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A Stream functions and flow around sphere/cylinder

  1. Oct 22, 2016 #1

    joshmccraney

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    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?
     
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  3. Oct 22, 2016 #2

    vanhees71

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    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
    $$\vec{v}=\vec{e}_r \frac{1}{r^2 \sin \vartheta} \partial_{\vartheta} \psi-\vec{e}_{\vartheta} \frac{1}{r \sin \vartheta} \partial_r \psi.$$
     
  4. Oct 22, 2016 #3

    joshmccraney

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