Meccanica

, Volume 47, Issue 5, pp 1261–1269 | Cite as

Numerical solutions of free convection boundary layer flow on a solid sphere with Newtonian heating in a micropolar fluid

Article

Abstract

In this paper, the problem of free convection boundary layer flow on a solid sphere in a micropolar fluid with Newtonian heating, in which the heat transfer from the surface is proportional to the local surface temperature, is considered. The transformed boundary layer equations in the form of partial differential equations are solved numerically using an implicit finite-difference scheme. Numerical solutions are obtained for the local wall temperature, the local skin friction coefficient, as well as the velocity, angular velocity and temperature profiles. The features of the flow and heat transfer characteristics for different values of the material or micropolar parameter K, the Prandtl number Pr and the conjugate parameter γ are analyzed and discussed.

Keywords

Boundary layer Free convection Micropolar fluid Newtonian heating Sphere 

Nomenclature

a

radius of the sphere

hs

heat transfer parameter for Newtonian heating

Cf

skin friction coefficient

f

dimensionless stream function

g

acceleration due to gravity

Gr

Grashof number

H

angular velocity of micropolar fluid

j

microinertia density

K

material parameter of micropolar fluid

k

thermal conductivity

Pr

Prandtl number

Re

Reynolds number

T

fluid temperature

T

ambient temperature

U

free stream velocity

u,v

velocity components along the x and y directions, respectively

x,y

Cartesian coordinates along the sphere and normal to it, respectively

Greek Letters

β

thermal expansion coefficient

γ

conjugate parameter for Newtonian heating

μ

dynamic viscosity

ν

kinematic viscosity

θ

dimensionless temperature

κ

vortex viscosity

φ

spin gradient viscosity

ρ

fluid density

ψ

stream function

Notes

Acknowledgement

The authors gratefully acknowledge the financial supports received from the Ministry of Higher Education, Malaysia (UKM-ST-07-FRGSS0036-2009) and the Universiti Malaysia Pahang (RDU110108). They also wish to express their very sincere thanks to the reviewers for the valuable comments and suggestions.

