, Volume 44, Issue 4, pp 369–375 | Cite as

Boundary layer flow and heat transfer over an unsteady stretching vertical surface

  • Anuar Ishak
  • Roslinda Nazar
  • Ioan Pop


The solution to the unsteady mixed convection boundary layer flow and heat transfer problem due to a stretching vertical surface is presented in this paper. The unsteadiness in the flow and temperature fields is caused by the time-dependent of the stretching velocity and the surface temperature. The governing partial differential equations with three independent variables are first transformed into ordinary differential equations, before they are solved numerically by a finite-difference scheme. The effects of the unsteadiness parameter, buoyancy parameter and Prandtl number on the flow and heat transfer characteristics are thoroughly examined. Both assisting and opposing buoyant flows are considered. It is observed that for assisting flow, the solutions exist for all values of buoyancy parameter, whereas for opposing flow, they exist only if the magnitude of the buoyancy parameter is small. Comparison with known results for steady-state flow is excellent.


Heat transfer Mixed convection Unsteady flow Stretching sheet Fluids mechanics 





skin friction coefficient


dimensionless stream function


acceleration due to gravity (m s−2)


local Grashof number


thermal conductivity (W m−1 K)


local Nusselt number


Prandtl number


local heat flux (W m−2)


local Reynolds number


fluid temperature (K)


surface temperature (K)


ambient temperature (K)


time (s)


velocity components in the x and y directions, respectively (m s−1)


velocity of the stretching surface (m s−1)


Cartesian coordinates along the surface and normal to it, respectively (m)

Greek letters


thermal diffusivity (m2 s−1)


thermal expansion coefficient (K−1)


similarity variable


dimensionless temperature


mixed convection or buoyancy parameter


kinematic viscosity (m2 s−1)


dynamic viscosity (kg m−1s−1)


fluid density (kg m−3)


wall shear stress (Pa)


stream function (m2 s−1)



condition at the surface

ambient condition


differentiation with respect to η


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  1. 1.
    Sakiadis BC (1961) Boundary-layer behavior on continuous solid surfaces: I. Boundary-layer equations for two-dimensional and axisymmetric flow. AIChE J 7:26–28 CrossRefGoogle Scholar
  2. 2.
    Sakiadis BC (1961) Boundary-layer behavior on continuous solid surfaces: II. The boundary-layer on a continuous flat surface. AIChE J 7:221–225 CrossRefGoogle Scholar
  3. 3.
    Sakiadis BC (1961) Boundary-layer behavior on continuous solid surfaces: II. The boundary-layer on a continuous cylindrical surface. AIChE J 7:467–472 CrossRefGoogle Scholar
  4. 4.
    Blasius H (1908) Grenzschichten in flüssigkeiten mit kleiner reibung. Z Math Phys 56:1–37 Google Scholar
  5. 5.
    Ishak A, Nazar R, Pop I (2007) Boundary layer on a moving wall with suction and injection. Chin Phys Lett 24:2274–2276 CrossRefGoogle Scholar
  6. 6.
    Crane LJ (1970) Flow past a stretching plate. Z Angew Math Phys 21:645–647 CrossRefGoogle Scholar
  7. 7.
    Hiemenz K (1911) Die grenzschicht an einem in den gleichförmigen flüssigkeitsstrom eingetauchten geraden kreiszylinder. Dingl Polytech J 326:321 Google Scholar
  8. 8.
    Crane LJ (1975) Boundary layer flow due to a stretching cylinder. Z Angew Math Phys 26:619–622 zbMATHCrossRefGoogle Scholar
  9. 9.
    Chen C-H (1998) Laminar mixed convection adjacent to vertical, continuously stretching sheets. Heat Mass Transf 33:471–476 CrossRefADSGoogle Scholar
  10. 10.
    Mahapatra TR, Gupta AS (2001) Magnetohydrodynamic stagnation-point flow towards a stretching sheet. Acta Mech 152:191–196 zbMATHCrossRefGoogle Scholar
  11. 11.
    Mahapatra TR, Gupta AS (2002) Heat transfer in stagnation-point flow towards a stretching sheet. Heat Mass Transf 38:517–521 CrossRefADSGoogle Scholar
  12. 12.
    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 39:1227–1235 zbMATHCrossRefGoogle Scholar
  13. 13.
    Ishak A, Nazar R, Pop I (2006) Mixed convection boundary layers in the stagnation-point flow toward a stretching vertical sheet. Meccanica 41:509–518 zbMATHCrossRefGoogle Scholar
  14. 14.
    Ishak A, Nazar R, Pop I (2008) Mixed convection stagnation point flow of a micropolar fluid towards a stretching sheet. Meccanica 43:411–418 zbMATHCrossRefGoogle Scholar
  15. 15.
    Ishak A, Nazar R, Pop I (2007) Mixed convection on the stagnation point flow towards a vertical, continuously stretching sheet. ASME J Heat Transf 129:1087–1090 CrossRefGoogle Scholar
  16. 16.
    Boutros YZ, Abd-el-Malek MB, Badran NA, Hassan HS (2006) Lie-group method of solution for steady two-dimensional boundary-layer stagnation-point flow towards a heated stretching sheet placed in a porous medium. Meccanica 41:681–691 zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Mahapatra TR, Dholey S, Gupta AS (2007) Momentum and heat transfer in the magnetohydrodynamic stagnation-point flow of a viscoelastic fluid toward a stretching surface. Meccanica 42:263–272 zbMATHCrossRefGoogle Scholar
  18. 18.
    El-Aziz MA (2007) Temperature dependent viscosity and thermal conductivity effects on combined heat and mass transfer in MHD three-dimensional flow over a stretching surface with Ohmic heating. Meccanica 42:375–386 zbMATHCrossRefGoogle Scholar
  19. 19.
    Devi CDS, Takhar HS, Nath G (1991) Unsteady mixed convection flow in stagnation region adjacent to a vertical surface. Heat Mass Transf 26:71–79 Google Scholar
  20. 20.
    Andersson HI, Aarseth JB, Dandapat BS (2000) Heat transfer in a liquid film on an unsteady stretching surface. Int J Heat Mass Transf 43:69–74 zbMATHCrossRefGoogle Scholar
  21. 21.
    Grubka LJ, Bobba KM (1985) Heat transfer characteristics of a continuous, stretching surface with variable temperature. ASME J Heat Transf 107:248–250 CrossRefGoogle Scholar
  22. 22.
    Ali ME (1994) Heat transfer characteristics of a continuous stretching surface. Heat Mass Transf 29:227–234 Google Scholar
  23. 23.
    Ali ME, Magyari E (2007) Unsteady fluid and heat flow induced by a submerged stretching surface while its steady motion is slowed down gradually. Int J Heat Mass Transf 50:188–195 zbMATHCrossRefGoogle Scholar
  24. 24.
    Abramowitz M, Stegun IA (1965) Handbook of mathematical functions. Dover, New York Google Scholar
  25. 25.
    Cebeci T, Bradshaw P (1988) Physical and computational aspects of convective heat transfer. Springer, New York zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.School of Mathematical SciencesUniversity Kebangsaan MalaysiaUKM BangiMalaysia
  2. 2.Faculty of MathematicsUniversity of ClujClujRomania

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