, Volume 44, Issue 2, pp 133–144 | Cite as

Radiation effects on combined convection over a vertical flat plate embedded in a porous medium of variable porosity

  • Dulal Pal
  • Hiranmoy Mondal


The present paper is concerned with the study of radiation effects on the combined (forced-free) convection flow of an optically dense viscous incompressible fluid over a vertical surface embedded in a fluid saturated porous medium of variable porosity with heat generation or absorption. The effects of radiation heat transfer from a porous wall on convection flow are very important in high temperature processes. The inclusion of radiation effects in the energy equation leads to a highly non-linear partial differential equations which are transformed to a system of ordinary differential equations using non-similarity transformation. These equations are then solved numerically using implicit finite-difference method subject to appropriate boundary and matching conditions. A parametric study of the physical parameters such as the particle diameter-based Reynolds number, the flow based Reynolds number, the Grashof number, the heat generation or absorption co-efficient and radiation parameter is conducted on temperature distribution. The effects of radiation and other physical parameters on the local skin friction and on local Nusselt number are shown graphically. It is interesting to observe that the momentum and thermal boundary layer thickness increases with the radiation and decrease with increase in the Prandtl number.


Convection Non-Darcy Thermal radiation Boundary layer flow Mechanics of fluids 



empirical constants


inertia co-efficient


fluid heat capacity


local skin-friction co-efficient


particle diameter


reduced stream function


acceleration due to gravity


Grashof number g β(T w T )L 3/ν 2


porous medium effective thermal conductivity


fluid thermal conductivity


thermal conductivity of the porous medium


porous medium permeability


characteristic plate length


local Nusselt number


dimensional heat generation or absorption coefficient


Prandtl number μ C p /k f


ratio of k e and k f


flow Reynolds number ρ U L/μ


Reynolds number based on the particle diameter, ρ U d/μ


local Reynolds number


thermal radiation parameter


radiative heat flux


mean absorption co-efficient


fluid temperature


wall temperature


free-stream temperature


Darcy’s number


x component of fluid velocity


y component of fluid velocity


free-stream velocity


vertical or tangential distance


normal distance

Greek symbols


coefficient of thermal expansion


heat-generation or absorption coefficient


transformed normal coordinate


stream function


Stefan-Boltzmann constant


porous medium porosity


free-stream porosity


fluid dynamic viscosity


fluid kinematic viscosity, μ/ρ


transformed tangential coordinate


fluid density


dimensionless temperature, (TT )/(T w T )


