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

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

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

[math]\displaystyle{ w = f(z) + \overline{f(z)}. }[/math]

Example

Consider a uniform irrotational flow [math]\displaystyle{ f(z) = Uz }[/math] with velocity [math]\displaystyle{ U }[/math] flowing in the positive [math]\displaystyle{ x }[/math] direction and place an infinitely long cylinder of radius [math]\displaystyle{ a }[/math] in the flow with the center of the cylinder at the origin. Then [math]\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} }[/math], hence using circle theorem,

[math]\displaystyle{ w(z) = U \left(z + \frac{a^2}{z}\right) }[/math]

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