Potential problems on surfaces: With or without boundary, subjected to discrete distributed sources

Abstract

Abstract In this paper we derive analytical solutions for potential problems on different types of surfaces, such as cylinders, cones and spheres. These solutions complement the solution obtained for a sphere. Besides examining modes of truncation for a sphere, we also provide a complete coverage of all modes of truncation for cylinders and cones. Our interest focuses on two-dimensional conducting surfaces, that are fed with point sources of current. Our solution gave explicitly the voltage distribution on these surfaces, results that can straight-forwardly be applied to pressure profiles of a fluid flow in a gap between surfaces. Yet since every physical set-up takes place in the R 3 space, some adjustments were required. We focused on three types of geometric surfaces: cylinder, cone and sphere. As being effectively two-dimensional problems, the use of analytic functions and conformal maps has been used to obtain analytical solutions for all these geometries, all are mapped into the complex plane. Special attention is paid to examine particular cases of truncated ends by taking into account boundary conditions and flux considerations. We used reflections on cylinder inversions on cone and explore a spherical version of inversion on a sphere in order to satisfy the boundary conditions.