A method for the calculation of the effective transport properties of suspensions of interacting particles

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Abstract
The early attempts at calculating effective transport properties of suspensions of interacting spherical particles resulted in non-absolutely convergent expressions. In this paper we provide a physical interpretation for these convergence difficulties and we present a new method of determining the effective transport properties which clarifies difficulties in existing methods.This method, which is described for simplicity in the context of the thermal conduction problem, is based on an expression that gives the temperature gradient ∇T at a point x in the matrix in terms of integrals over the surrounding particles and an integral over a large surface Γ which encloses x and which we term the ‘macroscopic boundary’. Without the integral over Γ, this expression for ∇T would be non-absolutely convergent, for the contribution to ∇T(x) from a distant particle is proportional to 1/r3, where r is the distance of the particle from x. On comparing the expression for ∇T with the formula used by Rayleigh (1892) in his investigation of the effective conductivity of a cubic array of spheres, we find that Rayleigh's convergence difficulties arose simply from an incorrect assessment of the macroscopic boundary integral.By combining the expression for ∇T(x) with a formula for the dipole strength of a sphere placed in an ambient temperature field, we obtain a convergent expression relating the dipole strength of a sphere to integrals over the surrounding particles. An expression for the effective conductivity of a random suspension of spheres correct to O(ϕ2) is obtained simply by averaging this expression for the thermal dipole strength. By a similar procedure we obtain expressions for the effective viscosity and effective elastic moduli correct to O(ϕ2). Most of these results have been obtained by earlier workers using a ‘renormalization’ procedure due to Batchelor; the method presented here has the advantage that the renormalization quantity arises naturally from the macroscopic boundary integral referred to earlier, so there is no uncertainty about its choice.

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