Homogenization of a polymer flow through a porous medium. (English) Zbl 0863.76082

The paper is a mathematical derivation of the laws governing polymer flows through porous media using the tools of the homogenization theory. In the case of Newtonian flow, the homogenization leads to the classical Darcy’s law. In this study some non-Newtonian flows are studied. The attention of the authors is focussed on the two most widely used laws in engineering practice: the “power law”, and the “Carreau law”. Depending on the value of the Reynolds number compared to the small parameter of the homogenization procedure, a relation connecting the filtration velocity to the mean pressure gradient is derived, which may be linear, or nonlinear and nonlocal. For each situation the uniqueness theorem is proved.
Reviewer: Th.Lévy (Paris)


76S05 Flows in porous media; filtration; seepage
76A05 Non-Newtonian fluids
35Q35 PDEs in connection with fluid mechanics
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
Full Text: DOI


[1] Bear, J., ()
[2] Whitaker, S., Flow in porous media I: A theoretical derivation of Darcy law, Transport porous media, 1, 3-25, (1990)
[3] Lions, J.L., ()
[4] Sanchez-Palencia, E., Non-homogeneous media and vibration theory, () · Zbl 0432.70002
[5] Lions, J.L.; Sanchez-Palencia, E., Ecoulement d’un fluide viscoplastique de Bingham dans un milieu poreux, J. math. pures appl., 60, 341-360, (1981) · Zbl 0484.76009
[6] Bourgeat, A.; Mikelić, A., Note on the homogenization of Bingham flow through porous medium, J. math pures appl., 72, 405-414, (1993) · Zbl 0836.76091
[7] Agassant, J.F.; Avenas, P.; Sergent, J.Ph.; Carreau, P.J., ()
[8] Bird, R.B.; Armstrong, R.C.; Hassager, O., ()
[9] Serrin, J., Mathematical principles of classical fluid mechanics, (), 125-263
[10] Cattabriga, L., Su un problema al contorno relativo al sistema di equazioni di Stokes, (), 308-340 · Zbl 0116.18002
[11] Temam, R., ()
[12] Allaire, G., Homogenization of the Stokes flow in a connected porous medium, Asymptotic analysis, 2, 203-222, (1989) · Zbl 0682.76077
[13] Tartar L., Convergence of the Homogenization Process, Appendix of [4]. · Zbl 1188.35004
[14] Mikelić, A., Homogenization of nonstationary Navier-Stokes equations in a domain with a grained boundary, Annali mat. pura appl.(IV), CLVIII, 167-179, (1991) · Zbl 0758.35007
[15] Mjasnikov, V.P.; Mosolov, P.P., A proof of korn inequality, Soviet math. dokl., 12, 1618-1622, (1971) · Zbl 0248.52011
[16] Lipton, R.; Avellaneda, M., A Darcy law for slow viscous flow past a stationiary array of bubbles, (), 71-79 · Zbl 0850.76778
[17] Nguetseng, G., A general convergence result for a functional related to the theory of homogenization, (), 3 · Zbl 0688.35007
[18] Allaire, G., Homogenization and two-scale convergence, SIAM J. math. analysis, 23, 1482-1518, (1992) · Zbl 0770.35005
[19] Allaire, G., Homogénéisation et convergence à deux échelles. application à un problème de convection diffusion, C. R. acad. sci. Paris, 312 t., 581-586, (1991) · Zbl 0724.46033
[20] Hornung, U.; Jäger, W.; Mikelić, A., Reactive transport through an array of cells with semi-permeable membranes, Rairo m_{2}an, 28, 59-94, (1994) · Zbl 0824.76083
[21] Wu, Y.S.; Pruess, K.; Witherspoon, A., Displacement of a newtonian fluid by a non-Newtonian fluid in a porous medium, Transport porous media, 6, 115-142, (1991)
[22] Baranger, J.; Najib, K., Analyse numérique des écoulements quasi-newtoniens dont la viscosité obeit a la loi puissance ou la loi de carreau, Num. math., 58, 35-49, (1990) · Zbl 0702.76007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.