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Exact solutions for equations of flow and heat transfer in fluid saturated porous media

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Abstract

In this paper we investigate the complete behavior of the viscous and incompressible fluid flow in an anisotropic saturated porous horizontal cavity. Taking into account the rigid/rigid and stress-free/stress-free horizontal boundaries, we obtain the exact solutions for flow and heat transfer equations, i.e. the velocity and temperature, depending on the Darcy–Rayleigh number, the Darcy number, the anisotropic permeability ratio and the inclination of the principal axis. These obtained variables allow us to compute the different expressions of the pressure gradient.

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Abbreviations

\(a, b, c\) :

Constants

\(C\) :

Dimensionless temperature gradient in \(x\) direction

\(Da\) :

Darcy number, \(K_1/{L'}^{2}\)

\(\overrightarrow{g}\) :

Gravitational acceleration

\(H'\) :

Depth of cavity

\(k\) :

Thermal conductivity

\(\overline{K}\) :

Flow permeability tensor

\(K_1, K_2\) :

Flow permeability along the principal axes

\(K^{\star }\) :

Anisotropic permeability ratio, \(K_1/K_2\)

\(L'\) :

Width of cavity

\(q'\) :

Uniform heat flux

\(Ra\) :

Darcy–Rayleigh number, \(g\beta K_1{L'}^2q'/k\nu \alpha \)

\(Ra^{\star }\) :

Rayleigh number for a fluid, \(Ra/Da\)

\(t\) :

Dimensionless time

\(\widetilde{T}\) :

Dimensionless temperature

\(T\) :

Dimensionless quasi-state temperature, \(\widetilde{T}-\xi t\)

\(\Delta T'\) :

Temperature scale, \(q'L'/k\)

\(\Delta T\) :

Wall to wall dimensionless temperature difference at \(x=0\)

\(\overrightarrow{V}\) :

Seepage velocity

\(u, v\) :

Dimensionless velocity components in \(x, y\) directions

\(x\) :

Dimensionless horizontal coordinate

\(y\) :

Dimensionless vertical coordinate Greek symbols

\(\alpha \) :

Thermal diffusivity

\(\beta \) :

Thermal expansion coefficient of the fluid

\(\gamma \) :

Inclination of the principal axis

\(\mu \) :

Dynamic viscosity of the fluid

\(\nu \) :

Kinematic viscosity of the fluid

\(\phi \) :

\(y\)-Dependent temperature term

\(\psi \) :

Dimensionless stream function, \(\psi '/\alpha \)

\(\rho \) :

Density of the fluid

\((\rho c)_f\) :

Heat capacity of the fluid

\((\rho c)_p\) :

Heat capacity of saturated porous medium

\(\tau \) :

Dimensionless Darcy parameter, \({Da}^{-\frac{1}{2}}\)

\(\sigma \) :

Heat capacity ratio, \((\rho c)_p/(\rho c)_f\)

\(\xi \) :

Dimensionless uniform heat sink Superscript

\('\) :

Dimensional quantities Subscript

\(o\) :

Refers to origin

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Authors and Affiliations

  1. International Chair of Mathematical Physics and Applications (ICMPA-UNESCO Chair) of Faculty of Sciences and Technology, University of Abomey-Calavi, 072 B.P. 50 , Cotonou, Republic of Benin

    V. Adanhounme

  2. Technological Institute of Lokossa, University of Abomey-Calavi, Cotonou, Republic of Benin

    E. T. Olodo

Authors
  1. V. Adanhounme
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  2. E. T. Olodo
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Correspondence to V. Adanhounme.

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Adanhounme, V., Olodo, E.T. Exact solutions for equations of flow and heat transfer in fluid saturated porous media. Afr. Mat. 26, 913–921 (2015). https://doi.org/10.1007/s13370-014-0258-7

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  • DOI: https://doi.org/10.1007/s13370-014-0258-7

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