On the Effective Thermal Conductivity of Dilute Dispersions: Highly Conducting Inclusions of Arbitrary Shape

The problem of determining the effective thermal conductivity Kij of dilute dispersions of highly conducting inclusions is considered in detail. Using the general results derived recently by the authors (1), this problem is shown to reduce to that of determining the solution to an integral equation of the first kind for an unknown surface distribution of singularities. The general expressions thus obtained for Kij are then applied to the case of very slender inclusions where relatively simple analytical results are found for particles of arbitrary cross-section. Finally, for inclusions that exhibit axial symmetry, the problem is shown to simplify to that of solving a one-dimensional integral equation for an unknown line distribution of singularities. Specific expressions for the effective thermal conductivity are also derived for a few representative examples.