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Heat Transfer in a Micropolar Fluid along a Non-linear Stretching Sheet with a Temperature-Dependent Viscosity and Variable Surface Temperature

  • Mohammad M. Rahman
  • M. A. Rahman
  • M. A. Samad
  • M. S. Alam
Article

Abstract

In this paper, heat transfer characteristics of a two-dimensional steady hydromagnetic natural convection flow of a micropolar fluid passed a non-linear stretching sheet taking into account the effects of a temperature-dependent viscosity and variable wall temperature are studied numerically for local similarity solutions by applying the Nachtsheim-Swigert iteration method. The results corresponding to the dimensionless temperature profiles and the local rate of heat transfer are displayed graphically for important material parameters. The results show that in modeling the thermal boundary layer flow with a temperature-dependent viscosity, consideration of the Prandtl number as a constant within the boundary layer produces unrealistic results and therefore it must be treated as a variable rather than a constant within the boundary layer. The results also show that the local rate of heat transfer strongly depends on the non-linear stretching index and temperature index.

Keywords

Heat generation or absorption Micropolar fluid Non-linear stretching sheet Variable viscosity Variable wall temperature 

List of Symbols

Variables

A

Constant

B

Constant

B0

Magnetic induction

c

Stretching coefficient

cp

Specific heat due to constant pressure

Ec

Eckert number

Fw

Dimensionless suction/injection velocity

f

Dimensionless stream function

Grx

Local Grashof number

g0

Acceleration due to gravity

g

Dimensionless microrotation

j

Micro-inertia per unit mass

M

Local magnetic field parameter

m

Stretching index

N

Microrotation component normal to the xy-plane

Nux

Local Nusselt number

Prv

Variable Prandtl number

Pr

Ambient Prandtl number

p

Temperature index

Q

Local heat source (or sink) parameter

Q0

Heat generation/absorption coefficient

Rex

Local Reynolds number

S

Coefficient of vortex viscosity

s

Microrotation parameter

T

Temperature of the fluid within the boundary layer

Tr

Reference temperature

Tw

Temperature at the surface of the plate

T

Temperature of the ambient fluid

u

The x-component of the velocity field

v

The y-component of the velocity field

v0 (x)

