next up previous
Next: Solution: 2D dipole with Up: 3.7 - Simple Potential Previous: Solution: Circulation .

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 $+x$ 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 $m$ a distance $2a$ 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 $+x$ direction.

(a)
Hint: write the superposition of a 3D source and a sink with strength $m$ a distance $2a$ from each other.
(b)
Hint: take the limit $a \rightarrow 0$.

Keyword Search


 
next up previous
Next: Solution: 2D dipole with Up: 3.7 - Simple Potential Previous: Solution: Circulation .