Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-28T01:32:18.420Z Has data issue: false hasContentIssue false
  • English
  • Français

Thermal convection in a Hele-Shaw cell

Published online by Cambridge University Press:  11 April 2006

Beverly K. Hartline  and
C. R. B. Lister
Beverly K. Hartline
Affiliation:
Geophysics Program, University of Washington, Seattle, Washington 98195
C. R. B. Lister
Affiliation:
Geophysics Program, University of Washington, Seattle, Washington 98195
Get access Rights & Permissions [Opens in a new window]

Abstract

We derive the Rayleigh number RHS for thermal convection in a Hele-Shaw cell with gap width d and full width (gap plus walls) Y. For the state of marginal stability, the system of equations is found to be formally identical to that describing flow through a uniform porous medium, if d3/12Y is identified as the Hele-Shaw permeability. Thus Lapwood's (1948) thermal-instability analysis should apply, and the critical Rayleigh number should be 4π2 when the cell has impermeable isothermal boundaries.

Baker's (1966) pH-indicator method for visualizing fluid flow has been adapted for use in a Hele-Shaw cell. In addition to revealing the convection pattern clearly, this technique proves to be an especially sensitive detector of incipient flow, and a highly accurate means of verifying the onset of convection. Our experiments confirm that the critical Hele-Shaw Rayleigh number is 40 ± 2, thereby validating our theoretically derived expression for the Rayleigh number. We also measure the vertical flow velocity wm and find that wm∝ (R2HS−402)½ closely fits our data for 40 < RHS < 140.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 79 , Issue 2 , 22 February 1977 , pp. 379 - 389
Copyright
© 1977 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

Asiz, K. & Combarnous, M. 1970 Transfert de chaleur par convection naturelle dans une couche poreuse horizontale. C. R. Acad. Sci., Paris, 271, 813–815.Google Scholar
Baker, D. J. 1966 A technique for the precise measurement of small fluid velocities. J. Fluid Mech. 26, 573–575.Google Scholar
Bear, J. 1972 Dynamics of Fluid in a Porous Medium. Elsevier.
Bories, S. 1970 Sur les mécanismes fondamentaux de la convection naturelle en milieu poreux. Rev. Gén. Therm. 108, 1377–1401.Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability, chap. 1. Oxford: Clarendon Press.
Darcy, H. 1856 Les Fontaines Publiques de la Ville de Dijon. Paris.
Elder, J. W. 1965 Physical processes in geothermal areas. Am. Geophys. Un. Mon. 8, 211–239.Google Scholar
Elder, J. W. 1967 Steady free convection in a porous medium heated from below. J. Fluid Mech. 27, 29–48.Google Scholar
Harned, H. S. & Owen, B. B. 1950 The Physical Chemistry of Electrolytic Solutions. Reinhold.
Hele-Shaw, H. S. J. 1898 Trans. Inst. Naval Archit. 40, 21.
Horne, R. N. & O'Sullivan, M. J. 1974 Oscillatory convection in a porous medium heated from below. J. Fluid Mech. 66, 339–352.Google Scholar
Lamb, H. 1932 Hydrodynamics, 6th edn. Cambridge University Press.
Lapwood, E. R. 1948 Convection of a fluid in a porous medium. Proc. Camb. Phil. Soc. 44, 508–521.Google Scholar
Lister, C. R. B. 1972 On the thermal balance of a mid-ocean ridge. Geophys. J. Roy. Astr. Soc. 26, 515–535.Google Scholar
Lister, C. R. B. 1974 On the penetration of water into hot rock. Geophys. J. Roy. Astr. Soc. 39, 465–509.Google Scholar
Palm, E., Weber, J. E. & Kvernvold, O. 1972 On steady convection in a porous medium. J. Fluid Mech. 54, 153–161.Google Scholar
Straus, J. M. 1974 Large amplitude convection in porous media. J. Fluid Mech. 64, 51–63.Google Scholar
Williams, D. L., Von Herzen, R. P., Sclater, J. G. & Anderson, R. N. 1974 The Galapagos spreading centre: lithospheric cooling and hydrothermal circulation. Geophys. J. Roy. Astr. Soc. 38, 587–608.Google Scholar