Integral representation of hydraulic permeability

Source abstract

In this paper, we show that the permeability of a porous material (Tartar (1980)) and that of a bubbly fluid (Lipton and Avellaneda. Proc. R. Soc. Edinburgh Sect. A: Math . 114 (1990), 71–79) are limiting cases of the complexified version of the two-fluid models posed in Lipton and Avellaneda ( Proc. R. Soc. Edinburgh Sect. A: Math . 114 (1990), 71–79). 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 similar to the one used in Bruno and Leo ( Arch. Ration. Mech. Anal . 121 (1993), 303–338) and Golden and Papanicolaou ( Commun. Math. Phys . 90 (1983), 473–491) 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 (3.17) with different values of contrast parameter $s:=1/(z-1)$ , as long as $s$ is outside the interval $\left [-\frac {2E_2^{2}}{1+2E_2^{2}},-\frac {1}{1+2E_1^{2}}\right ]$ , 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.