Hostname: page-component-745bb68f8f-b6zl4 Total loading time: 0 Render date: 2025-01-09T05:51:48.886Z Has data issue: false hasContentIssue false
  • English
  • Français

Natural convection in porous media: effect of weak dispersion on bifurcation

Published online by Cambridge University Press:  26 April 2006

Xiaowei S. He  and
John G. Georgiadis
Xiaowei S. He
Affiliation:
Department of Mechanical Engineering and Materials Science. Duke University, Durham, NC 27706, USA
John G. Georgiadis
Affiliation:
Department of Mechanical Engineering and Materials Science. Duke University, Durham, NC 27706, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

We use weakly nonlinear analysis via a two-parameter expansion to study bifurcation of conduction into cellular convection of an internally heated fluid in a porous medium that forms a horizontal layer between two isothermal walls. The Darcy–Boussinesq model of convection is enhanced by including two nonlinear terms: (i) quadratic (Forchheimer) drag; and (ii) hydrodynamic dispersion enhancement of the thermal conductivity described by a weak linear relationship between effective conductivity and local amplitude of filtration velocity. The impact of the second term on the shape of the bifurcation curve for two-dimensional rolls is profound in the presence of uniform volumetric heating. The resulting bifurcation structure is unlike any pitchfork bifurcations typical of the classical Bénard problem. Although direct experimental validation of the novel bifurcation is not available, we would like to register it as an alternative or a supplement to models of small imperfections, and as an attempt to account for the scatter of observed critical values for the first bifurcation.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 216 , July 1990 , pp. 285 - 298
Copyright
© 1990 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Buretta, R. J. & Berman, A. S. 1976 Convective heat transfer in a liquid saturated porous layer. Trans. ASME E: J. Appl. Mech. 98, 249–253.Google Scholar
Close, D. G., Symmons, J. G. & White, R. F. 1985 Convective heat transfer in shallow, gas-filled porous media: experimental investigation. Intl. J. Heat Mass Transfer 28, 2371–2378.Google Scholar
Ergun, S. 1952 Fluid flow through packed columns. Chem. Engng Prog. 48, 89–94.Google Scholar
Gartling, D. K. & Hickox, C. E. 1985 Numerical study of the application of the Boussinesq approximation for a fluid-saturated porous medium. Intl J. Numer. Meth. Fluids 5, 995–1013.Google Scholar
Gasser, R. D. & Kazimi, M. S. 1976 Onset of convection in a porous medium with internal heat generation. Trans. ASME C: J. Heat Transfer 98, 49–54.Google Scholar
Georgiadis, J. G & Catton, I. 1987 Stochastic modeling of unidirectional fluid transport in uniform and random packed beds. Phys. Fluids 30, 1017–1022.Google Scholar
Georgiadis, J. G & Catton, I. 1988a An effective equation governing convective transport in porous media. Trans. ASME C: J. Heat Transfer 110, 635–641.Google Scholar
Georgiadis, J. G & Catton, I. 1988b Dispersion in cellular thermal convection in porous media. Intl. J. Heat Mass Transfer 31, 1081–1091.Google Scholar
Hardee, H. C. & Nilson, R. H. 1977 Natural convection in porous media with heat generation. Intl J. Heat Mass Transfer 63, 119–132.Google Scholar
Hwang, I. 1971 Finite amplitude thermal convection in porous media with heat source and variable viscosity. Ph.D. thesis, University of Minnesota.
Joseph, D. D. 1976 Stability of Fluid Motions, Vols. 1 and 2. Springer.
Katto, Y. & Masuoka, T. 1967 Criterion for the onset of convection in a fluid in a porous medium. Intl J. Heat Mass Transfer 10, 296–309.Google Scholar
Koch, D. & Brady, J. F. 1985 Dispersion in fixed beds. J. Fluid Mech. 154, 399–427.Google Scholar
Koch, D., Cox, R., Brenner, H. & Brady, J. F. 1989 The effect of order on dispersion in porous media. J. Fluid Mech. 200, 173–188.Google Scholar
Kulacki, F. A. & Freeman, R. G. 1979 A note on thermal convection in a saturated, heat-generating porous layer. Trans. ASME C: J. Heat Transfer 101, 169–171.Google Scholar
Kvernvold, O. & Tyvand, P. 1980 Dispersion effects on thermal convection in porous media. J. Fluid Mech. 99, 673–686.Google Scholar
Levec, J. & Carbonell, R. G. 1985 Longitudinal and lateral thermal dispersion in packed beds — Part II: comparison between theory and experiment, AIChE J. 31, 591–602.Google Scholar
Neischloss, H. & Dagan, G. 1975 Convective currence in a porous layer heated from below: The influence of hydrodynamic dispersion. Phys. Fluids 18, 757–761.Google Scholar
Nield, D. A. & Joseph, D. D. 1985 Effects of quadratic drag on convection in a saturated porous medium. Phys. Fluids 28, 995–997.Google Scholar
Rees, D. A. S. & Riley, D. S. 1986 Free convection in an undulating saturated porous layer: resonant wavelength excitation. J. Fluid Mech. 166, 503–530.Google Scholar
Rees, D. A. S. & Riley, D. S. 1989 The effects of boundary imperfections on convection in a saturated porous layer: near-resonant wavelength excitation. J. Fluid Mech. 199, 133–154.Google Scholar
Rhee, S. J., Dhir, V. K. & Catton, I. 1978 Natural convection heat transfer in beds of inductively heated particles. Trans. ASME C: J. Heat Transfer 100, 78–85.Google Scholar
Rubin, H. 1974 Heat dispersion effect on thermal convection in a porous medium layer. J. Hydrol. 21, 173–185.Google Scholar
Rudraiah, N., Verrappa, B. & Balachandra, R. 1982 Convection in a fluid-saturated porous layer with non-uniform temperature gradient. Intl J. Heat Mass Transfer 25, 1147–1156.Google Scholar
Selimos, B. & Poulikakos, D. 1985 On double diffusion in a Brinkman heat generating porous layer. Intl Commun. Heat Mass Transfer 12, 149–158.Google Scholar