Abstract
Linear stability analysis is applied to the onset of convection due to internal heating in a porous medium with weak vertical and horizontal heterogeneity. It is found that the effect of horizontal heterogeneity of each of permeability and thermal conductivity is slightly destabilizing. Increase of permeability in the upward direction is destabilizing and increase in the downward direction is stabilizing, and the reverse is true for increase of conductivity.
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Abbreviations
- \(a\) :
-
Wavenumber
- \(A\) :
-
Aspect ratio (width to height)
- \(c\) :
-
Specific heat
- \(k\) :
-
\(k^*/k_{0}\)
- \(k^*\) :
-
Overall (effective) thermal conductivity
- \(k_{0}\) :
-
Mean value of \(k^*(x^*,y^*)\)
- \(K\) :
-
\(K^*/K_{0}\)
- \(K^{*}\) :
-
Permeability
- \(K_{0}\) :
-
Mean value of \(K^*(x^*,y^*)\)
- \(H\) :
-
Height of the enclosure
- \(P\) :
-
Dimensionless pressure, \(\frac{(\rho c)_{\mathrm{f}} K_0 }{\mu k_0 }P^*\)
- \(P^*\) :
-
Pressure
- \(Q\) :
-
Volumetric heat source strength
- \(R\) :
-
\(\text{ Ra }/\pi ^{2}\)
- Ra:
-
Internal Rayleigh number, \(\frac{(\rho c)_{\mathrm{f}} \rho _{0} g\beta K_{0} H^{3}Q}{2\mu k_0 ^{2}}\)
- \(t\) :
-
Dimensionless time, \(\frac{k_0 }{(\rho c)_\mathrm{m} L^{2}}t^*\)
- \(t^*\) :
-
Time
- \(T^*\) :
-
Temperature
- \(T_{0}\) :
-
Temperature at the upper and lower boundary
- \(u\) :
-
Dimensionless horizontal velocity, \(\frac{(\rho c)_{\mathrm{m}} H}{Ak_0 }u^*\)
- \(\mathbf{u}^*\) :
-
Vector of Darcy velocity, \((u^*,v^*)\)
- \(v\) :
-
Dimensionless vertical velocity, \(\frac{(\rho c)_{\mathrm{m}} H}{k_0 }v^*\)
- \(x\) :
-
Dimensionless horizontal coordinate, \(x^*/AH\)
- \(x^*\) :
-
Horizontal coordinate
- \(y\) :
-
Dimensionless upward vertical coordinate, \(y^*/H\)
- \(y^*\) :
-
Upward vertical coordinate
- \(\beta \) :
-
Fluid volumetric expansion coefficient
- \(\theta \) :
-
Dimensionless temperature, \(\frac{(\rho c)_{\mathrm{f}} \rho _0 g\beta KH}{\mu k_0 }(T^*-T_0 )\)
- \(\mu \) :
-
Fluid viscosity
- \(\rho \) :
-
Density
- \(\rho _{0}\) :
-
Fluid density at temperature \(T_{0}\)
- \(\sigma \) :
-
Heat capacity ratio, \(\frac{(\rho c)_{\mathrm{m}}}{(\rho c)_{\mathrm{f}}}\)
- \(\psi \) :
-
Streamfunction defined by Eqs. (10a,b)
- \(\mathrm{c}\) :
-
Critical value
- \(\mathrm{f}\) :
-
Fluid
- \(\mathrm{m}\) :
-
Overall porous medium
- \(^{*}\) :
-
Dimensional variable
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Nield, D.A., Kuznetsov, A.V. Onset of Convection with Internal Heating in a Weakly Heterogeneous Porous Medium. Transp Porous Med 98, 543–552 (2013). https://doi.org/10.1007/s11242-013-0158-6
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DOI: https://doi.org/10.1007/s11242-013-0158-6