Electrostatic fields of an infinite medium containing arbitrarily positioned coated cylinders

We propose a framework for evaluation of the electrostatic fields in an unbounded isotropic medium containing a number of arbitrarily dispersed circular cylinders or coated cylinders subjected to a remotely prescribed potential field. The cylinders or coated cylinders could be at most cylindrically orthotropic, and may have different radii with different conductivities. The approach is based on a multipole expansion formalism, together with a construction of consistency conditions and translation operators. This main procedure is inspired from an ingenious concept of the classic work of Lord Rayleigh [1], in which the effective conductivity of a periodic array of circular disks or spheres is considered. In the present formulation, we expand the potential field versus various local coordinates with origins positioned at the inclusions' centers. The key step is to link the potential data with the outer applied field, which is accomplished by the use of Green's second identity in the matrix domain. We show that the coefficients of field expansions are governed by an infinite set of linear algebraic equations. Numerical results are presented for a few different configurations. We have verified our numerical solutions for a simplified configuration with those obtained from the bipolar coordinate transformation.