Source added to a uniform stream
\( \psi = Ur\sin\theta + \dfrac{m}{2\pi}\theta \)
\( \phi = Ur\cos\theta + \dfrac{m}{2\pi}\ln r \)
A source placed in a uniform stream creates a stagnation point upstream of the source. The streamline through that point can be interpreted as an inviscid solid boundary, forming a Rankine half-body.
\(\psi=\) constant
stagnation streamline / body boundary
The plot uses normalized axes, \(x/b\) and \(y/b\), so the half-body shape stays readable as \(m/U\) changes.
\(b=m/(2\pi U)\)
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Stagnation point
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Far half-width
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\(\psi_\mathrm{stag}=m/2\)
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\( b=\dfrac{m}{2\pi U},\qquad x_\mathrm{stag}=-b \)
\( \psi_\mathrm{stag}=\dfrac{m}{2},\qquad y_{\infty}=\pm\pi b \)
The source is an idealized line source. In the half-body interpretation, that singularity lies inside the streamline that becomes the model body, so the exterior flow field remains finite.