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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
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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

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