Mixed Convection Boundary Layer Flows Induced by a Permeable Continuous Surface Stretched with Prescribed Skin Friction

The boundary layer flow and heat transfer on a stretched surface moving with prescribed skin friction is studied for permeable surface. The surface temperature is assumed to vary inversely with the vertical direction x for n = -1. The skin friction at the surface scales as (x-1/2) at m = 0. The constants m and n are the indices of the power law velocity and temperature exponent respectively. Similarity solutions are obtained for the boundary layer equations subject to power law temperature and velocity variation. The effect of various governing parameters, such as the buoyancy parameter λ and the suction/injection parameter fw for air (Pr = 0.72) are studied. The choice of n and m ensures that the used similarity solutions are x independent. The results show that, assisting flow (λ > 0) enhancing the heat transfer coefficient along the surface for any constant value of fw. Furthermore, injection increases the heat transfer coefficient but suction reduces it at constant λ.

Authors:

• Mohamed Ali

Keywords:

• Stretching surface
• Boundary layers
• Prescribed skin friction
• Suction or injection
• similarity solutions
• buoyancy effects.

References:

[1] Altan, T., Oh, S. and Gegel, H., Metal Forming Fundamentals and
Applications, American Society of Metals, Metals Park, OH., 1979.
[2] Fisher, E.G., Extrusion of plastics, Wiley, New York, 1976.
[3] Tadmor, Z. and Klein, I., Engineering Principles of Plasticating
Extrusion, Polymer Science and Engineering Series, Van Nostrand
Reinhold, New York, 1970.
[4] Sakiadis, B. C., " Boundary Layer Behavior on Continuous Solid
Surfaces: I. Boundary-Layer Equations for Two-Dimensional and
Axisymmetric Flow", A.I.Ch.E. Journal, Vol. 7, No. 1, 26-28, 1961.
[5] Tsou, F. K., Sparrow, E. M. and Goldstein, R. J., " Flow and Heat
Transfer in the Boundary Layer on a Continuous Moving Surface", Int.
J. Heat Mass Transfer, Vol. 10, 219-235, 1967.
[6] Crane, L. J., " Flow Past a Stretching Plane", Z. Amgew. Math. Phys.,
Vol. 21, 645-647, 1970.
[7] Grubka, L. G. and Bobba, K. M.," Heat Transfer Characteristics of a
Continuous Stretching Surface with Variable Temperature", ASME J.
Heat Transfer, Vol. 107, 248-250, 1985.
[8] Soundalgekar, V. M. and Ramana Murty, T. V., " Heat Transfer Past a
Continuous Moving Plate with Variable Temperature", Wärme- und
Stoff├╝bertragung, Vol. 14, 91-93, 1980.
[9] Vleggaar, J., " Laminar Boundary Layer Behavior on Continuous
Accelerating Surfaces", Chemical Engineering Science, Vol. 32, 1517-
1525, 1977.
[10] Ali, M. E.," Heat Transfer Characteristics of a Continuous Stretching
Surface", Wärme- und Stoffübertragung, Vol. 29, 227-234, 1994.
[11] Banks, W. H. H., "Similarity solutions of the boundary-layer equations
for a stretching wall", J. Mecanique Theorique et Appliquee, Vol. 2,
375-392, 1983.
[12] Ali, M. E., "The effect of suction or injection on the laminar boundary
layer development over a stretched surface", J. King Saud University,
Vol. 8, Eng. Sci. No. 1, 43-58, 1996.
[13] Magyari, E. and Keller, B., "Heat and mass transfer in the boundary
layers on an exponentially stretching continuous surface", J. Phys. D:
Appl. Phys. Vol. 32, 577-585, 1999.
[14] Magyari, E. and Keller, B., "Heat transfer characteristics of the
separation boundary flow induced by a continuous stretching surface", J.
Phys. D: Appl. Phys. Vol. 32, 2876-2881, 1999.
[15] Magyari, E. and Keller, B., "Heat transfer characteristics of boundarylayer
flows induced by continuous surfaces stretched with prescribed
skin friction", Heat and Mass Transfer. Vol. 42, 679-687, 2006.
[16] Erickson, L. E. , Fan, L. T. and Fox, V. G., " Heat and Mass Transfer on
a Moving Continuous Flat Plate with Suction or Injection.", Ind. Eng.
Chem. Fundamentals, Vol. 5, 19-25, 1966.
[17] Fox, V. G., Erickson, L. E. and Fan, L. T., " Methods for Solving the
Boundary Layer Equations for Moving Continuous Flat Surfaces with
Suction and Injection", A.I.Ch.E. Journal, Vol. 14, 726-736, 1968.
[18] Gupta, P. S. and Gupta, A. S., " Heat and Mass Transfer on a Stretching
Sheet with Suction or Blowing", Canadian Journal of Chemical
Engineering, Vol. 55, No. 6, 744-746, 1977.
[19] Chen, C. K. and Char, M. I., " Heat Transfer of a Continuous Stretching
Surface with Suction or Blowing", J. Math. Analysis and Applications,
Vol. 135, 568-580, 1988.
[20] Ali, M. E.," On Thermal Boundary Layer on a Power-Law Stretched
Surface with Suction or Injection", Int. J. Heat Fluid Flow, Vol. 16, No.
4, 280-290, 1995.
[21] Magyari, E., Ali, M. E., and Keller, B., "Heat and mass transfer
characteristics of the self-similar boundary-layer flows induced by
continuous surface stretched with rapidly decreasing velocities", Heat
and Mass Transfer, vol. 38, 65-74, 2001.
[22] Magyari, E. and Keller, B., "Exact solutions for self-similar boundarylayer
flows induced by permeable stretching walls", Eur. J. Mech. BFluids,
Vol. 19, 109-122, 2000.
[23] Ingham, D. B., "Singular and non-unique solutions of the boundarylayer
equations for the flow due to free convection near a continuously
moving vertical plate", Applied mathematics and physics (ZAMP), Vol.
37, 559-572, 1986.
[24] Ali, M. E. and Al-Yousef, F., " Laminar Mixed Convection from a
Continuously Moving Vertical Surface with Suction or Injection", Heat
and Mass Transfer, Vol. 33, No. 4, 301-306, 1998.
[25] Ali, M. and Al-Yousef, F., "Laminar mixed Convection boundary layers
induced by a linearly stretching permeable surface", Int. J. Heat and
Mass Transfer, Vol. 45, 4241- 4250, 2002
[26] Ali, M. E., "The buoyancy effects on the boundary layers induced by
continuous surfaces stretched with rapidly decreasing velocities", Heat
and Mass Transfer, vol. 40, pp. 285-291, 2004.
[27] Ali, M and Al-salem, K., "The effect of suction or injection on the
boundary layer flows induced by continuous surfaces stretched with
prescribed skin friction", Meccanica, published on line, DOI
10.1007/s11012-9687-2, 2013.