Milne-Thomson circle theorem
# Milne-Thomson circle theorem

In fluid dynamics the Milne-Thomson circle theorem or the circle theorem is a statement giving a new stream function for a fluid flow when a cylinder is placed into that flow. It was named after the English mathematician L. M. Milne-Thomson.

Let $f(z)$ be the complex potential for a fluid flow, where all singularities of $f(z)$ lie in $|z|>a$. If a circle $|z|=a$ is placed into that flow, the complex potential for the new flow is given by

    $w=f(z)+{\overline {f\left({\frac {a^{2}}{\bar {z}}}\right)}}=f(z)+{\overline {f}}\left({\frac {a^{2}}{z}}\right).$

with same singularities as $f(z)$ in $|z|>a$ and $|z|=a$ is a streamline. On the circle $|z|=a$, $z{\bar {z}}=a^{2}$, therefore

    $w=f(z)+{\overline {f(z)}}.$

## Example

Consider a uniform irrotational flow $f(z)=Uz$ with velocity $U$ flowing in the positive $x$ direction and place an infinitely long cylinder of radius $a$ in the flow with the center of the cylinder at the origin. Then $f\left({\frac {a^{2}}{\bar {z}}}\right)={\frac {Ua^{2}}{\bar {z}}},\ \Rightarrow \ {\overline {f\left({\frac {a^{2}}{\bar {z}}}\right)}}={\frac {Ua^{2}}{z}}$, hence using circle theorem,

    $w(z)=U\left(z+{\frac {a^{2}}{z}}\right)$

represents the complex potential of uniform flow over a cylinder.

