A Pair of Perfectly Conducting Disks in an External Field

Abstract

A pair of non-overlapping perfectly conducting equal disks embedded in a two-dimensional background was investigated by the classic method of images, by Poincaré series, by use of the bipolar coordinates and by the elliptic functions in the previous works. In particular, successive application of the inversions with respect to circles were applied to obtain the field in the form of a series. For closely placed disks, the previous methods yield slowly convergent series. In this paper, we study the local fields around closely placed disks by the elliptic functions. The problem of small gap is completely investigated since the obtained closed form solution admits a precise asymptotic investigation in terms of the trigonometric functions when the gap between the disks tends to zero. The exact and asymptotic formulae are extended to the case when a prescribed singularity is located in the gap. This extends applications of structural approximations to estimations of the local fields in densely packed fiber composites in various external fields.

Effective Conductivity of Densely Packed Disks and Energy of Graphs

“…The corresponding asymptotic analysis justifies the structural approximation method and demonstrates the precision of its applications. The local fields estimated by continuous methods confirm the derived approximations [6,7] and used as initial approximations to get accurate analytical expressions for the local fields [8].…”

“…for some numbers s 1 , s 2 = 0, ±1. This dependence corresponds to the quasi-periodicity of the potential U(x) satisfying the relation (7). The formula (9) determines the energy per cell.…”