Conductivity of a medium containing a dense array of perfectly conducting square cylinders
``In his landmark paper, \textit{J. B. Keller} [J. Appl. Phys. 34, 991--993 (1963; Zbl 0108.21005)] obtained an approximation for the effective conductivity of a composite medium made out of a densely packed square array of perfectly conducting circular cylinders embedded in a conducting medium.'' (cited from the abstract). In the present paper, the same problem is studied for square cylinders either perfectly packed or arranged in the checkerboard pattern. Narrow gaps between the inclusions are represented by a small parameter \(\epsilon\) and the effective conductivity approaches \(+\infty\) as \(\epsilon\to 0\). The main idea of the paper is to combine the technique of matched asymptotic expansions between the region of the narrow gaps and the rest of the medium with specially chosen conformal Schwarz-Christoffel maps. The former step is needed to transform the narrow gap region into the upper half plane plane where the underlying Laplace equation can be solved explicitly. The final results, roughly, is that the effective conductivity diverges like \(\epsilon^{-1}\) for the square array and like \(\log\frac{1}{\epsilon}\) for the checkerboard pattern.
scientific article; zbMATH DE number 7443361
scientific article