Integral Representation of Hydraulic Permeability

In this paper, we show that the permeability of a porous {material} and that of a bubbly fluid are limiting cases of the complexified version of the two-fluid models posed in {Lipton_Avellaneda_1990}. We assume the viscosity of the inclusion fluid is $z\mu_1$ and the viscosity of the hosting fluid is $\mu_1\in \mathbb{R}^+$, $z\in\mathbb{C}$. The proof is carried out by the construction of solutions for large $|z|$ and small $|z|$ with an iteration process and the analytic continuation. Moreover, we also show that for a fixed microstructure, the permeabilities of these three cases share the same integral representation formula (IRF) in Equation (99) with different values of contrast parameter $s:=1/(z-1)$, as long as $s$ is outside the interval $[-\frac{2E_2^2}{1+2E_2^2},-\frac{1}{1+2E_1^2}]$, where the positive constants $E_1$ and $E_2$ are the extension constants that depend only on the geometry of the periodic pore space of the material.