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

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.

A Unified Approach to Two-Dimensional Brinkman-Bénard Convection of Newtonian Liquids in Cylindrical and Rectangular Enclosures

“…We first present a detailed literature survey concerning RBC in a loosely-packed porous medium (Brinkman-Bénard convection-BBC) followed by that of a denselypacked porous medium (Darcy-Bénard convection-DBC) occupying a rectangular geometry and a cylindrical geometry with the LTNE assumption. The articles studying BBC with LTNE [1][2][3][4][5][6][7][8] and DBC with LTNE [9][10][11][12][13][14] in rectangular enclosures (RE) are covered below. Phanikumar and Mahajan [1] examined the heat transfer characteristics of natural convection flow in a fluid-saturated metal-foam porous medium heated from below.…”

“…The results of the LTE model are recovered for large values of the scaled interface heat transfer parameter. Bidin and Rees [12] made a weakly non-linear analysis of the classical DBC problem to analyse the effect of the LTNE assumption on different planforms of convection. Secondary instabilities like Eckhaus and zig-zag instabilities were also considered.…”

Heat and Mass Transfer Analysis on Magneto Micropolar Fluid Flow with Heat Absorption in Induced Magnetic Field

“…Gaffar et al [16] have developed a mathematical model to investigate the free convective flow in a third-grade viscoelastic micropolar fluid from a vertical isothermal inverted cone. Scholars in the references [17][18][19][20][21] have studied convective flows along plate embedded in a porous medium. Recently, Karvelas et al [22] have used a micropolar fluid model to study the auto rotation effect of human blood s microstructure on its flow.…”

Instability of Vertical Throughflows in Porous Media under the Action of a Magnetic Field

“…A wide literature can be found on different aspects of porous convection. In particular, in [2] the linear stability of a uniform parallel flow in a horizontal channel with open upper boundary, has been performed; in [3] the linear stability of an unsteady thermal boundary layer in a semi-infinite porous medium is considered; in [4] the thermal instability is analyzed on considering two different boundary conditions on the bounding walls. In [5] the onset of double-diffusive convection with heat generation and absorption effects, has been performed.…”