Double-diffusion models from a highly-heterogeneous medium. (English) Zbl 1049.35033

Summary: A distributed microstructure model is obtained by homogenization from an exact micro-model with continuous temperature and flux for heat diffusion through a periodically distributed highly-heterogeneous medium. This composite medium consists of two flow regions separated by a third region which forms the doubly-porous matrix structure. The homogenized system recognizes the multiple scale processes and the microscale geometry of the local structure, and it quantifies the distributed heat exchange across the internal boundaries. The classical double-diffusion models of Rubinstein (1948) and Barenblatt (1960) are obtained in non-isotropic form for the special case of quasi-static coupling in this homogenized system.


35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
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[1] Allaire, G., Homogenization and two-scale convergence, SIAM J. math. anal., 23, 1482-1518, (1992) · Zbl 0770.35005
[2] Arbogast, T.; Douglas, J.; Hornung, U., Derivation of the double porosity model of single phase flow via homogenization theory, SIAM J. math. anal., 21, 823-836, (1990) · Zbl 0698.76106
[3] Barenblatt, G.I.; Zheltov, I.P.; Kochina, I.N., Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks (strata), Prikl. mat. mekh., J. appl. math. mech., 24, 1286-1303, (1960) · Zbl 0104.21702
[4] Clark, G.W.; Showalter, R.E., Two-scale convergence of a model for flow in a partially fissured medium, Electron. J. differential equations, 28, (1999) · Zbl 0914.35013
[5] Douglas, J.; Peszyńska, M.; Showalter, R.E., Single phase flow in partially fissured media, Transport porous media, 28, 285-306, (1997)
[6] Hornung, U.; Showalter, R.E., Diffusion models for fractured media, J. math. anal. appl., 147, 69-80, (1990) · Zbl 0703.76080
[7] Hornung, U., Homogenization and porous media, (1997), Springer Berlin · Zbl 0872.35002
[8] Hornung, U., Models for flow and transport through porous media derived by homogenization, (), 201-222 · Zbl 0885.35010
[9] Lee, A.I.; Hill, J.M., On the general linear coupled system for diffusion in media with two diffusivities, J. math. anal. appl., 89, 530-557, (1982) · Zbl 0493.35051
[10] L.I. Rubinstein, On the problem of the process of propagation of heat in heterogeneous media, Izv. Akad. Nauk SSSR Ser. Geogr. 1, 1948
[11] Showalter, R.E., Monotone operators in Banach space and nonlinear partial differential equations, Mathematical surveys and monographs, vol. 49, (1997), American Mathematical Society Providence, RI · Zbl 0870.35004
[12] Showalter, R.E., Distributed microstructure models of porous media, (), 155-163 · Zbl 0805.76082
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