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

- 260 Downloads
- 5 Citations

## 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

- Ali, M.E.: The effect of lateral mass flux on the natural convection boundary layers induced by a heated vertical plate embedded in a saturated porous medium with internal heat generation. Int. J. Therm. Sci.
**46**, 157–163 (2007)CrossRefGoogle Scholar - Banu, N., Rees, D.A.S.: The onset of Darcy-Bénard convection using a thermal nonequilibrium model. Int. J. Heat Mass Transf.
**45**, 2221–2228 (2002)CrossRefGoogle Scholar - Baytaş, A.C., Pop, I.: Free convection in a square porous cavity using a thermal nonequilibrium model. Int. J. Therm. Sci.
**41**, 861–870 (2002)CrossRefGoogle Scholar - Cheng, P.: The influence of lateral mass flux on free convection boundary layers in a saturated porous medium. Int. J. Heat Mass Transf.
**20**, 201–206 (1977)CrossRefGoogle Scholar - Cheng, P., Chang, I.-D.: On buoyancy induced flows in a saturated porous medium adjacent to impermeable horizontal surfaces. Int. J. Heat Mass Transf.
**19**, 1267–1272 (1976)CrossRefGoogle Scholar - Cheng, P., Minkowycz, W.J.: Free convection about a vertical flat plate embedded in a porous medium with application to heat transfer from a dike. J. Geophys. Res.
**82**, 2040–2044 (1977)CrossRefGoogle Scholar - Combarnous, M., Bories, S.: Modelisation de la convection naturelle au sein d’une couche poreuse horizontal l’aide d’un coefficient de transfert solide–fluide. Int. J. Heat Mass Transf.
**17**, 505–515 (1974)CrossRefGoogle Scholar - Gupta, P.S., Gupta, A.S.: Heat and mass transfer on a stretching sheet with suction or blowing. Canad. J. Chem. Eng.
**55**, 744–746 (1977)CrossRefGoogle Scholar - Kuznetsov, A.V.: Thermal nonequilibrium forced convection in porous media. In: Ingham, D.B., Pop, I. (eds.) Transport Phenomena in Porous Media. Pergamon, Oxford (1998)Google Scholar
- Magyari, E., Keller, B.: Exact analytical solutions for free convection boundary layers on a heated vertical plate with lateral mass flux embedded in a saturated porous medium. Heat Mass Transf.
**36**, 109–116 (2000)CrossRefGoogle Scholar - Mohamad, A.A.: Nonequilibrium natural convection in a differentially heated cavity filled with a porous matrix. Trans. ASME J. Heat Transf.
**122**, 380–384 (2000)CrossRefGoogle Scholar - Rees, D.A.S.: Vertical free convective boundary-layer flow in a porous medium using a thermal nonequilibrium model: elliptical effects. J. Appl. Math. Phys. (ZAMP)
**54**, 437–448 (2003)CrossRefGoogle Scholar - Rees, D.A.S.: Microscopic modelling of the two-temperature model for conduction in heterogeneous media: three-dimensional media. In: Proceedings of the 4th International Conference on Applications of Porous Media, Paper 15, Istanbul (2009)Google Scholar
- Rees, D.A.S.: Microscopic modeling of the two-temperature model for conduction in heterogeneous media. J. Porous Media
**13**, 125–143 (2010)CrossRefGoogle Scholar - Rees, D.A.S., Bassom, A.P.: The radial injection of a hot fluid into a cold porous medium: the effects of local thermal non-equilibrium. Comput. Therm. Sci.
**2**(3), 221–230 (2010)CrossRefGoogle Scholar - Rees, D.A.S., Bassom, A.P., Siddheshwar, P.G.: Local thermal non-equilibrium effects arising from the injection of a hot fluid into a porous medium. J. Fluid Mech.
**594**, 379–398 (2008)CrossRefGoogle Scholar - Rees, D.A.S., Pop, I.: Free convective stagnation point flow in a porous medium using a thermal nonequilibrium model. Int. Commun. Heat Mass Transf.
**26**, 945–954 (1999)CrossRefGoogle Scholar - Rees, D.A.S., Pop, I.: Vertical free convective boundary-layer flow in a porous medium using a thermal nonequilibrium model. J. Porous Media
**3**, 31–44 (2000)Google Scholar - Rees, D.A.S., Pop, I.: Local thermal nonequilibrium in porous medium convection. In: Ingham, D.B., Pop, I. (eds.) Transport Phenomena in Porous Media III, pp.147–173. Pergamon, Oxford (2005)Google Scholar