Milne-Thomson circle theorem explained

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

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^[3] w = f(z) + \overline{f\left( (a ^{2})/(\bar{z) } \right)} = f(z) + \overline f\left(\frac \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} , thereforew = 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( (a ^{2})/(\bar z) \right) = (Ua ^{2})/(\bar z) , ⇒ \overline{f\left( (a ^{2})/(\bar{z) } \right)} = \frac, hence using circle theorem,w(z) = U \left(z + (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

Notes and References

1. Book: Batchelor, George Keith. George Batchelor

   . George Batchelor. An Introduction to Fluid Dynamics. 1967. Cambridge University Press. 0-521-66396-2. 422.

2. Book: Raisinghania, M.D.. Fluid Dynamics. December 2003. 9788121908696.
3. Tulu. Serdar. Vortex dynamics in domains with boundaries. 2011.