Heat Conduction of 2D Composite Materials with Symmetric Inclusions: a Model and Reduction to a Vector-Matrix Problem

Abstract

We consider steady potential heat conduction of a cylindrical composite material with the special geometry. The matrix is modelling by the unit disc with di®erent conductivity of six equal sectors. Inclusions (having di®erent conductivity too) are symmetrically situated discs non-intersecting boundary of sectors. Mixed boundary conditions on parts of the boundary of matrix and matrix-inclusions leads to di®erent model of composite materials. A new method to study the corresponding mathematical model is proposed. It is based on the reduction of the problem to the vector-matrix boundary value problem for analytic vectors. The method is connected with the approach by Zhorovina and Mityushev to the study of R-linear boundary value on a fan-shaped domain.