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 Poincar´e 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.
Highlights
The famous Villat-Dini formula [12,31] solves the Dirichlet problem for a circular annulus R−1 < |w| < R (R > 1) in terms of the Poisson type integral with the kernel expressed through the elliptic functions
In this paper, following [14, 19] we first construct an exact solution of the modified Dirichlet problem for a doubly connected circular domain which describes the field around two perfectly conducting disks when an external flux is given at infinity
Fields around two particles embedded in a two-dimensional background were investigated by various methods
Summary
The famous Villat-Dini formula [12,31] (see [2]) solves the Dirichlet problem for a circular annulus R−1 < |w| < R (R > 1) in terms of the Poisson type integral with the kernel expressed through the elliptic functions. In order to extend the fast method [22] to general boundary value problems, first, we need to construct simple asymptotic formulae for the local fields for a two-particle problem Such an investigation can be used in the hybrid basis scheme [9] as the basic elements in the series representations. In this paper, following [14, 19] we first construct an exact solution of the modified Dirichlet problem for a doubly connected circular domain which describes the field around two perfectly conducting disks when an external flux is given at infinity
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