Integral representation of hydraulic permeability

In this paper, we show that the permeability of a porous material (Tartar (1980)) and bubbly fluid (Lipton Avellaneda. Proc. R. Soc. Edinburgh Sect. A: Math . 114 (1990), 71–79) are limiting cases complexified version two-fluid models posed in Lipton Avellaneda ( 71–79). We assume viscosity inclusion is $z\mu _1$ hosting $\mu _1\in \mathbb {R}^{+}$ , $z\in {C}$ The proof carried out by construction solutions for large $|z|$ small with an iteration process similar to one used Bruno Leo Arch. Ration. Mech. Anal 121 (1993), 303–338) Golden Papanicolaou Commun. Math. Phys 90 (1983), 473–491) analytic continuation. Moreover, also fixed microstructure, permeabilities these three share same integral representation formula (3.17) different values contrast parameter $s:=1/(z-1)$ as long $s$ outside interval $\left [-\frac {2E_2^{2}}{1+2E_2^{2}},-\frac {1}{1+2E_1^{2}}\right ]$ where positive constants $E_1$ $E_2$ extension depend only on geometry periodic pore space material.