Source or sink at the origin
\( \phi = \dfrac{m}{2\pi}\ln r \)
\( \psi = \dfrac{m}{2\pi}\theta \)
\( v_r=\dfrac{\partial\phi}{\partial r}=\dfrac{1}{r}\dfrac{\partial\psi}{\partial\theta}=\dfrac{m}{2\pi r} \)
\( v_\theta=\dfrac{1}{r}\dfrac{\partial\phi}{\partial\theta}=-\dfrac{\partial\psi}{\partial r}=0 \)
Differentiating the velocity potential in polar coordinates gives the radial and tangential velocity components. The stream function gives the same components through \(v_r=(1/r)\partial\psi/\partial\theta\) and \(v_\theta=-\partial\psi/\partial r\). For a pure source or sink, flow is only radial, so \(v_\theta=0\).
Positive \(m\) gives outward radial flow. Negative \(m\) gives inward radial flow. The ideal model has a mathematical singularity at \(r=0\), so the small center disk marks a region where the model is not physical.
\(\phi=\) constant
\(\psi=\) constant
Flow type
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\(v_r=m/(2\pi r)\)
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φ at probe
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ψ at probe
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The streamline family is radial because \(\psi\) changes with angle only. The equipotential family is circular because \(\phi\) changes with distance from the origin only.