Suction/injection velocity

x

Axis in direction along the surface

y

Axis in direction normal to the surface

Greek

β

Volumetric coefficient of thermal expansion

γ

Richardson parameter

γ*

Constant

ρ

Fluid density

ρ

Density of the ambient fluid

μ

Coefficient of dynamic viscosity

μ

Coefficient of dynamic viscosity of the ambient fluid

υ

Coefficient of kinematic viscosity

υ

Coefficient of kinematic viscosity of the ambient fluid

υs

Spin-gradient viscosity

σ

Electrical conductivity of the fluid

ψ

Stream function

ξ

Micro-inertia density parameter

η

Similarity variable

θ

Dimensionless temperature

θr

Variable viscosity parameter

k

Thermal conductivity of fluid

Δ

Vortex viscosity parameter

Subscripts

w

Surface conditions

Conditions far away from the surface

Superscript

Differentiation with respect to η

References

  1. 1.
    Sakiadis B.C. (1961) AIChE J. 7: 26CrossRefGoogle Scholar
  2. 2.
    Erickson L.E., Fan L.T., Fox V.G. (1966) Ind. Eng. Chem. Fund. 5: 19CrossRefGoogle Scholar
  3. 3.
    Tsou F.K., Sparrow E.M., Goldstien R.J. (1967) Int. J. Heat Mass Transfer 10: 219CrossRefGoogle Scholar
  4. 4.
    Gupta P.S., Gupta A.S. (1977) Can. J. Chem. Eng. 55: 744CrossRefGoogle Scholar
  5. 5.
    Chakrabarti A., Gupta A.S. (1979) Quart. Appl. Math. 3: 73Google Scholar
  6. 6.
    Chiam T.C. (1995) Int. J. Eng. Sci. 33: 429CrossRefzbMATHGoogle Scholar
  7. 7.
    Chandran P., Sacheti N.C., Singh A.K. (1996) Int. Commun. Heat Mass Transfer 23: 889CrossRefGoogle Scholar
  8. 8.
    Vajravelu K., Hadjinicalaou A. (1997) Int. J. Eng. Sci. 35: 1237CrossRefzbMATHGoogle Scholar
  9. 9.
    Chen C.K., Char N.I. (1988) J. Math. Anal. Appl. 135: 568CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    Ali M.E. (1995) Int. J. Heat Fluid Flow 16: 280CrossRefGoogle Scholar
  11. 11.
    Chen C.-H. (2008) Int. J. Therm. Sci. 47: 954CrossRefGoogle Scholar
  12. 12.
    Cortell R. (2007) Appl. Math. Comput. 184: 864CrossRefMathSciNetzbMATHGoogle Scholar
  13. 13.
    Cortell R. (2008) Phys. Lett. A 372: 631CrossRefADSGoogle Scholar
  14. 14.
    Cortell R. (2008) J. Mater. Process. Tech. 203: 176CrossRefGoogle Scholar
  15. 15.
    Eringen A.C. (1966) J. Math. Mech. 16: 1MathSciNetGoogle Scholar
  16. 16.
    Eringen A.C. (1972) J. Math. Anal. Appl. 38: 480CrossRefzbMATHGoogle Scholar
  17. 17.
    Ahmadi G. (1976) Int. J. Eng. Sci. 14: 639CrossRefzbMATHGoogle Scholar
  18. 18.
    Jena S.K., Mathur M.N. (1981) Int. J. Eng. Sci. 19: 1431CrossRefzbMATHGoogle Scholar
  19. 19.
    Gorla R.S.R., Takhar H.S. (1987) Int. J. Eng. Sci. 25: 949CrossRefzbMATHGoogle Scholar
  20. 20.
    Gorla R.S.R. (1988) Int. J. Eng. Sci. 26: 385CrossRefzbMATHGoogle Scholar
  21. 21.
    Yucel A. (1989) Int. J. Eng. Sci. 27: 1593CrossRefGoogle Scholar
  22. 22.
    Gorla R.S.R., Lin P.P., Yang A.J. (1990) Int. J. Eng. Sci. 28: 525CrossRefzbMATHGoogle Scholar
  23. 23.
    Gorla R.S.R. (1992) Int. J. Eng. Sci. 30: 349CrossRefGoogle Scholar
  24. 24.
    Hossain M.A., Chaudhury M.K. (1998) Acta Mech. 131: 139CrossRefzbMATHGoogle Scholar
  25. 25.
    Rees D.A.S., Pop I. (1998) IMAJ Appl. Math. 61: 179CrossRefMathSciNetzbMATHGoogle Scholar
  26. 26.
    Rahman M.M., Sattar M.A. (2006) ASME J. Heat Transfer 128: 142CrossRefGoogle Scholar
  27. 27.
    Rahman M.M., Sattar M.A. (2007) Int. J. Appl. Mech. Eng. 12: 497MathSciNetGoogle Scholar
  28. 28.
    Rahman M.M., Sultana T. (2008) Nonlinear Anal. Model. Control 13: 71zbMATHGoogle Scholar
  29. 29.
    Rahman M.M. (2009) Commun. Nonlinear Sci. Numer. Sim. 14: 3018CrossRefGoogle Scholar
  30. 30.
    Rahman M.M., Eltayeb I.A., Rahman S.M.M. (2009) Therm. Sci. 13: 23CrossRefGoogle Scholar
  31. 31.
    Pop I., Gorla R.S.R., Rashidi M. (1992) Int. J. Eng. Sci. 30: 1CrossRefGoogle Scholar
  32. 32.
    Elbashbeshy E.M.A., Bazid M.A.A. (2000) J. Phys. D. Appl. Phys. 33: 2716CrossRefADSGoogle Scholar
  33. 33.
    Abel M.S., Khan S.K., Prasad K.V. (2002) Int. J. Non-linear Mech. 37: 81CrossRefzbMATHGoogle Scholar
  34. 34.
    Ali M.E. (2006) Int. J. Therm. Sci. 45: 60CrossRefGoogle Scholar
  35. 35.
    Pantokratoras A. (2002) Int. J. Heat Mass Transfer 45: 963CrossRefzbMATHGoogle Scholar
  36. 36.
    Pantokratoras A. (2004) Int. J. Eng. Sci. 42: 1891CrossRefGoogle Scholar
  37. 37.
    Mukhopadhyay S., Layek G.C., Samad Sk.A. (2005) Int. J. Heat Mass Transfer 48: 4460CrossRefGoogle Scholar
  38. 38.
    Alam M.S., Rahman M.M., Samad M.A. (2009) Nonlinear Anal. Model. Control 14: 3zbMATHGoogle Scholar
  39. 39.
    M.M. Rahman, K.M. Salahuddin, Commun. Nonlinear Sci. Numer. Sim. (2009) doi: 10.1016/j.cnsns.2009.08.012
  40. 40.
    Mohammadein A.A., Gorla R.S.R. (2001) Int. J. Numer. Meth. Heat Fluid Flow 11: 50CrossRefzbMATHGoogle Scholar
  41. 41.
    Desseaux A., Kelson N.A. (2000) ANZIAM J. 42(E): C536MathSciNetGoogle Scholar
  42. 42.
    Bhargava R., Sharma S., Takhar H.S., Beg O.A., Bhargava P. (2007) Nonlinear Anal. Model. Control 12: 45zbMATHGoogle Scholar
  43. 43.
    Hayat T., Abbas Z., Javed T. (2008) Phys. Lett. A 372: 637CrossRefADSGoogle Scholar
  44. 44.
    J.X.Ling, A. Dybbs, ASME Paper 87-WA/HT-23 (New York, 1987)Google Scholar
  45. 45.
    R.C. Weast, CRC Handbook of Chemistry and Physics, 71st edn. (CRC Press, Boca Raton, Florida, 1990)Google Scholar
  46. 46.
    Kays W.M., Crawford M.E. (1987) Convective Heat and Mass Transfer. McGraw-Hill, New YorkGoogle Scholar
  47. 47.
    Raptis A. (1998) Int. J. Heat Mass Transfer 41: 2865CrossRefzbMATHGoogle Scholar
  48. 48.
    El-Arabawy H.A.M. (2003) Int. J. Heat Mass Transfer 46: 1471CrossRefzbMATHGoogle Scholar
  49. 49.
    Aziz A. (2009) Commun. Nonlinear. Sci. Numer. Sim. 14: 1064CrossRefGoogle Scholar
  50. 50.
    P.R. Nachtsheim, P. Swigert, Satisfaction of the asymptotic boundary conditions in numerical solution of the system of non-linear equations of boundary layer type (NASA TND-3004, 1965)Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Mohammad M. Rahman
    • 1
  • M. A. Rahman
    • 2
  • M. A. Samad
    • 3
  • M. S. Alam
    • 4
  1. 1.Department of Mathematics and Statistics, College of ScienceSultan Qaboos UniversityMuscatSultanate of Oman
  2. 2.Department of MathematicsNational UniversityGazipurBangladesh
  3. 3.Department of MathematicsUniversity of DhakaDhakaBangladesh
  4. 4.Department of MathematicsDhaka University of Engineering and TechnologyGazipurBangladesh

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