Small Concentration Expansion for the Effective Heat Conductivity of a Random Disperse Two-Component Material: An Assessment of Batchelor's Renormalization Method

摘要

The difficulty in the expansion of the effective properties of random disperse media in powers of the volume concentration c of the disperse phase presented by the divergence of certain integrals that perform averaging of two-particle approximations is considered. The random heat conduction problem analyzed by Jeffrey (1974) is treated using Batchelor's (1974) renormalization method. Batchelor's two-particle equation is extended to a hierarchical set of n-particle equations for arbitrary n. The solution of the hierarchy is seen to consist of a sequence of two, three, and more particle terms. The two and three-particle terms are calculated. It is proved that all i-particle terms (i greater than or = 2) can be averaged convergently showing that the hierarchical approach yields a well-defined expansion in integer powers of c of the effective conductivity. It follows that Jeffrey's expression for the effective conductivity is 0(c sq) - accurate.