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 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[3]
- [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
References
- ↑ Batchelor, George Keith (1967). An Introduction to Fluid Dynamics. Cambridge University Press. p. 422. ISBN 0-521-66396-2. https://books.google.com/books?id=Rla7OihRvUgC&q=an+introduction+to+fluid+dynamics.
- ↑ Raisinghania, M.D. (December 2003). Fluid Dynamics. ISBN 9788121908696. https://books.google.com/books?id=wq3TU5tArTkC&dq=milne+thomson+circle+theorem&pg=PA211.
- ↑ Tulu, Serdar (2011). Vortex dynamics in domains with boundaries (PDF) (Thesis).
 |  |
