Conformal Mapping and flow normal to ellipse

[nickthequick]

Hi,

Given that the flow normal to a thin disk or radius r is given by

$\phi = -\frac{2rU}{\pi}\sqrt{1-\frac{x^2+y^2}{r^2}}$

where U is the speed of the flow normal to the disk, find the flow normal to an ellipse of major axis a and minor axis b.

I can only find the answer in the literature in one place, where it's stated

$\phi = -\frac{U b}{E(e)} \sqrt{1-\frac{x^2}{a^2}-\frac{y^2}{b^2}}$

where E(e) is the complete elliptical integral of the second kind and e is the eccentricity of the disk.

I have been trying to use the Joukowski map to send lines of equipotential of the disk to those of the ellipse, but I'm not sure how the complete elliptical integral of the second kind enters this picture.

Any suggestions, references, would be appreciated!

Nick

 

[nickthequick]

On second thought, the Joukowski map seems inappropriate here. I think the map I want is

$z\to a \cosh(\xi + i \eta)$ so that
$x=a\sinh (\xi) \cos(\eta)$ and $y = a\cosh (\xi) \sin(\eta)$.

This will effectively give me the change in functional form that I expect; however, I still don't see how this will modify the coefficient in the appropriate way.