Application of Kelvin's Inversion Theorem to the Solution of Laplace's Equation Over a Domain That Includes the Unbounded Exterior of a Sphere

J. Russell[1]
[1]Florida Institute of Technology, Melbourne, FL, USA
Published in 2013

In simulating physical systems modeled as solutions of partial differential equations over an unbounded domain, a challenge arises owing to the impossibility to cover an unbounded domain with a mesh. A classical problem in fluid dynamics, a solid sphere of radius a_s is submerged in a tank of water (at rest) of much larger size. The water is disturbed by the sphere’s sudden acceleration in the z-direction. The sphere may initially translate only a short distance compared to its radius, and the fluid motion just afterwards is irrotational. The fluid velocity vector is the gradient of a potential, phi. Treating the water as incompressible, the fluid velocity is divergence-free and phi becomes a solution of Laplace's equation. One may generate this solution in COMSOL Multiphysics®.