# Effects of internal heat generation, thermal radiation and buoyancy force on a boundary layer over a vertical plate with a convective surface boundary condition

Keywords:
thermal radiation, buoyancy force, internal heat generation, vertical plate, Biot number, boundary layer

### Abstract

In this paper we analyse the effects of internal heat generation, thermal radiation and buoyancy force on the laminar boundary layer about a vertical plate in a uniform stream of fluid under a convective surface boundary condition. In the analysis, we assumed that the left surface of the plate is in contact with a hot fluid whilst a stream of cold fluid flows steadily over the right surface; the heat source decays exponentially outwards from the surface of the plate. The similarity variable method was applied to the steady state governing non-linear partial differential equations, which were transformed into a set of coupled non-linear ordinary differential equations and were solved numerically by applying a shooting iteration technique together with a sixth-order Runge–Kutta integration scheme for better accuracy. The effects of the Prandtl number, the local Biot number, the internal heat generation parameter, thermal radiation and the local Grashof number on the velocity and temperature profiles are illustrated and interpreted in physical terms. A comparison with previously published results on similar special cases showed excellent agreement.### References

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16. Hayat T, Abbas Z, Sajid M, Asghar S, The influence of thermal radiation on MHD flow of a second grade fluid. Int J Heat Mass Transfer. 2007;50:931–941. doi:10.1016/j.ijheatmasstransfer.2006.08.014

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19. Aziz A. Hydrodynamic and thermal slip flow boundary layers over a flat plate with constant heat flux boundary condition. Commun Nonlinear Sci Numer Simulat. 2010;15:573–580. doi:10.1016/j.cnsns.2009.04.026

20. Makinde OD, Olanrewaju PO. Buoyancy effects on thermal boundary layer over a vertical plate with a convective surface boundary condition. J Fluids Eng. 2010;132:044502–0044505. doi:10.1115/1.4001386

21. Chamkha AJ. Hydromagnetic natural convection from an isothermal inclined surface adjacent to a thermally stratified porous medium. Int J Eng Sci. 1997;37:975–986. doi:10.1016/S0020-7225(96)00122-X

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2. Crane IJ. Flow past a stretching plate. Z Angew Math Phys. 1970;21(56):1–37.

3. Gupta PS, Gupta AS. Heat and mass transfer on a stretching sheet with suction and blowing. Can J Chem Eng. 1977;55:744–746. doi:10.1002/cjce.5450550619

4. Carragher P, Crane IJ. Heat transfer on a continuous stretching sheet. Z Angew Math Mech. 1982;62:564–565. doi:10.1002/zamm.19820621009

5. Danberg JE, Fansler KS. A nonsimilar moving wall boundary-layer problem. Quart Appl Math. 1976;34:305–309.

6. Chakrabarti A, Gupta AS. Hydromagnetic flow and heat transfer over a stretching sheet. Quart Appl Math. 1979;37:73–78.

7. Vajravelu K. Hydromagnetic flow and heat transfer over a continuous moving porous, flat surface. Acta Mech. 1986;64:179–185. doi:10.1007/BF01450393

8. Dutta BK. Heat transfer from a stretching sheet with uniform suction and blowing. Acta Mech. 1986;78:255–262. doi:10.1007/BF01179221

9. Lee SL, Tsai JS. Cooling of a continuous moving sheet of finite thickness in the presence of natural convection. Int J Heat Mass Transfer. 1990;33:457–464. doi:10.1016/0017-9310(90)90181-S

10. Wanous KJ, Sparrow EM. Heat transfer for flow longitudinal to a cylinder with surface mass transfer. J Heat Transf ASME Ser C. 1965;87(1):317–319.

11. Catherall D, Stewartson K, Williams PG. Viscous flow past a flat plate with uniform injection. Proc R Soc A. 1965;284:370–396. doi:10.1098/rspa.1965.0069

12. Sparrow EM, Quack H, Boerner CJ. Local non-similarity boundary layer solutions. J AIAA 1970;8(11):1936–1942.

13. Sparrow EM, Yu HS. Local non-similarity thermal boundary-layer solutions. J Heat Transf ASME. 1971;93:328–334.

14. Massoudi M. Local non-similarity solutions for the flow of a non-Newtonian fluid over a wedge. Int J Non-Linear Mech. 2001;36:961–976. doi:10.1016/S0020-7462(00)00061-5

15. Raptis A, Perdikis C, Takhar HS. Effects of thermal radiation on MHD flow. Appl Math Comput. 2004;153:645–649. doi:10.1016/S0096-3003(03)00657-X

16. Hayat T, Abbas Z, Sajid M, Asghar S, The influence of thermal radiation on MHD flow of a second grade fluid. Int J Heat Mass Transfer. 2007;50:931–941. doi:10.1016/j.ijheatmasstransfer.2006.08.014

17. Ishak A. Similarity solutions for flow and heat transfer over a permeable surface with convective boundary condition. Appl Math Comput. 2010;217:837–842. doi:10.1016/j.amc.2010.06.026

18. Aziz A. A similarity solution for laminar thermal boundary layer over a flat plate with a convective surface boundary condition. Commun Nonlinear Sci Numer Simulat. 2009;14:1064–1068.

19. Aziz A. Hydrodynamic and thermal slip flow boundary layers over a flat plate with constant heat flux boundary condition. Commun Nonlinear Sci Numer Simulat. 2010;15:573–580. doi:10.1016/j.cnsns.2009.04.026

20. Makinde OD, Olanrewaju PO. Buoyancy effects on thermal boundary layer over a vertical plate with a convective surface boundary condition. J Fluids Eng. 2010;132:044502–0044505. doi:10.1115/1.4001386

21. Chamkha AJ. Hydromagnetic natural convection from an isothermal inclined surface adjacent to a thermally stratified porous medium. Int J Eng Sci. 1997;37:975–986. doi:10.1016/S0020-7225(96)00122-X

22. Heck A. Introduction to Maple. 3rd ed. New York: Springer-Verlag; 2003.

Published

2011-09-07

Section

Research Letters