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The Onset of Convection in a Layered Porous Medium with Vertical Throughflow

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 iso-flux and iso-temperature 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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Fig. 1

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

\(k\) :

Non-dimensional wave-number 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

\(\zeta \) :

\(\tilde{Q}(1-z)\)

\(\theta \) :

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

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  10. Sutton, F.M.: Onset of convection in a porous channel with net through flow. Phys. Fluids 13, 1931–1934 (1970)

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Correspondence to A. V. Kuznetsov.

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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/s11242-013-0148-8

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Keywords

  • Heterogeneity
  • Throughflow
  • Thermal instability
  • Horizontal layers