Integral representation of hydraulic permeability

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 zμ1 and the viscosity of the hosting fluid is μ1∈R+ , z∈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 [−2E221+2E22,−11+2E21] , where the positive constants E1 and E2 are the extension constants that depend only on the geometry of the periodic pore space of the material.