On the Degenerate Scale of an Infinite Plane Containing Two Unequal Circles

Abstract

In this paper, we analytically derived the degenerate scale of an infinite plane containing two unequal circles and numerically implemented by using the BEM. We provide two methods to analytically derive the degenerate scale. One is using the degenerate kernel and the other is using the conformal mapping. The closed-form fundamental solution of the two-dimensional Laplace equation is expanded to the degenerate kernel form in order to analytically study the degenerate scale in the BIE. Moreover, we used the technique of the conformal mapping in order to analytically study the degenerate scale in the complex variables. Then, a boundary value problem can be transformed into a Green's function. Finally, we prove the equivalence of the two analytical formulas derived by using the degenerate kernel and by using the complex variables. They are also examined by using the BEM. Good agreement is made. Besides, the case of the two equal circles is just a special one of the present formula.