×

zbMATH — the first resource for mathematics

A systematic approximation for the equations governing convection-diffusion in a porous medium. (English) Zbl 1194.35336
Summary: In order to take into account thermal effects in flows through porous media, one makes ad hoc modifications to Darcy’s equation by appending a term that is similar to the one that is obtained in the Oberbeck-Boussinesq approximation for a fluid. In this short paper, we outline a systematic procedure for obtaining an Oberbeck-Boussinesq type of approximation for the flow of a fluid through a porous medium. In addition to establishing the appropriate equation for a flow governed by Darcy’s equation, we proceed to obtain the approximations for flows governed by equations due to Forchheimer and Brinkman.

MSC:
35Q35 PDEs in connection with fluid mechanics
76S05 Flows in porous media; filtration; seepage
35A35 Theoretical approximation in context of PDEs
80A20 Heat and mass transfer, heat flow (MSC2010)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] H. Darcy, Les Fontaines Publiques de La Ville de Dijon, Victor Dalmont, 1856
[2] Forchheimer, P., Wasserbewegung durch boden, Z. ver. dtsch. ing., 45, 1736-1741, (1901), and 1781-1788
[3] Brinkman, H.C., A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles, Appl. sci. res. A, 1, 27-34, (1947) · Zbl 0041.54204
[4] Brinkman, H.C., On the permeability of media consisting of closely packed porous particles, Appl. sci. res. A, 1, 81-86, (1947)
[5] Oberbeck, A., Ueber die Wärmleitung der flüssigkeiten bei berücksichtigung der strömungen infolge von temperaturdifferenzen, Ann. phys. chem., 7, 271-292, (1879) · JFM 11.0787.01
[6] Oberbeck, A., Uber die bewegungsercheinungen der atmosphere, Sitz. ber. K. preuss. akad. miss., (1888), 383 and 1120
[7] Boussinesq, J., Théorie analytique de la chaleur, (1903), Gauthier-Villars Paris · JFM 34.0887.05
[8] Spiegel, E.A.; Veronis, G., On the Boussinesq approximation for a compressible fluid, Astrophys. J., 131, 442-447, (1960)
[9] Mihaljan, J.M., A rigorous exposition of the Boussinesq approximations applicable to a thin layer of fluid, Astrophys. J., 136, 1126-1133, (1962)
[10] Hills, R.N.; Roberts, P.H., On the motion of a fluid that is incompressible in a generalized sense and its relationship to the Boussinesq approximation, Stability appl. anal. continua, 1, 3, 205-212, (1991)
[11] Gray, G.G.; Giorgini, A., The validity of the Boussinesq approximation for liquids and gases, Int. J. heat mass transfer, 19, 545-551, (1976) · Zbl 0328.76066
[12] Rajagopal, K.R.; Ruzika, M.; Srinivasa, A.R., On the oberbeck – boussinesq approximation, Math. models methods appl. sci., 6, 8, 1157-1167, (1996) · Zbl 0883.76078
[13] Rajagopal, K.R.; Saccomandi, G.; Vergori, L., On the oberbeck – boussinesq approximation in fluids with pressure-dependent viscosities, Nonlinear anal. RWA, 10, 2, 1139-1150, (2009) · Zbl 1167.76368
[14] Qin, Y.; Kaloni, P.N., A nonlinear stability problem of convection in a porous vertical slab, Phys. fluids A, 5, 2067-2069, (1993) · Zbl 0784.76029
[15] Qin, Y.; Kaloni, P.N., Nonlinear stability problem of a rotating porous layer, Quart. appl. math., 53, 129-142, (1995) · Zbl 0816.76035
[16] Qin, Y.; Guo, J.L.; Kaloni, P.N., Double diffusive penetrative convection in porous media, Internat. J. engrg. sci., 33, 3, 303-312, (1995) · Zbl 0899.76161
[17] Guo, J.; Kaloni, P.N., Double-diffusive convection in a porous medium, nonlinear stability, and the Brinkman effect, Stud. appl. math., 94, 341-358, (1995) · Zbl 0822.76035
[18] Payne, L.E.; Straughan, B., Stability in the initial-time geometry problem for the Brinkman and Darcy equations of flows in porous media, J. math. pures et. appl., 75, 225-271, (1996) · Zbl 0848.76089
[19] Fick, A., Uber diffusion, Ann. phys., 94, 59-86, (1855)
[20] Truesdell, C., Sulle base Della termomeccanica, Rend. lincei, 22, 33-38, (1957) · Zbl 0098.21002
[21] Truesdell, C., Sulle basi Della termomeccanica, Rend. lincei, 22, 158-166, (1957) · Zbl 0098.21002
[22] Truesdell, C., Mechanical basis of diffusion, J. chem. phys., 37, 2336-2344, (1962)
[23] Bowen, R.M.; Eringen, A.C., Continuum physics, vol. III, (1976), Academic Press
[24] Atkin, R.J.; Craine, R.E., Continuum theories of mixtures: basic theory and historical developments, Quart. J. mech. appl. math., 29, 209-244, (1976) · Zbl 0339.76003
[25] Atkin, R.J.; Craine, R.E., Continuum theories of mixtures: applications, J. inst. appl. math., 17, 153-207, (1976) · Zbl 0355.76004
[26] Rajagopal, K.R.; Tao, L., Mechanics of mixtures, (1995), World Scientific · Zbl 0941.74500
[27] I. Samohyl, Thermodynamics of Irreversible Processes in Fluid Mixtures, Teubner, 1987
[28] Rajagopal, K.R., On implicit constitutive theories for fluids, J. fluid mech., 550, 243-249, (2006) · Zbl 1097.76009
[29] Munaf, D.; Wineman, A.S.; Rajagopal, K.R.; Lee, D.W., A boundary value problem in groundwater motion analysis – comparison of the predictions based on darcy’s law and the continuum theory of mixtures, Math. models methods appl. sci., 3, 2, 231-248, (1993) · Zbl 0773.76067
[30] Rajagopal, K.R., On a hierarchy of approximate models for flows of incompressible fluids through porous solids, Math. models methods appl. sci., 17, 2, 215-252, (2007) · Zbl 1123.76066
[31] Dunn, J.E.; Fosdick, R.L., Thermodynamics, stability and boundedness of fluid of complexity 2 and fluids of second grade, Arch. rat. mech. anal., 6, 191-252, (1974) · Zbl 0324.76001
[32] K.R. Rajagopal, Thermodynamics and stability of non-Newtonian fluids, Ph.D. Dissertation: University of Minnesota, Minneapolis, Minnesota, 1978
[33] Ward, J.C., Turbulent flow in porous media, ASCE J. hydraul. div., 90, 1-12, (1964)
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.