What is the complex potential of a uniform flow around an ellipse?

Question

I have the complex potential of a uniform flow around a circle which is:

$Ω = Uz+\frac{Ua^2}{z}$

I was told that the complex potential of a uniform flow around an ellipse is:

$Ω = Uz+\frac{U(a^2+b^2)}{z}$

where $a$ is the semi-minor axis and $b$ is the semi-major axis.

Is this correct? I was wondering since I noticed that if I had an ellipse with a semi-minor axis of 3 and a semi-major axis of 4, the entire flow field would act the same as if it were a circle with radius 5.

Answer 1

• Elliptic cylinder in uniform (horizontal) flow:

$$f(z)=U \left( \frac{az-b\sqrt{z^2-a^2+b^2}}{a-b} \right)$$

• Oblique flow around an elliptic cylinder:

$$f(z)=U \left( \frac{z+\sqrt{z^2-a^2+b^2}}{2} \right)+ \overline{U} \left( \frac{a+b}{a-b} \right) \left( \frac{z-\sqrt{z^2-a^2+b^2}}{2} \right)$$