DOI: 10.1017/s0022112089002211. Flow in random porous media: mathematical formulation, variational principles, and rigorous bounds

Flow in random porous media: mathematical formulation, variational principles, and rigorous bounds

10.1017/s0022112089002211

Crossref journal-article

Cambridge University Press (CUP)

Journal of Fluid Mechanics (56)

Abstract

The problem of the slow viscous flow of a fluid through a random porous medium is considered. The macroscopic Darcy's law, which defines the fluid permeabilityk, is first derived in an ensemble-average formulation using the method of homogenization. The fluid permeability is given explicitly in terms of a random boundary-value problem. General variational principles, different to ones suggested earlier, are then formulated in order to obtain rigorous upper and lower bounds onk. These variational principles are applied by evaluating them for four different types of admissible fields. Each bound is generally given in terms of various kinds of correlation functions which statistically characterize the microstructure of the medium. The upper and lower bounds are computed for flow interior and exterior to distributions of spheres.

Bibliography

Rubinstein, J., & Torquato, S. (1989). Flow in random porous media: mathematical formulation, variational principles, and rigorous bounds. Journal of Fluid Mechanics, 206, 25â46.

Authors 2

Jacob Rubinstein (first)
S. Torquato (additional)

References 37 Referenced 142

Rubinstein, J. & Torquato, S. 1988 Diffusion-controlled reactions: mathematical formulation, variational principle, and rigorous bounds.J. Chem. Phys. 88,6372.
Berryman, J. G. 1983 Random close packing of hard spheres and disks..Phys. Rev. A27,1053.
Torquato, S. & Rubinstein, J. 1988 Diffusion-controlled reactions. II. Further bounds on the rate constant.J. Chem. Phys. 90,1644.
10.1063/1.447497
Sangani, A. & Acrivos, A. 1982 Slow flow through a cubic array of spheres.Intl J. Multiphase Flow 8,343.
Childress, S. 1972 Viscous flow past a random array of spheres.J. Chem. Phys. 56,2527.
Tartar, L. 1980 Nonhomogeneous Media and Vibration Theory, Appendix 2. Lecture Notes in Physics, vol. 127.Springer.
Widom, B. 1966 Random sequential addition of hard spheres to a volume.J. Chem. Phys. 44,3888.
Rubinstein, J. & Keller, J. B. 1987 Lower bounds on permeability.Phys. Fluids 30,2919.
10.1017/S0022112059000222
Prager, S. 1961 Viscous flow through porous media.Phys. Fluids 4,1477.
10.1103/PhysRevB.33.6428
Happel, J. & Brenner, H. 1983 Low Reynolds Number Hydrodynamics. Nijhoff. (10.1007/978-94-009-8352-6)
Torquato, S. & Stell, G. 1982 Microstructure of two-phase random media. I. The n-point probability functions.J. Chem. Phys. 77,2071.
Haan, S. W. & Zwanzig, R. 1977 Series expansions in a continuum percolation problem..J. Phys. A10,1547.
Zick, A. A. & Homsy, G. M. 1982 Stokes flow through periodic arrays of spheres.J. Fluid Mech. 115,13.
Lundgren, T. S. 1972 Slow flow through stationary random beds and suspension of spheres.J. Fluid Mech. 51,273.
Torquato, S. & Beasley, J. D. 1987 Bounds on the permeability of a random array of partially penetrable spheres.Phys. Fluids 30,633.
Torquato, S. 1986a Bulk properties of two-phase disordered media: III. New bounds on the effective conductivity of dispersions of penetrable spheres.J. Chem. Phys. 84,6345. (10.1063/1.450727)
Neuman, S. P. 1977 Theoretical derivation of Darcy's law.Acta. Mech. 25,153.
Caflisch, R. E. & Rubinstein, J. 1986 Lectures on the Mathematical Theory of Multiphase Flow. Courant Institute Notes Series, New York University.
Hinch, E. J. 1977 An averaged-equation approach to particle interactions in a fluid suspension.J. Fluid Mech. 83,695.
10.1063/1.444274
Elam, W. T. , Kerstein, A. R. & Rehr, J. J. 1984 Critical properties of the void percolation problem for spheres.Phys. Rev. Lett. 52,1516.
Scheidegger, A. E. 1960 The Physics of Flow through Porous Media. Macmillan.
Berryman, J. G. & Milton, G. W. 1985 Normalization constraint for variational bounds on fluid permeability.J. Chem. Phys. 83,754.
Sanchez-Palencia, E. 1980 Nonhomogeneous media and vibration theory. Lectures Notes in Physics, vol. 127.Springer.
Berryman, J. G. 1986 Variational bounds on Darcy's constant. In Homogenization and Effective Moduli of Materials. IMA Volumes in Mathematics and Its Application, vol. 1,Springer. (10.1007/978-1-4613-8646-9_3)
Keller, J. B. , Rubenfeld, L. & Molyneux, J. 1967 Extremum principles for slow viscous flows with applications to suspensions.J. Fluid Mech. 30,97.
Keller, J. B. 1980 Darcy's law for flow in porous media and the two-scale method. In Nonlinear P.D.E. in Engineering and Applied Sciences (ed. R. L. Sternberg , A. J. Kalinowski & J. S. Papadakis ).Marcel Dekker. (10.21236/ADA095629)
Weissberg, H. L. & Prager, S. 1970 Viscous flow through porous media. III. Upper bounds on the permeability for a simple random geometry.Phys. Fluids 13,2958.
Brinkman, H. C. 1947 A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles.Appl. Sci. Res. A1,27.
Rubinstein, J. 1987 Hydrodynamic screening in random media. In Hydrodynamic Behavior and Interacting Particle Systems. IMA Volumes in Mathematics and Its Application. vol. 9 (ed. G. Papanicolao )Springer. (10.1007/978-1-4684-6347-7_12)
Whitaker, S. 1986 Flow in porous media: a theoretical derivation of Darcy's law.Trans. Porous Media 1,3. (10.1007/BF01036523)
Dullien, F. A. L. 1979 Porous Media. Academic.
Doi, M. 1976 A new variational approach to the diffusion and flow problem in porous media.J. Phys. Soc. Japan 40,567.
Torquato, S. 1986b Microstructure characterization and bulk properties of disordered two-phase media.J. Statist. Phys. 45,843.

Dates

Type	When
Created	19 years, 4 months ago (April 26, 2006, 9:31 a.m.)
Deposited	2 years, 3 months ago (May 7, 2023, 2:20 a.m.)
Indexed	3 weeks, 1 day ago (Aug. 6, 2025, 9:02 a.m.)
Issued	35 years, 11 months ago (Sept. 1, 1989)
Published	35 years, 11 months ago (Sept. 1, 1989)
Published Online	19 years, 4 months ago (April 26, 2006)
Published Print	35 years, 11 months ago (Sept. 1, 1989)

Funders 0

None

BibTeX

@article{Rubinstein_1989, title={Flow in random porous media: mathematical formulation, variational principles, and rigorous bounds}, volume={206}, ISSN={1469-7645}, url={http://dx.doi.org/10.1017/s0022112089002211}, DOI={10.1017/s0022112089002211}, journal={Journal of Fluid Mechanics}, publisher={Cambridge University Press (CUP)}, author={Rubinstein, Jacob and Torquato, S.}, year={1989}, month=sep, pages={25–46} }

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