Abstract
The onset of convection in a horizontal porous layer is investigated theoretically. The permeability of the porous medium is a continuous periodic function of the horizontal x coordinate. Floquet theory has been employed to determine the favoured two-dimensional mode of convection. For a wide range of periods of the permeability variation, a matrix eigenvalue technique with eighth order accuracy has been employed to find the critical Darcy– Rayleigh number. This is supplemented by a multiple-scales analysis of the large-period limit, and a brief consideration of the anisotropic limit for very short periods.
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Abbreviations
- a :
-
Central difference coefficients for a first derivative
- b :
-
Central difference coefficients for a second derivative
- f, g:
-
Reduced forms of the perturbations
- \({\underline{f},\underline{g}}\) :
-
Vector forms of the perturbations
- F :
-
Nondimensional permeability variation
- H :
-
Height of the porous layer
- k :
-
Wave number of the permeability variations
- K :
-
Permeability
- K 0 :
-
Mean permeability
- M1–M4:
-
Matrices used in the numerical method
- N :
-
Number of intervals in the numerical scheme
- p :
-
Pressure
- P :
-
Period of the permeability variation
- R0, R1, R2:
-
Terms in the small-k expansion for Ra
- Ra:
-
Darcy–Rayleigh number
- T :
-
Temperature
- Tc, Th:
-
Cold and hot boundary temperatures, respectively
- x :
-
Horizontal coordinate
- y :
-
Horizontal (spanwise) coordinate
- z :
-
Vertical coordinate
- u :
-
Horizontal velocity
- v :
-
Horizontal (spanwise) velocity
- w :
-
Vertical velocity
- β :
-
Coefficient of thermal expansion
- ε :
-
Relative amplitude of the permeability variation
- θ :
-
Nondimensional temperature
- κ m :
-
Thermal diffusivity of the porous medium
- μ :
-
Dynamic viscosity
- ν :
-
Floquet exponent
- ξ :
-
Anisotropy parameter
- \({\hat\chi}\) :
-
Scaled form of χ
- ρ 0 :
-
Reference density
- ρ :
-
Density
- σ :
-
Heat capacity ratio of the porous medium to that of the fluid
- χ :
-
Slow x-variable
- ψ :
-
Streamfunction
- ω :
-
Scaled Floquet exponent
- c:
-
Critical conditions
- i, j:
-
Denoting grid points
- 0,1,2:
-
Terms in the small-k expansion
- χ :
-
Partial derivatives with respect to χ
- ¯:
-
Dimensional variables
- ˆ:
-
Reduced variables
- ′:
-
Derivative with respect to x
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Rees, D.A.S., Tyvand, P.A. Onset of Convection in a Porous Layer with Continuous Periodic Horizontal Stratification. Part I. Two-Dimensional Convection. Transp Porous Med 77, 187–205 (2009). https://doi.org/10.1007/s11242-008-9324-7
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DOI: https://doi.org/10.1007/s11242-008-9324-7