Abstract
An analytical investigation of the effect of vertical throughflow on the onset of convection in a composite porous medium consisting of two horizontal layers has been made. The cases of isoflux and isotemperature boundaries are both investigated. The critical Rayleigh number depends on a Péclet number \(Q\), a permeability ratio \(K_{r}\), a thermal conductivity ratio \(k_{r}\), and a depth ratio \(\delta \). For the case of small \(Q\) an approximate solution is obtained, which shows that in general throughflow has a stabilizing effect whose magnitude may be increased or decreased by the heterogeneity. This solution is supplemented by an asymptotic solution valid for large \(Q.\)
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Abbreviations
 \(a\) :

Horizontal overall wavenumber
 \(D\) :

\({\text{ d}}/{\text{ d}}z\)
 \(\tilde{D}\) :

\({\text{ d}}/{\text{ d}}\zeta \)
 \(F_{1}\) :

Function defined by Eq. (30)
 \(F_{2}\) :

Function defined by Eq. (31)
 \(g\) :

Gravitational acceleration
 \(H\) :

Layer depth
 \(k\) :

Effective thermal conductivity
 \(k\) :

Nondimensional wavenumber in the \(x\)direction, used in Eq. (42)
 \(k_{r}\) :

Conductivity ratio, \(k_{2}/k_{1}\)
 \(\hat{{k}}\) :

Parameter defined in Eq. (13)
 \(K\) :

Permeability
 \(K_{r}\) :

Permeability ratio, \(K_{2}/K_{1}\)
 \(\hat{{K}}\) :

Parameter defined in Eq. (13)
 \(P\) :

Dimensionless pressure, \(\frac{(\rho c)_\mathrm{f} K_1 }{\mu k_1 }P^{*}\)
 \(P^{*}\) :

Pressure
 \(Q\) :

Péclet number, \(\frac{(\rho c)_\mathrm{f} HV}{k_1 }\)
 \(\tilde{Q}\) :

Parameter defined by Eq. (32)
 \(Ra\) :

Rayleigh number \(\frac{(\rho c)_\mathrm{f} \rho _0 g\beta K_1 H(T_1 T_0 )}{\mu k_1 }\)
 \(t\) :

Dimensionless time, \(\frac{k_1 }{(\rho c)_\mathrm{m} H^{2}}t^{*}\)
 \(t^{*}\) :

Time
 \(T^{*}\) :

Temperature
 \(T_{0}\) :

Temperature at the upper boundary
 \(T_{1}\) :

Temperature at the lower boundary
 \((u, v, w)\) :

Dimensionless velocity components, \(\frac{(\rho c)_\mathrm{f} H}{k_1}(u^{*},v^{*},w^{*})\)
 u \(^{*}\) :

Darcy velocity, \((u^{*},v^{*},w^{*})\)
 \(V\) :

Upward vertical throughflow velocity
 \((x, y, z)\) :

Dimensionless Cartesian coordinates, \(\frac{1}{H}(x^{*},y^{*},z^{*})\)
 \((x^{*},y^{*},z^{*})\) :

Cartesian coordinates (\(z^{*}\) is the vertically upward coordinate)
 \(\alpha \) :

\(a/\tilde{Q}\)
 \(\beta \) :

Volumetric expansion coefficient
 \(\delta \) :

Dimensionless layer depth ratio (interface position)
 \(\varepsilon \) :

Dimensionless small quantity
 \(\zeta \) :

\(\tilde{Q}(1z)\)
 \(\theta \) :

Dimensionless temperature, \(\frac{T^{*}T_0 }{T_1 T_0 }\)
 \(\mu \) :

Fluid viscosity
 \(\rho _{0}\) :

Fluid density at temperature \(T_{0}\)
 \((\rho c)_\mathrm{f}\) :

Heat capacity of the fluid
 \((\rho c)_\mathrm{m} \) :

Heat capacity of the overall porous medium
 \(\sigma \) :

Heat capacity ratio, \(\frac{(\rho c)_\mathrm{m} }{(\rho c)_\mathrm{f} }\)
 \(\omega \) :

Parameter defined in Eq. (32)
 B:

Basic state
 c:

Critical value
 1:

The region \(0\le z^{*}<\delta H\)
 2:

The region \(\delta H\le z^{*}\le H\)
 \(r\) :

Relative quantity, introduced in Eq. (14)
 \(\prime \) :

Perturbation variable
 \(^{*}\) :

Dimensional variable
References
Homsy, G.M., Sherwood, A.E.: Convective instabilities in porous media with throughflow. AIChE J 22, 168–174 (1976)
Jones, M.C., Persichetti, J.M.: Convective instability in packed beds with throughflow. AIChE J 32, 1555–1557 (1986)
Kuznetsov, A.V., Nield, D.A.: The effects of combined horizontal and vertical heterogeneity on the onset of convection in a porous medium with vertical throughflow. Transp. Porous Media 90, 465–478 (2011)
Kuznetsov, A.V., Nield, D.A.: The effect of strong heterogeneity and strong throughflow on the onset of convection in a porous medium: periodic and localized variation. Transp. Porous Media 92, 289–298 (2012a)
Kuznetsov, A.V., Nield, D.A.: The onset of doublediffusive convection in a vertical cylinder occupied by a heterogeneous porous medium with vertical throughflow. Transp. Porous Media 95, 327–336 (2012b)
Nield, D.A.: Convective instability in porous media with throughflow. AIChE J 33, 1222–1224 (1987)
Nield, D.A., Bejan, A.: Convection in Porous Media, 4th edn. Springer, New York (2013)
Nield, D.A., Kuznetsov, A.V.: The onset of convection in a heterogeneous porous medium with vertical throughflow. Transp. Porous Media 88, 347–355 (2011)
Nield, D.A., Kuznetsov, A.V.: The effect of strong heterogeneity and strong throughflow on the onset of convection in a porous medium: nonperiodic global variation. Transp. Porous Media 91, 927–938 (2012)
Sutton, F.M.: Onset of convection in a porous channel with net through flow. Phys. Fluids 13, 1931–1934 (1970)
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Nield, D.A., Kuznetsov, A.V. The Onset of Convection in a Layered Porous Medium with Vertical Throughflow. Transp Porous Med 98, 363–376 (2013). https://doi.org/10.1007/s1124201301488
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Keywords
 Heterogeneity
 Throughflow
 Thermal instability
 Horizontal layers