec u} = 0 $$ 
The incompressible continuity equation in spherical coordinates is:
$$ \nabla\cdot{\vec u} = \frac{1}{r^2}\frac{\partial}{\partial r}(r^2q_r)+\frac{1}{r}\frac{\partial}{\partial \theta}(q_\theta \sin\theta) = 0 $$
The condition a stream function must satisfy is that the mixed partial derivatives should be equal:
$$ \frac{\partial^2 \psi}{\partial r \partial \theta} = \frac{\partial^2 \psi}{\partial \theta \partial r}  $$
The stream function's relation to the radial and tangential components of the velocity as given in your question are:
$$ u_r = \frac{1}{r^2 \sin\theta}\frac{\partial \psi}{\partial \theta}, \hspace{5 pt} u_\theta = -\frac{1}{r\sin\theta}\frac{\partial \psi}{\partial r} $$
Substituting these into the continuity equation gives:
$$ \frac{1}{r^2}\frac{\partial}{\partial r}\left(\frac{r^2}{r^2 \sin\theta}\frac{\partial \psi}{\partial \theta}\right)+\frac{1}{r}\frac{\partial}{\partial \theta}\left(-\frac{\sin\theta}{r\sin\theta}\frac{\partial \psi}{\partial r}\right) = 0 $$
$$ \frac{1}{r^2\sin\theta}\frac{\partial}{\partial r}\left(\frac{\partial \psi}{\partial \theta}\right)-\frac{1}{r^2\sin\theta}\frac{\partial}{\partial \theta}\left(\frac{\partial \psi}{\partial r}\right) = 0 $$
$$ \frac{\partial^2 \psi}{\partial r \partial \theta} = \frac{\partial^2 \psi}{\partial \theta \partial r} $$
Which satisfies the required mathematical condition.
Note: I've just shown you how this satisfies the continuity equation. You should reason it out for yourself why the stream function satisfies the mixed partial derivatives condition. 
Hint: You're attempting to solve a famous partial differential equation, which I suggest you try to solve in Cartesian coordinates first, and then transform into other coordinate systems. You'll get the corresponding result in cylindrical coordinates, which is axisymmetric 3D flow.

