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 United Kingdom 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

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

with same singularities as $f(z)$ in $|z|>a$ and $|z|=a$ is a streamline. On the circle $|z|=a$, $z\overline{z}=a^2$, 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 $f\left(\frac{a^2}{\overline{z}}\right)=\frac{U{a}^2}{\overline{z}},\Rightarrow \overline{f\left(\frac{a^2}{\overline{z}}\right)}=\frac{U{a}^2}{z}$, hence using circle theorem,

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

represents the complex potential of uniform flow over a cylinder.

See also

• Potential flow
• Conformal mapping
• Velocity potential
• Milne-Thomson method for finding a holomorphic function