# The onset of convection in a shallow box occupied by a heterogeneous porous medium with constant flux boundaries

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## Abstract

The effects of hydrodynamic and thermal heterogeneity, for the case of variation in both the horizontal and vertical directions, on the onset of convection in a horizontal layer of a saturated porous medium uniformly heated from below, are studied analytically using linear stability theory for the case of weak heterogeneity. Attention is focused on the case of constant flux upper and lower boundaries, a case for which the critical horizontal wavenumber is zero, and attention is also concentrated on the case of a shallow layer. It is found that the effect of such heterogeneity on the critical value of the Rayleigh number Ra based on mean properties is of second order if the properties vary in a piecewise constant or linear fashion. The effects of horizontal heterogeneity and vertical heterogeneity are then comparable once the aspect ratio is taken into account, and to a first approximation are independent. The combination of permeability heterogeneity and conductivity heterogeneity can be either stabilizing or destabilizing for the present case.

## Keywords

Natural convection Heterogeneity Instability Horton–Rogers–Lapwood problem## Nomenclature

*A*Aspect ratio (height to width)

*c*Specific heat

*H*Height of the enclosure

*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*^{*})*L*Width of the enclosure

*P*Dimensionless pressure, \(\frac{(\rho c)_{\rm f} K_0}{\mu k_0}P^* \)

*P*^{*}Pressure

- Ra
Rayleigh number, \(\frac{(\rho c)_{\rm f} \rho_0 g\beta K_0 L(T_1 - T_0)}{\mu k_0}\)

*t*^{*}Time

*t*Dimensionless time, \(\frac{k_0}{(\rho c)_{\rm m} L^2}t^* \)

*T*^{*}Temperature

*T*_{0}Temperature at the upper boundary

*T*_{1}Temperature at the lower boundary

*u*Dimensionless horizontal velocity, \(\frac{(\rho c)_{\rm m} L}{k_0}u^* \)

**u**^{*}Vector of Darcy velocity, (

*u*^{*},*v*^{*})*v*Dimensionless vertical velocity, \(\frac{(\rho c)_{\rm m} L}{k_0}v^* \)

*x*Dimensionless horizontal coordinate,

*x*^{*}/*L**x*^{*}Horizontal coordinate

*y*Dimensionless upward vertical coordinate,

*y*^{*}/*H**y*^{*}Upward vertical coordinate

*Greek symbols*- β
Fluid volumetric expansion coefficient

- θ
Dimensionless temperature, \(\frac{T^* -T_0}{T_1 -T_0}\)

- μ
Fluid viscosity

- ρ
Density

- ρ
_{0} Fluid density at temperature

*T*_{0}- σ
Heat capacity ratio, \(\frac{(\rho c)_{\rm m}}{(\rho c)_{\rm f}}\)

- ψ
Streamfunction defined by Eqs. (10a,b)

*Subscripts*- f
Fluid

- m
Overall porous medium

*Superscripts*- *
Dimensional variable

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## References

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