Abstract
We consider convection in a uniform fluid-saturated porous layer which is bounded by conducting plates and heated from below. The primary aim is to determine the identity of the postcritical convection planform as a function of the thicknesses and conductivities of the bounding plates relative to that of the porous layer. This work complements and extends an early paper by Riahi (J Fluid Mech 129:153–171, 1983) who considered a situation where the porous layer is bounded by infinitely thick conducting media. We present regions in parameter space wherein convection in the form of rolls is unstable and within which cells with square planform form the preferred pattern.
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Abbreviations
- \({{\mathcal A}}\) :
-
Arbitrary constant
- A, B, C:
-
Roll amplitudes
- c1, c2, c3:
-
Constants in amplitude equations
- c.c.:
-
Complex conjugate
- C :
-
Heat capacity
- d :
-
Conductivity ratio
- f, g:
-
Functions in weakly nonlinear theory
- \({\mathcal F}\) :
-
Dispersion relation
- \({{\bar g}}\) :
-
Gravity
- h :
-
Height of sublayer
- k :
-
Disturbance wavenumber
- K :
-
Permeability
- p :
-
Pressure
- Ra :
-
Darcy–Rayleigh number
- t :
-
Time
- T :
-
Dimensional temperature
- u, v:
-
Horizontal velocities
- w :
-
Vertical velocity
- x, y:
-
Horizontal coordinates
- z :
-
Vertical coordinate
- α :
-
Disturbance wavenumber
- β :
-
Expansion coefficient
- γ :
-
Equal to d/δ
- δ :
-
Sublayer thickness ratio
- ΔT :
-
Reference temperature drop
- \({\epsilon}\) :
-
Small quantity
- θ :
-
Temperature
- Θ:
-
Disturbance temperature
- κ :
-
Thermal diffusivity ratio
- λ :
-
Function of Ra and α
- μ :
-
Dynamic viscosity
- ρ :
-
Density
- σ :
-
Function of Ra and α
- τ :
-
Slow time scale
- \({\phi}\) :
-
Relative orientation of two rolls
- Ω :
-
Coupling coefficient
- c:
-
Critical conditions
- f:
-
Fluid
- m :
-
Iteration number
- ref:
-
Reference value
- 1, 2, 3, …:
-
Context-dependent meanings
- \({\hat {}}\) :
-
Dimensional quantity
- \({\bar {}}\) :
-
Reduced disturbance
- ′:
-
Derivative with respect to z
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Rees, D.A.S., Mojtabi, A. The Effect of Conducting Boundaries on Weakly Nonlinear Darcy–Bénard Convection. Transp Porous Med 88, 45–63 (2011). https://doi.org/10.1007/s11242-011-9722-0
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DOI: https://doi.org/10.1007/s11242-011-9722-0