Abstract
The effect of vertical throughflow on the onset of convection in a rectangular box occupied by a saturated porous medium uniformly heated from below, is studied using linear stability theory. It is found that, for small values of the throughflow, the stabilizing effect of the throughflow and the stabilizing effect of the confining lateral walls of the box are approximately independent of each other.
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Abbreviations
- A :
-
Aspect ratio (height to width)
- c :
-
Specific heat
- G :
-
Pressure gradient
- H :
-
Layer depth
- k :
-
Overall (effective) thermal conductivity
- K :
-
Permeability
- L x :
-
Length of the domain in the x-direction
- L y :
-
Length of the domain in the y-direction
- P :
-
Dimensionless pressure, \({\frac{(\rho c)_f K}{\mu k}P^{\ast}}\)
- P*:
-
Pressure
- Q :
-
Péclet-Darcy number, \({\frac{(\rho c)_f HGK}{k\mu }}\)
- Ra :
-
Rayleigh number, \({\frac{(\rho c)_f \rho _0 g\beta KH(T_1 -T_0 )}{\mu k}}\)
- t*:
-
Time
- t :
-
Dimensionless time, \({\frac{k}{(\rho c)_m H^{2}}t^{\ast}}\)
- T*:
-
Temperature
- T 0 :
-
Temperature at the upper boundary
- T 1 :
-
Temperature at the lower boundary
- u*:
-
Vector of Darcy velocity, (u*, v*)
- x :
-
Dimensionless horizontal coordinate, x*/L x
- x*:
-
Horizontal coordinate
- y :
-
Dimensionless upward vertical coordinate, y*/L y
- y*:
-
Upward vertical coordinate
- β :
-
Fluid volumetric expansion coefficient
- θ :
-
Dimensionless temperature, \({\frac{T^{\ast}-T_0 }{T_1 -T_0 }}\)
- μ :
-
Fluid viscosity
- ρ :
-
Density
- ρ 0 :
-
Fluid density at temperature T 0
- σ :
-
Heat capacity ratio, \({\frac{(\rho c)_m }{(\rho c)_f }}\)
- B:
-
Basic solution
- *:
-
Dimensional variable
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Nield, D.A., Kuznetsov, A.V. The Effect of Vertical Throughflow on the Onset of Convection in a Porous Medium in a Rectangular Box. Transp Porous Med 90, 993–1000 (2011). https://doi.org/10.1007/s11242-011-9828-4
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DOI: https://doi.org/10.1007/s11242-011-9828-4