assisting λ=1 or opposing λ=−1 flow constant


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  1. 1.
    Cheng P (1978) Heat transfer in geothermal systems. In: Advances in heat transfer, vol 14. Academic Press, New York, pp 1–105 Google Scholar
  2. 2.
    Cheng P, Minkowycz WJ (1977) Free convection about a vertical plate embedded in a porous medium with application to heat transfer from a dike. J Geophys Res 82:2040–2044 CrossRefADSGoogle Scholar
  3. 3.
    Benenati RF, Brosilow CB (1962) Void fraction distribution in beds of spheres. AIChE J 8:359–361 CrossRefGoogle Scholar
  4. 4.
    Vafai K (1984) Convective flow and heat transfer in variable-porosity media. J Fluid Mech 147:233–259 MATHCrossRefADSGoogle Scholar
  5. 5.
    Vafai K, Alkire RL, Tien CL (1985) An experimental investigation of heat transfer in variable porosity media. J Heat Transf 107:642–947 Google Scholar
  6. 6.
    Plumb O, Huenefeld JC (1981) Non-Darcy natural convection from heated surfaces in saturated porous medium. Int J Heat Mass Transf 24:765–768 MATHCrossRefGoogle Scholar
  7. 7.
    Ranganathann P, Viskanta R (1984) Mixed convection boundary layer flow along a vertical surface in a porous medium. Numer Heat Transf 7:305–317 CrossRefADSGoogle Scholar
  8. 8.
    Takhar HS, Soundalgekar VM, Gupta AS (1990) Mixed convection of an incompressible viscous fluid in a porous medium past a hot vertical plate. Int J Non-Linear Mech 25:723–728 MATHCrossRefGoogle Scholar
  9. 9.
    Nakayama A, Pop I (1991) A unified similarity transformation for free, forced and mixed convection flows in Darcy and non-Darcy porous media. Int J Heat Mass Transf 34:357–367 MATHCrossRefGoogle Scholar
  10. 10.
    Hsieh JC, Chen TS, Armaly BF (1993) Nonsimilarity solutions for mixed convection from vertical surfaces in porous media. Int J Heat Mass Transf 36:485–1493 Google Scholar
  11. 11.
    Chen CH, Chen TS, Chen CK (1996) Non-Darcy mixed convection along non-isothermal vertical surfaces in porous media. Int J Heat Mass Transf 39:1157–1164 CrossRefGoogle Scholar
  12. 12.
    Amiri A, Vafai K (1994) Analysis of dispersion effects and non-thermal equilibrium, non-Darcian, variable porosity incompressible flow through porous media. Int J Heat Mass Transf 37:936–954 CrossRefGoogle Scholar
  13. 13.
    Fried JJ, Combarnous M (1976) Dispersion in porous media. Adv Hydrosc 11:169–282 Google Scholar
  14. 14.
    Cheng P (1981) Thermal dispersion effects in non-Darcian convective flows in a saturated porous medium. Lett Heat Mass Transf 8:267–270 CrossRefGoogle Scholar
  15. 15.
    Lai FC, Kulacki FA (1989) Thermal dispersion effect on non-Darcy convection from horizontal surface in saturated porous media. Int J Heat Mass Transf 32:971–976 CrossRefGoogle Scholar
  16. 16.
    Murthy PVSN, Singh P (1997) Thermal dispersion effects on non-Darcy natural convection with lateral mass flux. Heat and Mass Transf 33:1–5 CrossRefADSGoogle Scholar
  17. 17.
    Schwartz CE, Smith JM (1953) Flow distribution in packed beds. Ind Eng Chem 45:1209–1218 CrossRefGoogle Scholar
  18. 18.
    Vafai K, Tien CL (1981) Boundary and inertia effects on flow and heat transfer in porous media. Int J Heat Mass Transf 24:195–203 MATHCrossRefGoogle Scholar
  19. 19.
    Chandrasekhara BC, Namboodiri PMS (1985) Influence of variable permeability on combined vertical surfaces in porous medium. Int J Heat Mass Transf 28:199–206 MATHCrossRefGoogle Scholar
  20. 20.
    Hong JT, Tien CL, Kaviany M (1985) Non-Darcy effect on vertical plate natural convection in porous media with high porosity. Int J Heat Mass Transf 28:2149–2157 CrossRefGoogle Scholar
  21. 21.
    Hong JT, Tien CL (1987) Analysis of thermal dispersion effect on vertical plate natural convection in porous media. Int J Heat Mass Transf 30:143–150 MATHCrossRefGoogle Scholar
  22. 22.
    Boutros YZ, Abd-el-Malek MB, Badran NA, Hassan HS (2006) Lie-group method of solution for steady two-dimension boundary layer stagnation-point flow towards a heated stretching sheet placed in a porous medium. Meccanica 41:681–691 CrossRefMathSciNetMATHGoogle Scholar
  23. 23.
    Whitaker S (1980) Radiant energy transport in porous media. Int Eng Chem Fund 19:210–218 CrossRefGoogle Scholar
  24. 24.
    Plumb OA, Huenfeld JS, Eschbach EJ (1981) The effect of crossflow and radiation on natural convection from vertical heated surfaces in saturated porous media. In: AIAA 16th thermophysics conference, Palo Alto, CA, USA, June 23–25 Google Scholar
  25. 25.
    Ibrahiem F.S., Hady FM (1990) Mixed convection-radiation interaction in boundary layer flow over a horizontal surface. Astrophys Space Sci 168:263–276 CrossRefADSGoogle Scholar
  26. 26.
    Ishak A, Nazar R, Pop I (2006) Mixed convection boundary layer in the stagnation-point flow toward a stretching vertical sheet. Meccanica 41:509–518 CrossRefMATHGoogle Scholar
  27. 27.
    Gorla RSR, Pop I (1993) Conjugate heat transfer with radiation from a vertical circular pin in non-Newtonian ambient medium. Warme Toffubertragung 28:11–15 CrossRefADSGoogle Scholar
