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

Bourgeat, Alain; Mikelić, Andro

Nonlinear Anal., Theory Methods Appl. 26, No. 7, 1221-1253 (1996).

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)

MSC:

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

Keywords:

power law; Carreau law; filtration velocity; mean pressure gradient; uniqueness

References:

[1]	Bear, J., (Dynamics of Fluids in Porous Media (1972), Elsevier: Elsevier New York) · Zbl 1191.76001
[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., (Quelques méthodes résolution des problèmes aux limites non linéaires (1969), Dunod Gauthier-Villars: Dunod Gauthier-Villars New York) · Zbl 0189.40603
[4]	Sanchez-Palencia, E., Non-homogeneous media and vibration theory, (Lecture Notes in Physics, 127 (1980), Springer Verlag: Springer Verlag Paris) · 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., (Polymer Processing, Principles and Modelling. (1991), Haser Publishers: Haser Publishers Munich)
[8]	Bird, R. B.; Armstrong, R. C.; Hassager, O., (Fluid Mechanics. Fluid Mechanics, Dynamics of Polymeric Liquid, Vol. 1. (1987), Wiley and Sons: Wiley and Sons Munich)
[9]	Serrin, J., Mathematical Principles of Classical Fluid Mechanics, (Handbuch der Physik, Vol. 8/1 (1959), Springer: Springer Berlin), 125-263
[10]	Cattabriga, L., Su un problema al contorno relativo al sistema di equazioni di Stokes, (R. Semin. mat. Padova, 31 (1961)), 308-340 · Zbl 0116.18002
[11]	Temam, R., (Navier-Stokes Equations (1984), Elsevier Science Publishers B.V.: Elsevier Science Publishers B.V. Berlin) · Zbl 0572.35083
[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].; 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, (Proc. R. Soc. Edinb., 114A (1990)), 71-79 · Zbl 0850.76778
[17]	Nguetseng, G., A general convergence result for a functional related to the theory of homogenization, (SIAM J. math. Analysis, 20 (1989)), 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