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Transport in Porous Media

, Volume 98, Issue 1, pp 223–238 | Cite as

Natural Convection Induced by a Heated Vertical Plate Embedded in a Porous Medium with Transpiration: Local Thermal Non-equilibrium Similarity Solutions

  • Mohsen Nazari
  • Esmaeel Shakerinejad
  • Morteza Nazari
  • D. Andrew S. ReesEmail author
Article

Abstract

This paper is concerned with the thermal non-equilibrium free convection boundary layer, which is induced by a vertical heated plate embedded in a saturated porous medium. The effect of suction or injection on the free convection boundary layer is also studied. The plate is assumed to have a linear temperature distribution, which yields a boundary layer of constant thickness. On assuming Darcy flow, similarity solutions are obtained for governing the steady laminar boundary layer equations. The reduced Nusselt numbers for both the solid and fluid phases are calculated for a wide range of parameters, and compared with asymptotic analyses.

Keywords

Thermal non-equilibrium Natural convection Porous medium  Similarity solution 

List of symbols

\(A\)

Constant

\(C\)

Constant

\(f\)

Reduced streamfunction

\(f_\mathrm{{w}}\)

Suction parameter

\(F\)

Reduced streamfunction in Appendix A

\(g\)

Gravity

\(h\)

Dimensional interstitial heat transfer coefficient

\(H\)

Nondimensional interstitial heat transfer coefficient

\(k\)

Thermal conductivity

\(K\)

Permeability

LTE

Local thermal equilibrium

LTNE

Local thermal non-equilibrium

Nu

Local Nusselt number

\(q\)

Surface rate of heat flux

\(\text{ Ra}_x\)

Local Darcy-Rayleigh number

\(T\)

Dimensional temperature

\(u\)

Vertical velocity

\(v\)

Horizontal velocity

\(x\)

Vertical coordinate

\(y\)

Horizontal coordinate

Greek Characters

\(\alpha \)

Thermal diffusivity

\(\beta \)

Thermal expansion coefficient

\(\gamma \)

Porosity-modified conductivity ratio

\(\delta \)

Constant

\(\epsilon \)

Porosity

\(\zeta \)

Scaled similarity variable

\(\eta \)

Similarity variable

\(\theta \)

Nondimensional fluid temperature

\(\varTheta \)

Inner-layer fluid temperature

\(\nu \)

Kinematic viscosity

\(\xi \)

Scaled similarity variable

\(\rho \)

Density

\(\tau \)

Constant

\(\phi \)

Nondimensional solid temperature

\(\varPhi \)

Inner-layer solid temperature

\(\psi \)

Streamfunction

Superscripts and Subscripts

\(\infty \)

Ambient/initial conditions

\(\dot{~}\)

Derivative with respect to \(\zeta \)

\(~^{\prime }\)

Derivative with respect to \(\eta \)

f

Fluid

p

Constant pressure

s

Solid

w

Wall

Notes

Acknowledgments

The authors would like to thank the anonymous reviewer for his/her very useful comments which have served to improve the paper.

References

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Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • Mohsen Nazari
    • 1
  • Esmaeel Shakerinejad
    • 1
  • Morteza Nazari
    • 2
  • D. Andrew S. Rees
    • 3
    Email author
  1. 1.Department of Mechanical EngineeringShahrood University of TechnologyShahroodIran
  2. 2.Department of MathematicsShahrood University of TechnologyShahroodIran
  3. 3.Department of Mechanical EngineeringUniversity of BathBathUK

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