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.