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Application of the finite volume method in the simulation of saturated flows of binary mixtures with particular emphasis on non-Darcy motions through porous media. (English) Zbl 0785.76080

This well-written paper utilizes mixture theory approach to the modelling and simulation of saturated flows in porous media with significant boundary and inertia effects. Adoption of the constitutive relation proposed by W. O. Williams [Quart. Appl. Math. 36, 255-268 (1978; Zbl 0392.76086)] leads to a system of elliptic partial differential equations which is parametrized by two numbers. The problem is discretized by the finite volume method.
Use of the model and the discretization technique is illustrated by numerically simulating two non-Darcy flows. A relatively simpler boundary layer flow on a flat plate is obtained. And secondly, developing flow in a porous parallel plate channel is investigated.

MSC:

76S05 Flows in porous media; filtration; seepage
76V05 Reaction effects in flows
76M25 Other numerical methods (fluid mechanics) (MSC2010)

Citations:

Zbl 0392.76086
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Full Text: DOI

References:

[1] AtkinR. J. and CraineR. E., ’Continuum theories of mixtures, basic theory and historical development, Q. J. Mech. Appl. Math., XXIX (Part 2) (1976) 209–243. · Zbl 0339.76003 · doi:10.1093/qjmam/29.2.209
[2] BeaversG. and SparrowE. M., ’Non-Darcy flow through fibrous porous media’, J. Appl. Mech., 36 (1970) 711–714.
[3] BeaversG., SparrowE. M. and RodenzD. E., ’Influence of bed size on the flow characteristics and porosity of randomly packed beds of spheres’, J. Appl. Mech., 40 (1973) 655–660. · doi:10.1115/1.3423067
[4] BeaversG. and JosephD. D., ’Boundary conditions at a naturally permeable wall’, J. Fluid Mech., 30 (1967) 197–208. · doi:10.1017/S0022112067001375
[5] BedfordA. and DrumkelerD. S., ’Recent advance-theories of immiscible and structured mixtures’, Int. J. Engng. Sci., 21 (1983) 803–960. · Zbl 0534.76105 · doi:10.1016/0020-7225(83)90071-X
[6] BejanA., Convection Heat Transfer, Wiley, New York, 1983.
[7] BiotM. A., ’General theory of three-dimensional consolidation’, J. Appl. Phys., 12 (1941) 155–164. · JFM 67.0837.01 · doi:10.1063/1.1712886
[8] BodoiaJ. R., ’Finite difference analysis of plane Poiseuille and Couette flow’, Appl. Sci. Res., A 10 (1961) 265–276. · Zbl 0100.22701 · doi:10.1007/BF00411919
[9] Bowen, R. M., Theory of Mixtures, Continuum Physics, III, Academic Press, 1976.
[10] BrinkmanH. C., ’A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles’, Appl. Sci. Res., A 1 (1947) 27–34. · Zbl 0041.54204 · doi:10.1007/BF02120313
[11] CollinsR. E., Flow of Fluids through Porous Materials, Reinhold Pub. Corp., New York, 1961.
[12] CoulaudO., MorelP. and CaltagironeJ. P., ’Numerical modelling of nonlinear effects in laminar flow through a porous medium’, J. Fluid Mech., 190 (1988) 393–407. · Zbl 0644.76107 · doi:10.1017/S0022112088001375
[13] CrochetM. J. and NaghdiP. M., ’On constitutive equations for flow of fluid through an elastic solid’, Int. J. Engng. Sci., 4 (1966) 383–401. · doi:10.1016/0020-7225(66)90038-3
[14] JosephD. D., NieldD. A. and PapanicolaouG., ’Non-linear equation governing flow in a saturated porous media’, Water Resources Res., 18 (1982) 1049–1052. · doi:10.1029/WR018i004p01049
[15] KatsubeN. and CarrolM. M., ’The modified mixture theory for fluid-filled porous materials: theory’, J. Appl. Mech., 54 (1987) 35–40. · Zbl 0604.73005 · doi:10.1115/1.3172991
[16] KatsubeN. and CarrolM. M., ’The modified mixture theory for fluid-filled porous materials: applications’, J. Appl. Mech., 54 (1987) 41–46. · Zbl 0604.73006 · doi:10.1115/1.3172992
[17] PatankarS. V., Numerical Heat Transfer and Fluid Flow, McGraw-Hill, New York, 1980.
[18] SampaioR. and WilliamsW. O., ’On the viscosities of liquid mixtures’, Z. Angew. Math. Phys., 28 (1977) 607–614. · Zbl 0379.76033 · doi:10.1007/BF01601339
[19] SchilichtingH., Boundary Layer Theory (6th edn), McGraw-Hill, New York, 1968.
[20] VafaiK. and TienC. L., ’Boundary and inertia effects on flow and heat transfer in porous media’, Int. J. Heat Mass Transfer, 24 (1981) 195–203. · Zbl 0464.76073 · doi:10.1016/0017-9310(81)90027-2
[21] WangY. L. and LongwellP. A., ’Laminar flow in the inlet section of parallel plates’, Amer. Inst. Chem. Engrs. J., 10 (1964) 323–329.
[22] WilliamsW. O., ’Constitutive equations for flow of an incompressible viscous fluid through a porous medium’, Quart. J. Appl. Math., 31 (1978) 255–267.
[23] WhitakerS. ’Flow in porous media: Part I, A theoretical derivation of Darcy’s law’, Transport in Porous Media, 1 (1986) 3–25. · doi:10.1007/BF01036523
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.