The overall sink strength of an inhomogeneous lossy medium. Part I: self-consistent estimates

Steady-state diffusion in a medium containing a continuous distribution of sinks is considered. The medium comprises a matrix, with sink strength $\text{k}{}_2{}^2$, which contains a random array of identical spherical inclusions, with sink strength $\text{k}{}_1{}^2$. On a macroscopic scale, the medium appears homogeneous, with uniform sink strength $\text{k}\text{?}{}^2$. This work is devoted to the estimation of the overall sink strength $\text{k}\text{?}{}^2$, in terms of $\text{k}{}_1{}^2\text{, k}{}_2{}^2$ and the statistics of the distribution of the inclusions. Part I discusses three distinct schemes of self-consistent type. One is based upon a simple embedding procedure and makes no explicit allowance for spatial correlations. The other two make use, in different ways, of the quasicristalline approximation (QCA). Part II develops variational principles which yield bounds for $\text{k}\text{?}{}^2$. The self-consistent estimates are interpreted relative to the variational formulation and explicit numerical results are presented.