References

  1. 1.
    Eringen AC (1966) Theory of micropolar fluids. J Math Mech 16:1–18 MathSciNetGoogle Scholar
  2. 2.
    Ariman T, Turk MA, Sylvester ND (1974) Application of microcontinuum fluid mechanics. Int J Eng Sci 12:273–293 MATHCrossRefGoogle Scholar
  3. 3.
    Nazar R, Amin N, Grosan T, Pop I (2002) Free convection boundary layer on an isothermal sphere in a micropolar fluid. Int Commun Heat Mass Transf 29:377–386 CrossRefGoogle Scholar
  4. 4.
    Nazar R, Amin N, Grosan T, Pop I (2002) Free convection boundary layer on a sphere with constant surface heat flux in a micropolar fluid. Int Commun Heat Mass Transf 29:1129–1138 CrossRefGoogle Scholar
  5. 5.
    Merkin JH (1994) Natural convection boundary-layer flow on a vertical surface with Newtonian heating. Int J Heat Fluid Flow 15:392–398 CrossRefGoogle Scholar
  6. 6.
    Chaudhary RC, Jain P (2007) An exact solution to the unsteady free convection boundary-layer flow past an impulsively started vertical surface with Newtonian heating. J Eng Phys Thermophys 80:954–960 CrossRefGoogle Scholar
  7. 7.
    Luikov AV, Aleksashenko VA, Aleksashenko AA (1971) Analytic methods of solution of conjugate problems in convection heat transfer. Int J Heat Mass Transf 14:1047–1056 MATHCrossRefGoogle Scholar
  8. 8.
    Martynenko OG, Khramtsov PP (2005) Free-convective heat transfer. Springer, Berlin Google Scholar
  9. 9.
    Kimura S, Kiwata T, Okajima A, Pop I (1997) Conjugate natural convection in porous media. Adv Water Resour 20:111–126 CrossRefGoogle Scholar
  10. 10.
    Salleh MZ, Nazar R, Pop I (2009) Forced convection boundary layer flow at a forward stagnation point with Newtonian heating. Chem Eng Commun 196:987–996 CrossRefGoogle Scholar
  11. 11.
    Salleh MZ, Nazar R, Pop I (2010) Mixed convection boundary layer flow about a solid sphere with Newtonian heating. Arch Mech 62:283–303 MathSciNetGoogle Scholar
  12. 12.
    Salleh MZ, Nazar R, Pop I (2010) Modeling of free convection boundary layer flow on a sphere with Newtonian heating. Acta Appl Math 112:263–274 MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Salleh MZ, Nazar R, Pop I (2010) Modelling of boundary layer flow and heat transfer over a stretching sheet with Newtonian heating. J Taiwan Inst Chem Eng 41(6):651–655 CrossRefGoogle Scholar
  14. 14.
    Salleh MZ, Nazar R, Arifin NM, Merkin JH, Pop I (2011) Forced convection heat transfer over a horizontal circular cylinder with Newtonian heating. J Eng Math 69(1):101–110 MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Merkin JH, Nazar R, Pop I (2011) The development of forced convection heat transfer near a forward stagnation point with Newtonian heating. J Eng Math. doi: 10.1007/s10665-011-9487-z MathSciNetGoogle Scholar
  16. 16.
    Cheng CY (2008) Natural convection heat and mass transfer from a sphere in micropolar fluids with constant wall temperature and concentration. Int Commun Heat Mass Transf 35:750–755 CrossRefGoogle Scholar
  17. 17.
    Cheng CY (2010) Nonsimilar solutions for double-diffusion boundary layers on a sphere in micropolar fluids with constant wall heat and mass fluxes. Appl Math Model 34:1892–1900 MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Miraj M, Alim MA, Andallah LS (2011) Effects of pressure work and radiation on natural convection flow around a sphere with heat generation. Int Commun Heat Mass Transf 38:911–916 CrossRefGoogle Scholar
  19. 19.
    Prasad VR, Vasu B, Anwar Beg O, Parshad RD (2012) Thermal radiation effects on magnetohydrodynamic free convection heat and mass transfer from a sphere in a variable porosity regime. Commun Nonlinear Sci Numer Simul 17:654–671 MathSciNetADSCrossRefGoogle Scholar
  20. 20.
    Turkyilmazoglu M (2011) Numerical and analytical solutions for the flow and heat transfer near the equator of an MHD boundary layer over a porous rotating sphere. Int J Therm Sci 50:831–842 CrossRefGoogle Scholar
  21. 21.
    Cebeci T, Bradshaw P (1988) Physical and computational aspects of convective heat transfer. Springer, New York MATHCrossRefGoogle Scholar
  22. 22.
    Jena SK, Mathur MN (1981) Similarity solutions for laminar free convection flow of a thermo-micropolar fluid past a nonisothermal flat plate. Int J Eng Sci 19:1431–1439 MATHCrossRefGoogle Scholar
  23. 23.
    Guram GS, Smith AC (1980) Stagnation flows of micropolar fluids with strong and weak interactions. Comput Math Appl 6:213–233 MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Ahmadi G (1976) Self-similar solution of incompressible micropolar boundary layer flow over a semi-infinite flat plate. Int J Eng Sci 14:639–646 MATHCrossRefGoogle Scholar
  25. 25.
    Peddieson J (1972) An application of the micropolar fluid model to the calculation of turbulent shear flow. Int J Eng Sci 10:23–32 CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Faculty of Industrial Science and TechnologyUniversiti Malaysia PahangUMP KuantanMalaysia
  2. 2.School of Mathematical Sciences, Faculty of Science and TechnologyUniversiti Kebangsaan MalaysiaUKM BangiMalaysia
  3. 3.Solar Energy Research InstituteUniversiti Kebangsaan MalaysiaUKM BangiMalaysia
  4. 4.Faculty of MathematicsUniversity of ClujClujRomania

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