×

zbMATH — the first resource for mathematics

Effect of anisotropic permeability on Darcy’s law. (English) Zbl 0984.76024
The authors study the structural stability of non-Boussinesq (penetrative) convective flows in porous media with anisotropic permeability. It is shown, by means of a priori estimates, that the solution exists in a natural functional framework and depends continuously on changes in the permeability.

MSC:
76E06 Convection in hydrodynamic stability
76S05 Flows in porous media; filtration; seepage
35Q35 PDEs in connection with fluid mechanics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Non-Standard and Improperly Posed Problems, Mathematics in Science and Engineering Series, vol. 194 Academic Press: New York, 1997.
[2] Modelling Mathematical Methods and Scientific Computation. CRC Press: Boca Raton, FL, 1995. · Zbl 0871.65001
[3] Qualitative Estimates for Partial Differential Equations. CRC Press: Boca Raton, FL, 1995.
[4] Applied Analysis of the Navier-Stokes Equations. Cambridge University Press: Cambridge, 1995. · Zbl 0838.76016
[5] Franchi, Mathematical Methods in the Applied Sciences 19 pp 1335– (1996) · Zbl 0859.76065
[6] Payne, Continuum Mechanics and Thermodynamics 9 pp 175– (1997) · Zbl 0893.76090
[7] Payne, Proceedings of the Royal Society of London A 455 pp 2173– (1999) · Zbl 0933.76091
[8] Payne, Journal de Mathematiques Pures et Appliquees 75 pp 225– (1996)
[9] Payne, Journal de Mathematiques Pures et Appliquees 77 pp 317– (1998) · Zbl 0906.35067
[10] Payne, Proceedings of the Royal Society of London A 454 pp 1691– (1998) · Zbl 0912.76088
[11] Payne, Studies in Applied Mathematics 102 pp 419– (1999) · Zbl 1136.76448
[12] Qin, Quarterly of Applied Mathematics 56 pp 71– (1998) · Zbl 0951.35116
[13] Straughan, Proceedings of the Royal Society of London A 455 pp 767– (1999) · Zbl 0935.76084
[14] Les fontaines publiques de la ville de Dijon. Dalmont: Paris, 1856
[15] Etudes thèoriques et pratiques sur le mouvement des eaux. Dunod: Paris, 1863
[16] Firdaouss, Journal of Fluid Mechanics 343 pp 331– (1997) · Zbl 0897.76091
[17] Forchheimer, Zeitschrift für Vereines Deutscher Ingnieure 50 pp 1781– (1901)
[18] Givler, Journal of Fluid Mechanics 258 pp 355– (1994)
[19] Giorgi, Transport in Porous Media 29 pp 191– (1997)
[20] Kladias, Journal of Thermophysics 5 pp 560– (1991)
[21] Convection in Porous Media. Springer: New York, 1992.
[22] Whitaker, Transport in Porous Media 25 pp 27– (1996)
[23] Straughan, Proceedings of the Royal Society of London A 452 pp 97– (1996) · Zbl 0868.76033
[24] Brinkman, Applied Scientific Research 1 pp 27– (1957)
[25] Ames, Mathematical Models and Methods in Applied Sciences 4 pp 733– (1994) · Zbl 0819.76077
[26] Ames, Quarterly of Applied Mathematics 55 pp 769– (1997) · Zbl 0892.35079
[27] Ames, Mathematical Models and Methods in Applied Sciences 8 pp 187– (1998) · Zbl 0982.35119
[28] Bennett, Differential Integral Equations 4 pp 1311– (1991)
[29] Lin, SIAM Journal on Mathematical Analysis 25 pp 1242– (1994)
[30] On stabilizing ill-posed problems against errors in geometry and modeling. In Proceedings of the Conference on Inverse and Ill-posed Problems: Strobhl, (eds). Academic Press: New York, 1987;399-416.
[31] On geometric and modeling perturbations in partial differential equations. In Proceedings of the L.M.S. Symposium on Non-Classical Continuum Mechanics, (eds). Cambridge University Press: Cambridge, 1987;108-128.
[32] Continuous dependence on geometry with applications in continuum mechanics. In Continuum Mechanics and its Applications, (eds). Hemisphere Publ. Co.: Washington, DC, 1989;877-890.
[33] Gilman, Transport in Porous Media 23 pp 275– (1996) · Zbl 0868.76089
[34] Payne, Journal de Mathematiques Pures et Appliquees 78 pp 609– (1999) · Zbl 0935.76020
[35] Beavers, Journal of Fluid Mechanics 30 pp 197– (1967)
[36] Caviglia, Journal of the Acoustical Society of America 92 pp 1113– (1992)
[37] Haber, International Journal of Multiphase Flow 9 pp 561– (1983) · Zbl 0539.76094
[38] Hennenberg, Transport in Porous Media 27 pp 327– (1997)
[39] Jones, Proceedings of the Cambridge Philosophical Society 73 pp 231– (1973)
[40] Nield, Journal of Fluid Mechanics 128 pp 37– (1983) · Zbl 0512.76101
[41] Nield, International Journal of Heat and Fluid Flow 12 pp 269– (1991)
[42] Nield, Transport in Porous Media 31 pp 365– (1998) · Zbl 0993.76024
[43] Ybarra, Geophysical and Astrophysical Fluid Dynamics 13 pp 83– (1979)
[44] Rothmeyer, Solar Energy 25 pp 567– (1980)
[45] Tabor, Philosophical Transactions of the Royal Society of London A 295 pp 423– (1980)
[46] Zangrando, Solar Energy 46 pp 323– (1991)
[47] Ly, Journal of Nonlinear Science 9 pp 333– (1999) · Zbl 0964.76082
[48] Oliver, Journal of Differential Equations 163 pp 292– (2000) · Zbl 0949.35025
[49] Rodrigues, Continuum Mechanics and Thermodynamics 11 pp 181– (1999) · Zbl 0947.76084
[50] Rodrigues, Annali di Matematica Pura ed Applicata 144 pp 203– (1986) · Zbl 0631.35083
[51] Variational Methods in the Stefan Problem. Lecture Notes in Mathematics, vol. 1584 Springer: Berlin, 1994;147-212. · Zbl 0819.35154
[52] Qin, Studies in Applied Mathematics 91 pp 189– (1994) · Zbl 0792.76027
[53] Tyvand, Journal of Fluid Mechanics 226 pp 371– (1991) · Zbl 0718.76100
[54] Storesletten, Transport in Porous Media 12 pp 19– (1993)
[55] Payne, Studies in Applied Mathematics 103 pp 267– (1999) · Zbl 1136.76310
[56] Payne, Pacific Journal of Mathematics 8 pp 551– (1958) · Zbl 0093.10901
[57] Stability analyses of multi-component convection-diffusion problems. Ph.D. Thesis, University of Glasgow, 1997.
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.