Stream functions and flow around sphere/cylinder

[member 428835]

Hi PF!

I am wondering why we define velocity for polar coordinates as $$\vec{V} = \nabla \times \frac{\psi(r,z)}{r} \vec{e_\theta}$$ and why we define velocity in spherical coordinates as $$\vec{V} = \nabla \times \frac{\psi(r,\phi)}{r \sin \phi} \vec{e_\theta}$$

The only thing I don't understand is why we divide by $r$ and then $r \sin \phi$ (which are really sort of the same dimension, just the spherical is a projection). Is this simply to make the equations that follow nicer?

 

[vanhees71]

Yes, it's because your equations become simpler, although your spherical coordinates are strange. I'd expect (with $\vartheta$ the polar and $\varphi$ the azimuthal angles)
$$\vec{v}=\vec{\nabla} \times \left (\frac{\psi(r,\vartheta)}{r \sin \vartheta} \vec{e}_{\varphi} \right).$$
Then you get
$$\vec{v}=\vec{e}_r \frac{1}{r^2 \sin \vartheta} \partial_{\vartheta} \psi-\vec{e}_{\vartheta} \frac{1}{r \sin \vartheta} \partial_r \psi.$$

 

[member 428835]

Shoot, we may be using a different symbols for different angles. Awesome, thanks for your response!