Conduction through a random suspension of spheres

Abstract

Abstract The conduction of heat (or electricity) through a stationary random suspension of spheres is studied for a volume fraction of the spheres (c) which is small. The work of Maxwell (1873) is extended to calculate the flux of heat exactly to order c2 by using the method of Batchelor (1972), which reduces the problem to a consideration of interactions between pairs of spheres while avoiding the usual convergence difficulties. The result depends on the way in which pairs of spheres are distributed with respect to each other; for the case of all possible pair configurations being equally probable the coefficient of c2 is found explicitly for all values of the ratio of conductivities of the two phases. The results also apply to permittivities and permeabilities of suspensions.