Heat and Mass Transfer

, Volume 46, Issue 4, pp 413–419 | Cite as

Heat and mass transfer for micropolar flow with radiation effect past a nonlinearly stretching sheet

  • Kai-Long HsiaoEmail author


In this study, an analysis has been performed for heat and mass transfer with radiation effect of a steady laminar boundary-layer flow of a micropolar flow past a nonlinearly stretching sheet. Parameters n, K, k 0, Pr, Ec, and Sc represent the dominance of the nonlinearly effect, material effect, radiation effect, heat and mass transfer effects which have presented in governing equations, respectively. The similar transformation, the finite-difference method and Runge–Kutta method have been used to analyze the present problem. The numerical solutions of the flow velocity distributions, temperature profiles, the wall unknown values of θ′(0) and ϕ′(0) for calculating the heat and mass transfer of the similar boundary-layer flow are carried out as functions of n, Ec, k 0, Pr, Sc. The value of n, k 0, Pr and Sc parameters are important factors in this study. It will produce greater heat transfer efficiency with a larger value of those parameters, but the viscous dissipation parameter Ec and material parameter K may reduce the heat transfer efficiency. On the other hand, for mass transfer, the value of Sc parameter is important factor in this study. It will produce greater heat transfer efficiency with a larger value of Sc.


Prandtl Number Boundary Layer Thickness Schmidt Number Thermal Boundary Layer Micropolar Fluid 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

List of symbols




Parameter related to the surface stretching speed


Concentration, kg/m3


Specific heat at a constant pressure, J/kg K


Mass diffusing, m2/s


Eckert number


Dimensionless stream function


Microrotation parameter


The micro-inertial per unit mass, N/kg

\( k_{0} = {\frac{{3N_{R} }}{{3N_{R} + 4}}} \)

Radiation parameter


Mean absorption coefficient


Fluid thermal conductivity, W m/K


Vortex viscosity or the material parameter


Reference length, m


Parameters related to the surface stretching speed


Microrotation component

\( N_{R} = {\frac{{k_{T} k^{*} }}{{4\sigma^{*} T_{\infty }^{3} }}} \)

Radiation parameter


Prandtl number


Radiative heat flux, J/m2

\( Sc = \upsilon / {\text{D}} \)

Schmidt number


Temperature across the thermal boundary layer, K


Temperature of the fluid far away from the plate, K


Temperature of the plate, K

u, v

Velocity components along x and y directions, respectively, m/s

x, y

Cartesian coordinates along the plate and normal to it, respectively, m


Thermal diffusivity, m2/s


Spin gradient viscosity


Dimensionless similarity variable


Dimensionless temperature


Dynamic viscosity, kg m/s


Kinematic viscosity, m2/s


Fluid density, kg/m3


Shear stress, N/m2


Stefan Boltzmann constant


Non-dimensional concentration variable



The author would like to thank the good comments which provided by the reviewers and would like to thank National Science Council R.O.C for the financial support through Grant. NSC 98-2221-E-434-009-.


