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

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