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The Onset of Prandtl–Darcy–Prats Convection in a Horizontal Porous Layer

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Abstract

We consider the effect of finite Prandtl–Darcy numbers of the onset of convection in a porous layer heated isothermally from below and which is subject to a horizontal pressure gradient. A dispersion relation is found which relates the critical Darcy–Rayleigh number and the induced phase speed of the cells to the wavenumber and the imposed Péclet and Prandtl–Darcy numbers. Exact numerical solutions are given and these are supplemented by asymptotic solutions for both large and small values of the governing nondimensional parameters. The classical value of the critical Darcy–Rayleigh number is \(4\pi ^2\), and we show that this value increases whenever the Péclet number is nonzero and the Prandtl–Darcy number is finite simultaneously. The corresponding wavenumber is always less than \(\pi \) and the phase speed of the convection cells is always smaller than the background flux velocity.

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Abbreviations

\(c\) :

Specific heat, phase speed of convection cells

\(c_a\) :

Acceleration coefficient scalar

\(f(z)\) :

Reduced streamfunction

\(F\) :

Dispersion relation

\(g(z)\) :

Reduced temperature

\(\mathbf{g}\) :

Gravity vector

\(H\) :

Height of the layer

\(k\) :

Wavenumber

\(k_\mathrm{{m}}\) :

Thermal conductivity of the porous medium

\(K\) :

Permeability

\(n\) :

Iteration number

\(P\) :

Pressure

Pd:

Prandtl–Darcy number

Pe:

Péclet number

\(\text{ Pr }_\mathrm{{m}}\) :

Prandtl number

\(\text{ Ra }\) :

Darcy–Rayleigh number

\(t\) :

Time

\(T\) :

Dimensional temperature

\(T_0\) :

Dimensional temperature of lower boundary

\(T_1\) :

Dimensional temperature of upper boundary

\(\mathbf{v}\) :

Darcy velocity vector

\(x,y,z\) :

Coordinate sysem; see Fig. 1

\(\alpha _\mathrm{{m}}\) :

Thermal diffusivity of the porous medium

\(\beta \) :

Volumetric coefficient of thermal expansion

\(\gamma \) :

Reciprocal of the Prandtl–Darcy number

\(\Delta T\) :

\(T_0-T_1\)

\(\theta \) :

Nondimensional temperature

\(\mu \) :

Dynamic viscosity

\(\rho _0\) :

Density

\(\sigma \) :

Heat capacity ratio

\(\psi \) :

Nondimensional streamfunction

\(\hat{ }\) :

Dimensional quantity

\(\tilde{ }\) :

Perturbation

\({}^{\prime }\) :

Ordinary derivative with respect to \(z\)

c:

Critical value

f:

Fluid

m:

Porous medium

\(1,2,3\ldots \) :

Terms in an asymptotic expansion

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Authors and Affiliations

  1. Department of Mechanical Engineering, University of Bath, Bath, BA2 7AY, UK

    Emily Dodgson & D. Andrew S. Rees

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  1. Emily Dodgson
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  2. D. Andrew S. Rees
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Correspondence to D. Andrew S. Rees.

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Dodgson, E., Rees, D.A.S. The Onset of Prandtl–Darcy–Prats Convection in a Horizontal Porous Layer. Transp Porous Med 99, 175–189 (2013). https://doi.org/10.1007/s11242-013-0180-8

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  • DOI: https://doi.org/10.1007/s11242-013-0180-8

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