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Milne-Thomson circle theorem

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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 English mathematician L. M. Milne-Thomson.

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

w = f ( z ) + f ( a 2 z ¯ ) ¯ = f ( z ) + f ¯ ( a 2 z ) . {\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 f ( z ) {\displaystyle f(z)} in | z | > a {\displaystyle |z|>a} and | z | = a {\displaystyle |z|=a} is a streamline. On the circle | z | = a {\displaystyle |z|=a} , z z ¯ = a 2 {\displaystyle z{\bar {z}}=a^{2}} , therefore

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

Example

Consider a uniform irrotational flow f ( z ) = U z {\displaystyle f(z)=Uz} with velocity U {\displaystyle U} flowing in the positive x {\displaystyle x} direction and place an infinitely long cylinder of radius a {\displaystyle a} in the flow with the center of the cylinder at the origin. Then f ( a 2 z ¯ ) = U a 2 z ¯ ,   ⇒   f ( a 2 z ¯ ) ¯ = U a 2 z {\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,

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

represents the complex potential of uniform flow over a cylinder.

See also

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. (December 2003). Fluid Dynamics. S. Chand. ISBN 9788121908696.
  3. Tulu, Serdar (2011). Vortex dynamics in domains with boundaries (PDF) (Thesis).
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