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Fluids 2016, 1(4), 41; doi:10.3390/fluids1040041

The Onset of Convection in an Unsteady Thermal Boundary Layer in a Porous Medium

1
Institute of Engineering Mathematics, University Malaysia Perlis, 02600 Arau, Malaysia
2
Department of Mechanical Engineering, University of Bath, Bath BA2 7AY, UK
*
Author to whom correspondence should be addressed.
Academic Editor: Mehrdad Massoudi
Received: 5 November 2016 / Revised: 5 November 2016 / Accepted: 25 November 2016 / Published: 8 December 2016
(This article belongs to the Special Issue Convective Instability in Porous Media)
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Abstract

In this study, the linear stability of an unsteady thermal boundary layer in a semi-infinite porous medium is considered. This boundary layer is induced by varying the temperature of the horizontal boundary sinusoidally in time about the ambient temperature of the porous medium; this mimics diurnal heating and cooling from above in subsurface groundwater. Thus if instability occurs, this will happen in those regions where cold fluid lies above hot fluid, and this is not necessarily a region that includes the bounding surface. A linear stability analysis is performed using small-amplitude disturbances of the form of monochromatic cells with wavenumber, k. This yields a parabolic system describing the time-evolution of small-amplitude disturbances which are solved using the Keller box method. The critical Darcy-Rayleigh number as a function of the wavenumber is found by iterating on the Darcy-Rayleigh number so that no mean growth occurs over one forcing period. It is found that the most dangerous disturbance has a period which is twice that of the underlying basic state. Cells that rotate clockwise at first tend to rise upwards from the surface and weaken, but they induce an anticlockwise cell near the surface at the end of one forcing period, which is otherwise identical to the clockwise cell found at the start of that forcing period. View Full-Text
Keywords: porous medium; convection; linear instability; Keller box method; subharmonic; boundary layer porous medium; convection; linear instability; Keller box method; subharmonic; boundary layer
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Bidin, B.; Rees, D.A.S. The Onset of Convection in an Unsteady Thermal Boundary Layer in a Porous Medium. Fluids 2016, 1, 41.

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