Dipole (doublet flow)

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Dipole (doublet flow)

A Dipole is a superposition of a sink and a source with the same strength.

$\begin{figure} \centering\epsfig{file=lfig1013.eps,height=2in,clip=}\end{figure}$

2D dipole with orientation angle $\alpha$ .

$\begin{displaymath}\phi = \frac{ - \mu }{2\pi }\frac{x\cos \alpha + y\sin \alpha... ... }\frac{\cos \theta \cos \alpha + \sin \theta \sin \alpha }{r} \end{displaymath}$

$\begin{figure} \centering\epsfig{file=lfig1014.eps,height=1.5in,clip=}\end{figure}$

3D dipole at the origin oriented in the direction.

$\begin{displaymath}\phi = -\frac{\mu }{4\pi }\frac{x}{\left( {x^2 + y^2 + z^2} \right)^{3/2}} = - \frac{\mu }{4\pi }\frac{x}{r^3} \notag \end{displaymath}$

1.

Question: derive the expression for a 2D dipole with orientation angle $\alpha$ from the superposition of a source and a sink.

(a): Hint: write the superposition of a 2D source and a sink with strength a distance from each other.
(b): Hint: take the limit $a \rightarrow 0$ .

2.

Question: derive the expression for a 3D dipole at the origin with orientation in the

direction.

(a): Hint: write the superposition of a 3D source and a sink with strength a distance from each other.
(b): Hint: take the limit $a \rightarrow 0$ .

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