  1. 1.
    Eringen AC (1966) Theory of micropolar fluids. J Math Mech 6:1–18MathSciNetGoogle Scholar
  2. 2.
    Eringen AC (1972) Theory of micropolar fluids. J Math Anal Appl 38:469–480CrossRefGoogle Scholar
  3. 3.
    Khonsari MM, Brewe D (1989) On the performance of finite journal bearings lubricated with micropolar fluids. STLE Tribol Trans 32:155–160CrossRefGoogle Scholar
  4. 4.
    Khonsari MM (1990) On the self-excited whirl orbits of a journal in a sleeve lubricated with micropolar fluids. Acta Mech 81:235–244zbMATHCrossRefGoogle Scholar
  5. 5.
    Hudimoto B, Tokuoka T (1969) Two-dimensional shear flows of linear micropolar fluids. Int J Eng Sci 7:515–522CrossRefGoogle Scholar
  6. 6.
    Lee JD, Eringen AC (1971) Boundary effects of orientation of noematic liquid crystals. J Chem Phys 55:4509–4512CrossRefGoogle Scholar
  7. 7.
    Lockwood F, Benchaita M, Friberg S (1987) Study of lyotropic liquid crystals in viscometric flow and elastohydrodynamic contact. ASLE Tribol Trans 30:539–548CrossRefGoogle Scholar
  8. 8.
    Cortell R (2008) Effects of viscous dissipation and radiation on the thermal boundary layer over a nonlinearly stretching sheet. Phys Lett A 372(5):631–636CrossRefGoogle Scholar
  9. 9.
    Awang Kechil S, Hashim I (2008) Series solution of flow over nonlinearly stretching sheet with chemical reaction and magnetic field. Phys Lett A 372(13):2258–2263CrossRefGoogle Scholar
  10. 10.
    Bataller RC (2008) Similarity solutions for flow and heat transfer of a quiescent fluid over a nonlinearly stretching surface. J Mater Process Technol 203(1–3):176–183CrossRefGoogle Scholar
  11. 11.
    Cortell R (2007) Viscous flow and heat transfer over a nonlinearly stretching sheet. Appl Math Comput 184(2):864–873zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Vajravelu K (2001) Viscous flow over a nonlinearly stretching sheet. Appl Math Comput 124(3):281–288zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Sanjayanand E, Khan SK (2006) On heat and mass transfer in a viscoelastic boundary layer flow over an exponentially stretching sheet. Int J Therm Sci 45(8):819–828CrossRefGoogle Scholar
  14. 14.
    Cortell R (2007) MHD flow and mass transfer of an electrically conducting fluid of second grade in a porous medium over a stretching sheet with chemically reactive species. Chem Eng Process 46(8):721–728CrossRefGoogle Scholar
  15. 15.
    Seddeek MA (2007) Heat and mass transfer on a stretching sheet with a magnetic field in a visco-elastic fluid flow through a porous medium with heat source or sink. Comput Mater Sci 38(4):781–787CrossRefGoogle Scholar
  16. 16.
    Liu C-M, Liu I-C (2006) A note on the transient solution of Stokes’ second problem with arbitrary initial phase. J Mech 22(4):349–354Google Scholar
  17. 17.
    Nazar R, Amin N, Filip D, Pop I (2004) Stagnation point flow of a micropolar fluid towards a stretching sheet. Int J Non Linear Mech 39:1227–1235zbMATHCrossRefGoogle Scholar
  18. 18.
    Brewster MQ (1972) Thermal radiative transfer properties. Wiley, New YorkGoogle Scholar
  19. 19.
    Hsiao K-L, Chen GB (2007) Conjugate heat transfer of mixed convection for viscoelastic fluid past a stretching sheet. Math Probl Eng Article ID 17058, 21 pages. doi: 10.1155/2007/17058
  20. 20.
    Hsiao K-L (2007) Conjugate heat transfer of magnetic mixed convection with radiative and viscous dissipation effects for second-grade viscoelastic fluid past a stretching sheet. Appl Therm Eng 27(11–12):1895–1903Google Scholar
  21. 21.
    Hsiao K-L (2008) Heat and mass transfer for electrical conducting mixed convection with radiation effect for viscoelastic fluid past a stretching sheet. J Mech 24(2):N21–N27Google Scholar
  22. 22.
    Hsiao K-L (2008) MHD mixed convection of viscoelastic fluid over a stretching sheet with ohmic dissipation. J Mech 24(3):N29–N34Google Scholar
  23. 23.
    Hsiao K-L, Hsu CH (2009) Conjugate heat transfer of mixed convection for viscoelastic fluid past a horizontal flat-plate fin. Appl Therm Eng 29(1):28–36CrossRefMathSciNetGoogle Scholar
  24. 24.
    Hsiao K-L, Hsu CH (2009) Conjugate heat transfer of mixed convection for visco-elastic fluid past a triangular fin. Nonlinear Anal Ser B Real World Appl 10(1):130–143zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Hsiao K-L (2010) Heat and mass mixed convection for MHD visco-elastic fluid past a stretching sheet with ohmic dissipation, Commun Nonlinear Sci Numer Simulat. 15(7):1803–1812Google Scholar
  26. 26.
    Vajravelu K (1994) Convection heat transfer at a stretching sheet with suction and blowing. J Math Anal Appl 188:1002–1011zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Chapra SC, Canale RP (1990) Numerical methods for engineers, 2nd edn. McGraw-Hill, New YorkGoogle Scholar

Copyright information

© Springer Verlag 2010

Authors and Affiliations

  1. 1.Department of the Electrical EngineeringDiwan UniversityT’ainanTaiwan, Republic of China

Personalised recommendations