The Physical Significance of Non-convergent Integrals in Expressions for Effective Transport Properties

The idea of assigning effective properties to an inhomogeneous medium, for example a composite material or a suspension of particles in fluid, is well known [4, 9, 14], but some methods used to calculate these effective properties have been sources of controversy over the years. In particular, the appearance of non-convergent integrals (or sums) in several calculational schemes has generated many discussions on the relative merits of the different ways of calculating effective properties. Unfortunately these discussions remain largely unappreciated, with the result that new-comers to the field usually reinterpret for themselves the non-convergence, to the despair of those who have heard it all before. This paper presents a discussion of the significance of the non-convergent integrals which may help new workers understand the existing literature more easily. There is space here for only the briefest descriptions of the original papers, and consequently the details in this article will be intelligible only to those who have read or will read the original papers. For the general reader, the main conclusions of this paper are that