Milne-Thomson Circle Theorem

MILNE-THOMSON CIRCLE THEOREM Let f (z ) be the complex potential for a flow having no rigid boundaries and such that there are no singularities within the circle z = a . Then on introducing the solid cylinder z = a , with impermeable boundary, into the flow, the new complex potential for the fluid outside the cylinder is given by W = f ( z ) + f (a 2 / z ) for z ≥ a . Proof All singularities of f (z ) occur in the region z > a . Hence the singularities of

f (a 2 / z ) occur in the region a 2 / z > a , i.e., z < a . Thus the singularities of f (a 2 / z ) also lie in the region z < a . It follows that in the region z > a , the functions f (z ) and f ( z ) + f (a 2 / z ) both have the same analytical singularities. Thus both functions considered as complex potentials represent the same hydrodynamical distributions in the region z > a . The proof of the theorem is now completed by considering what happens on the circular boundary z = a . To this end, we write z = ae iθ on the boundary where θ is real. Then a 2 / z = ae −iθ = z on the circular boundary. Thus, on the boundary z = a , W = f ( z ) + f (a 2 / z ) = f ( z ) + f ( z ) ,

which is entirely real. Hence on the boundary, ψ = ImW = 0 . This shows that the circular boundary is a streamline across which no fluid flows. Hence z = a is a possible boundary for the new flow specified by the complex potential W = f ( z ) + f (a 2 / z ) .