A general circle theorem

Abstract

Sunm~aryA general circle theorem is obtained which not only unifies all existing circle theorems, but allows new ones to be deduced. The results axe extended to allow for certain other finite boundaries, thus providing simple solutions for problems involving difficult boundary shapes. w 1. Introducti6.n. A general form of the circle theorem is here presented which no,~ only allows all the existing circle theorems to be quickly derived, but also permits-the deduction of new ones. This theorem is also extended to include certain other boundaries along which either the potential function or the current function takes a constant value. This greatly facilitates the solution of problems previously treated by ad hoc methods.The theorem was first enunciated by Milne-Thomson 1) in a hydrodynamical context and a corresponding theorem has been stated for electrical theory 9.), but a much more general circle theorem is now given which permits extension into other fields.

Generalized Circle and Sphere Theorems for Inviscid and Viscous Flows

“…The theorem can also be used for non-circular boundaries if one can find conformal transformation that maps the given boundary to a circle. The circle theorem of Milne-Thomson has also its analogue in electrostatics [29], Stokes flows [1,35] and in isotropic elasticity [20]. Furthermore, the circle theorem has also been extended to include surface singularity distributions [33,2].…”

“…The two-dimensional counterpart of the Weiss sphere theorem was obtained earlier by Milne-Thomson [23,24] which is widely known as the circle theorem. These basic theorems were extended by several authors in order to satisfy various boundary conditions that arise in various fields such as hydrodynamics, heat, magnetism and electrostatics [19,28,29,30,38,39]. The Kelvin's inversion was the key idea in those works involving a single spherical or a circular boundary.…”