Abstract
Prager’s variational method of obtaining upper bounds on the fluid permeability (Darcy’s constant) for slow flow through porous media is reexamined. By exploiting the freedom one has in choosing the trial stress distributions, several new results are derived. One result is a phase interchange relation for permeability; when the fluid-phase and particle-phase are interchanged for a fixed geometry, we find an upper bound on a linear combination of the complementary permeabilities. Another result is a proof of the monotone properties of the bounds. The optimal two-point bounds from this class of variational principles are evaluated numerically and compared to exact results of low density expansions for assemblages of spheres.
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References
H. Darcy, “Les fontaines publique de a ville de Dijon,” Paris, 1856.
M, Poreh and C. Elata, “An analytical derivation of Darcy’s law,” Israel J. Tech. 4, 214–217 (1966).
S.P. Neuman, “Theoretical derivation of Darcy’s law,” Acta Mech. 25, 153–170 (1977).
J.B. Keller, “Darcy’s law for flow in porous media and the two-space method,” in Nonlinear Partial Differential Equations in Engineering and Applied Science, ed. by R.L. Sternberg, A.J. Kalinowski, and J. S. Papadakis (Marcel Dekker, New York, 1980), pp. 429–443.
H.C. Brinkman, “A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles,” Appl. Sci. Res. Al, 27–34 (1947).
S. Childress, “Viscous flow past a random array of spheres,” J. Chem. Phys. 56, 2527–2539 (1972).
I.D. Howells, “Drag due to the motion of a Newtonian fluid through a sparse random array of small fixed rigid objects,” J. Fluid Mech. 64, 449–475 (1974).
E.J. Hinch, “An averaged-equation approach to particle interactions in a fluid suspension,” J. Fluid Mech. 83, 695–720 (1977).
A.A. Zick and G.M. Homsy, “Stokes flow through a periodic array of spheres,” J. Fluid Mech. 115, 13–26 (1982).
A.S. Sangani and A. Acrivos, “Slow flow through a periodic array of spheres,” Int. J. Multiphase Flow 8, 343–360 (1982).
S. Prager, “Viscous flow through porous media,” Phys. Fluids 4, 1477–1482 (1961).
M. Doi, “A new variational approach to the diffusion and the flow problem in porous media,” J. Phys. Soc. Japan 40, 567–572 (1976).
S. Torquato, “Microscopic approach to transport in two-phase random media,” Ph.D. thesis (State University of New York at Stony Brook, 1980).
J.G. Berryman, “Computing variational bounds for flow through random aggregates of spheres,” J. Comput. Phys. 52, 142–162 (1983).
J.G. Berryman, “Bounds on fluid permeability for viscous flow through porous media,” J. Chem. Phys. 82, 1459–1467 (1985).
J.G. Berryman, “Measurement of spatial correlation functions using image processing techniques,” J. Appl. Phys. 57, 2374–2384 (1985).
R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. I, (Interscience, New York, 1953), pp. 252–257 and
R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. I, (Interscience, New York, 1953), pp. 268–272.
M.J. Beran, Statistical Continuum Theories (Interscience, New York, 1968), Chapter 6.
H.L. Weissberg and S. Prager, “Viscous flow through porous media. III. Upper bounds on the permeability for a simple random geometry,” Phys. Fluids 13, 2958–2965 (1970).
J.G. Berryman and G.W. Milton, “Normalization constraint for variational bounds on fluid permeability”, J. Chem. Phys., July, 1985.
L.D. Landau and E.M. Lifshitz, Fluid Mechanics (Pergamon Press, London, 1959), p. 54.
H. Lamb, Hydrodynamics (Dover, New York, 1945), pp. 617–619.
J.B. Keller, “A theorem on the conductivity of a composite medium,” J. Mathematical Phys. 5, 548–549 (1964).
K. Schulgasser, “On a phase interchange relationship for composite materials,” J. Mathematical Phys. 17, 378–381 (1976).
H.L. Weissberg, “Effective diffusion coefficient in porous media,” J. Appl. Phys. 34, 2636–2639 (1963).
W. Strieder and R. Aris, Variational Methods Applied to Problems of Diffusion and Reaction (Springer-Verlag, New York, 1973), pp. 4–8.
S. Torquato and G. Stell, “Microstructure of two-phase random media. III. The n-point matrix probability functions for fully penetrable spheres,” J. Chem. Phys. 79, 1505–1510 (1983).
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Berryman, J.G. (1986). Variational Bounds on Darcy’s Constant. In: Ericksen, J.L., Kinderlehrer, D., Kohn, R., Lions, JL. (eds) Homogenization and Effective Moduli of Materials and Media. The IMA Volumes in Mathematics and its Applications, vol 1. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-8646-9_3
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DOI: https://doi.org/10.1007/978-1-4613-8646-9_3
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