Group Structure in Circle and Sphere Theorems

Abstract

In this paper, we study five different types of problems in potential and viscous flows past intersecting circles or spheres in two and three dimensions, respectively, with different angles of intersection. We observe a striking resemblance in the form of the solutions in all these cases by introducing certain operators L and M which generate a group G. By introducing a procedure called ‘closure’ which determines the order of the group G, we give a general method to discuss the problem of flow past two intersecting circles or spheres in potential and Stokes flows with different angles of intersection.

Viability of circle and sphere theorems in potential theory and hydrodynamics via Maxwell's conjecture

“…We begin Section 3 with Ranger's approach 12 to the problem of two intersecting circles or spheres and go on to explicitly state the errors made in Ranger's work and the subsequent fallacious circle and sphere theorems in Refs. 13, 14. Maxwell's conjecture on intersecting spheres, which contradicts the claims of these works, is presented and briefly discussed in the latter part of the same section.…”

“…As will be seen later, the results reported in Refs. 13, 14, which address all angles of intersection that are rational multiples of π, are built upon Ranger's faulty assumption and are therefore not valid. In the case that the angle of intersection is $$\frac{\pi }{n}$$, however, circle and sphere theorems were generalized to obtain viable solutions for inviscid and viscous flows in Ref.…”

“…However, the lead actors in Maxwell's conjecture are images of singularities under spherical inversions, and the problems in Refs. 12–14 provide the reader with some simple concrete examples to keep in mind, so we provide some intermediate sections laying out the pertinent constructions of these works. We begin Section 3 with Ranger's approach 12 to the problem of two intersecting circles or spheres and go on to explicitly state the errors made in Ranger's work and the subsequent fallacious circle and sphere theorems in Refs.…”

“…A family of counterexamples refuting claims in Refs. 12–14 is given in Section 5. The construction of image points for special cases used in Ranger's work is detailed in the same section to make the direct implications of Maxwell's conjecture transparent and straightforward.…”

“…In the concluding Section 7, we restate the validity of Kelvin's inversion method for solving problems involving intersecting circles or spheres and reiterate the implications of Maxwell's conjecture for the results given in Refs. 12–14. The numerical values of image points for some additional simple examples are listed in the Appendix.…”