  28. 28.
    Hossain MA, Takhar HS (1996) Radiation effect on mixed convection along a vertical plate with uniform surface temperature. Heat Mass Transf 31:243–248 CrossRefADSGoogle Scholar
  29. 29.
    Takhar HS, Gorla RSR, Soundalgekar VM (1996) Radiation effects on MHD free convection flow of a gas past a semi-infinite vertical plate. Int J Numer Methods Heat Fluid Flow 6:77–83 MATHCrossRefGoogle Scholar
  30. 30.
    Bakier AY, Gorla RSR (1996) Thermal radiation effect on mixed convection from horizontal surfaces in saturated porous media. Transp Porous Media 23:357–363 CrossRefGoogle Scholar
  31. 31.
    Mansour MA (1997) Forced convection-radiation interaction heat transfer in boundary layer over a flat plate submerged in a porous medium. Appl Mech Eng 3:405–413 Google Scholar
  32. 32.
    Raptis A (1998) Radiation and free convection flow through a porous medium. Int Commun Heat Mass Transf 25:289–295 CrossRefGoogle Scholar
  33. 33.
    Chamkha AJ (1997) solar radiation assisted convection in uniform porous medium supported by a vertical flat plate. Trans ASME J Heat Transf 119:89–96 CrossRefGoogle Scholar
  34. 34.
    Chamka AJ, Khanafer K (1999) Nonsimilar combined convection flow over a vertical surface embedded in a variable porosity medium. J Porous Media 2(3):231–249 Google Scholar
  35. 35.
    Hossain MA, Alim MA, Rees DA (1999) The effect of radiation on free convection from a porous vertical plate. Int J Heat Mass Transf 42:181–191 MATHCrossRefGoogle Scholar
  36. 36.
    Mohammadein AA, El-Amin MF (2000) Thermal dispersion-radiation effects on non-Darcy natural convection in a fluid saturated porous medium. Transp Porous Medium 40(2):153–163 CrossRefGoogle Scholar
  37. 37.
    El-Hakiem MA, El-Amin MF (2001) Thermal radiation effects on non-Darcy natural convection with lateral mass flux. Heat Mass Transf 37:161–165 CrossRefADSGoogle Scholar
  38. 38.
    Mezhericher M, Levy A, Borde I (2008) The influence of thermal radiation on drying of single droplet/wet particle. Dry Technol 26:78–89 CrossRefGoogle Scholar
  39. 39.
    Mezhericher M, Levy A, Borde I (2007) Theoretical drying model of single droplets containing insoluble or dissolved solids. Dry Technol 25(6):1035–1042 CrossRefGoogle Scholar
  40. 40.
    Dolinsky AA (2001) High-temperature spray drying. Dry Technol 19(5):785–806 CrossRefGoogle Scholar
  41. 41.
    Chamkha AJ, Issa C, Khanafer K (2001) Natural convection due to solar radiation from a vertical plate embedded in a porous medium with variable porosity. J Porous Media 4:69–77 MATHGoogle Scholar
  42. 42.
    Lawrence PS, Rao BN (1994) Heat transfer in a viscoelastic boundary layer flow over a stretching sheet. J Phys D: Appl Phys 27:1323–1327 CrossRefADSGoogle Scholar
  43. 43.
    Nithiarasu P, Seetharamu KN, Sundararajan T (1997) Natural convective heat transfer in a fluid saturated variable porosity medium. Int J Heat Mass Transf 40:3955–3967 MATHCrossRefGoogle Scholar
  44. 44.
    Ramachandran N, Armaly BF, Chen TS (1985) Measurements and predictions of laminar mixed convection flow adjacent to a vertical surface. ASME J Heat Transf 107:636–641 CrossRefGoogle Scholar
  45. 45.
    Kuznetsov AV, Nield DA (2006) Boundary layer treatment of forced convection over a wedge with an attached porous substrate. J Porous Media 9(7):683–694 CrossRefGoogle Scholar
  46. 46.
    Aydin OR, Kaya AH (2007) Non-Darcian forced convection flow of viscous dissipating fluid over a flat plate embedded in a porous medium. Transp Porous Medium. doi: 10.1007/s11242-007-9166-8 Google Scholar
  47. 47.
    Lin HT, Lin LK (1987) Similarity solutions for laminar forced convection heat transfer from wedges to fluids of any Prandtl number. Int J Heat Mass Transf 30:1111–1118 CrossRefADSGoogle Scholar
  48. 48.
    Yih KA (1999) MHD forced convection flow adjacent to a non-isothermal wedge. Int Commun Heat Mass 26(6):819–827 CrossRefGoogle Scholar
  49. 49.
    Chamkha AJ, Mujtaba M, Quadri A, Issa C (2003) Thermal radiation effects on MHD forced convection flow adjacent to a non-isothermal wedge in the presence of a heat source or sink. Heat Mass Transf 39:305–312 ADSGoogle Scholar
  50. 50.
    Takhar HS, Kumare M, Nath G (2001) Buoyancy effect in boundary layer on a continuously moving vertical surface with a parallel free stream. Arch Mech 53(2):151–166 MATHGoogle Scholar
  51. 51.
    Acharya M, Singh LP, Dash GC (1999) Heat ans Mass transfer over an accelerating surface with heat source in presence of suction and blowing. Int J Eng Sci 37:189–211 CrossRefGoogle Scholar
  52. 52.
    Hossain MA, Rees DAS, Pop I (1998) Free convection-radiation interaction from an isothermal plate inclined at a small angle to the horizontal. Acta Mech 127:63–73 MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Department of Mathematics, Siksha BhavanaVisva-Bharati UniversitySantiniketanIndia
  2. 2.Department of MathematicsBengal Institute of Technology and ManagementSantiniketanIndia

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