The Potential Exterior to Close-To-Touching Discs

Abstract

There are many two-dimensional physical systems governed by potentials that satisfy the Laplace’s equation and Dirichlet boundary condition. An elegant approach is to set up the problem in the complex plane and use the freedom of conformal mappings to map the region and equation to different domains. In this paper, I focus on the problem of a uniform flow past two close-to-touching discs and the goal is to determine the potential and velocity field. The problem becomes very singular if the separation of the two discs gets smaller and it poses a challenge to both numerical and analytical solutions. Numerically, we come up with a new method that combines the method of images with conformal mappings and it performs better than all existing numerical methods in terms of efficiency and robustness. Analytically, we have a “limit solution” that matches the true solution uniformly and the “limit solution” is more accurate when the discs are closer. This solution also gives explicit relations between the behavior of the flow and the separation of the two discs.