Spectral Representation of Permeability.

In this paper, we show that the permeability of porous media and that of a bubbly fluid are limiting cases of the complexified version of the two-fluid models posed in the 1990 paper by Lipton and Avellaneda. We assume the viscosity of the inclusion fluid is $z\mu_1$ and the viscosity of the hosting fluid is $\mu_1$, $z\in\field{C}$. The proof is carried out by construction of solutions for large $|z|$ and small $|z|$ by an iteration process similar with the one used in the 1993 paper by Bruno and Leo. Moreover, we also show that for a fixed microstructure, the permeabilities of these three cases share the same integral representation formula (IRF) with different values of $s'$, as long as the 'contrast parameter' $s':=\frac{1}{z-1}$ is not in the interval $[-\frac{2E_2^2}{1+2E_2^2},-\frac{1}{1+2E_1^2}]$, where the constants $E_1$ and $E_2$ are the extension constants that depend on the geometry of $Q_1$, $Q_2$ and $Q$.