Convection with temperature dependent viscosity in a porous medium: nonlinear stability and the Brinkman effect.

Lorna Richardson; Brian Straughan

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1993)

  • Volume: 4, Issue: 3, page 223-230
  • ISSN: 1120-6330

Abstract

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We establish a nonlinear energy stability theory for the problem of convection in a porous medium when the viscosity depends on the temperature. This is, in fact, the situation which is true in real life and has many applications to geophysics. The nonlinear analysis presented here would appear to require the presence of a Brinkman term in the momentum equation, rather than just the normal form of Darcy's law.

How to cite

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Richardson, Lorna, and Straughan, Brian. "Convection with temperature dependent viscosity in a porous medium: nonlinear stability and the Brinkman effect.." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 4.3 (1993): 223-230. <http://eudml.org/doc/244260>.

@article{Richardson1993,
abstract = {We establish a nonlinear energy stability theory for the problem of convection in a porous medium when the viscosity depends on the temperature. This is, in fact, the situation which is true in real life and has many applications to geophysics. The nonlinear analysis presented here would appear to require the presence of a Brinkman term in the momentum equation, rather than just the normal form of Darcy's law.},
author = {Richardson, Lorna, Straughan, Brian},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Convection; Porous medium; Variable viscosity; energy stability theory; Brinkman term},
language = {eng},
month = {9},
number = {3},
pages = {223-230},
publisher = {Accademia Nazionale dei Lincei},
title = {Convection with temperature dependent viscosity in a porous medium: nonlinear stability and the Brinkman effect.},
url = {http://eudml.org/doc/244260},
volume = {4},
year = {1993},
}

TY - JOUR
AU - Richardson, Lorna
AU - Straughan, Brian
TI - Convection with temperature dependent viscosity in a porous medium: nonlinear stability and the Brinkman effect.
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1993/9//
PB - Accademia Nazionale dei Lincei
VL - 4
IS - 3
SP - 223
EP - 230
AB - We establish a nonlinear energy stability theory for the problem of convection in a porous medium when the viscosity depends on the temperature. This is, in fact, the situation which is true in real life and has many applications to geophysics. The nonlinear analysis presented here would appear to require the presence of a Brinkman term in the momentum equation, rather than just the normal form of Darcy's law.
LA - eng
KW - Convection; Porous medium; Variable viscosity; energy stability theory; Brinkman term
UR - http://eudml.org/doc/244260
ER -

References

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  2. HSU, C. T. - CHENG, P., The Brinkman model for natural convection about a semi-infinite vertical flat plate in a porous medium. Int. J. Heat Mass Transfer, 28, 1985, 683-697. Zbl0576.76079
  3. KATTO, Y. - MASUOKA, T., Criteria for the onset of convective flow in a fluid in a porous medium. Int. J. Heat Mass Transfer, 10, 1967, 297-309. 
  4. MULONE, G. - RIONERO, S., On the nonlinear stability of the magnetic Bénard problem with rotation. ZAMM, 72, 1992, in press. Zbl0798.76028MR1291193DOI10.1002/zamm.19930730112
  5. NIELD, D. A., The boundary correction for the Rayleigh-Darcy problem: limitations of the Brinkman equation. J. Fluid Mech., 128, 1983, 37-46. Zbl0512.76101MR704274DOI10.1017/S0022112083000361
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  8. QIN, Y. - KALONI, P. N., Steady convection in a porous medium based upon the Brinkman model. IMA J. Appl. Math., 35, 1992, 85-95. Zbl0765.76085MR1150892DOI10.1093/imamat/48.1.85
  9. QIN, Y. - KALONI, P. N., Nonlinear stability problem of a rotating porous layer. Quart. Appl. Math., to appear. Zbl0816.76035MR1315452
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  12. RIONERO, S., Sulla stabilità asintotica in media in magnetoidrodinamica. Ann. Mat. Pura Appl., 76, 1967, 75-92. Zbl0158.45104MR220478
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  14. RUDRAIAH, N. - MASUOKA, T., Asymptotic analysis of natural convection through a horizontal porous layer. Int. J. Engng. Sci., 20, 1982, 27-40. Zbl0484.76103
  15. STRAUGHAN, B., The Energy Method, Stability, and Nonlinear Convection. Springer-Verlag, Ser. in Appl. Math. Sci., vol. 91, 1992. Zbl0743.76006MR1140924
  16. TIPPELSKIRCH, H., Über Konvektionszellen, insbesondere im flüssigen Schwefel. Beit. zur Physik der Atmosphäre, 29, 1956, 37-54. 
  17. VAFAI, K. - KIM, S. J., Forced convection in a channel filled with a porous medium: an exact solution. Trans. ASME J. Heat Transfer, 112, 1989, 1103-1106. 
  18. VAFAI, K. - TIEN, C. L., Boundary and inertia effects on flow and heat transfer in porous media. Int. J. Heat Mass Transfer, 24, 1981, 195-203. Zbl0464.76073
  19. WALKER, K. - HOMSY, G. M., A note on convective instability in Boussinesq fluids and porous media. J. Heat Transfer, 99, 1977, 338-339. 
  20. WEAST, R. C., Handbook of Chemistry and Physics. 69th ed. C.R.C. Press, Inc., Boca Raton, Florida, 1988. 

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