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 $w = f(z)$ be the complex stream function for a fluid flow with no rigid boundaries and no singularities within $|z| = a$. If a circular cylinder $|z| = a$ is placed into that flow, the complex potential for the new flow is given by^[3]

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

See also

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

References

1. Batchelor, George Keith (1967). An Introduction to Fluid Dynamics. Cambridge University Press. p. 422. ISBN 0-521-66396-2.
2. Raisinghania, M.D. Fluid Dynamics.
3. Tulu, Serdar (2011). Vortex dynamics in domains with boundaries (PDF) (Thesis).