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.^[1][2] 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^[3]

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

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

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

Example

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

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

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