Axisymmetric Flow of an Ideal Incompressible Fluid About a Solid Torus

Abstract

Abstract This paper contains an exact solution in series form to the problem presented by the irrotational, axisymmetric flow of an ideal, incompressible fluid past a solid torus of circular cross section. At infinity the fluid is assumed to be in a state of uniform motion parallel to the axis of the torus. The solution is based on the use of toroidal coordinates, and is given in terms of Legendre functions of fractional order as well as complete and incomplete elliptic integrals of the first and second kind. The individual component solutions employed here are interpreted physically through their relation to the basic ring singularities represented by the source ring and the vortex ring. The problem is first approached via Stokes’s stream function exclusively and is subsequently re-solved independently in terms of the velocity potential alone. The convergence of the pertinent series developments is found to be unusually favorable, and a complete streamline pattern, corresponding to an illustrative numerical example, is included.