Mixed Convection Boundary Layer Flow over a Vertical Flat Plate with Radiative Heat Transfer

Affiliation(s)

Department of Mathematics, COMSATS Institute of Information Technology, Park Road, Chak Shahzad, Islamabad, Pakistan.

Former Professor of Mathematics, University of Dhaka, Dhaka, Bangladesh.

Department of Mathematics, COMSATS Institute of Information Technology, Park Road, Chak Shahzad, Islamabad, Pakistan.

Former Professor of Mathematics, University of Dhaka, Dhaka, Bangladesh.

ABSTRACT

In this article, the problem of mixed convection boundary layer flow of viscous fluid along a heated vertical plate is examined. In the analysis radiative component of heat flux emulates the surface temperature. Appropriate set of variables are embraced here which reduces the governing boundary layer equations into dimensionless form. Subsequently, a group of continuous transformation is applied on the dimensionless equations in order to obtain the parabolic partial differential equations for the regimes where modified Richardson number,*Ri**, is 1) small *i.e.* when *Ri** 1, 2) large *i.e.* when *Ri** 1, and 3) covers all its values i.e. when 0 ≤ *Ri** ≤ ∞. The system of equation for the corresponding regimes are thus integrated numerically via straightforward finite difference method along with Gaussian elimination technique. Its worth mentioning that results obtained here are valid particularly for the liquid metals for which Pr 1. Moreover, the numerical results are demonstrated graphically by showing the effects of important physical parameters, namely, the modified Richardson number (or mixed convection parameter), *Ri**, surface radiation parameter, R, and Prandtl number, Pr, in terms of local skin friction and local Nusselt number coefficients. In addition, comprehensive interpretation of thermal energy distributions is also given in terms of heatlines which is termed as good tool to visualize the flow patterns.

In this article, the problem of mixed convection boundary layer flow of viscous fluid along a heated vertical plate is examined. In the analysis radiative component of heat flux emulates the surface temperature. Appropriate set of variables are embraced here which reduces the governing boundary layer equations into dimensionless form. Subsequently, a group of continuous transformation is applied on the dimensionless equations in order to obtain the parabolic partial differential equations for the regimes where modified Richardson number,

Cite this paper

S. Siddiqa and M. Hossain, "Mixed Convection Boundary Layer Flow over a Vertical Flat Plate with Radiative Heat Transfer,"*Applied Mathematics*, Vol. 3 No. 7, 2012, pp. 705-716. doi: 10.4236/am.2012.37104.

S. Siddiqa and M. Hossain, "Mixed Convection Boundary Layer Flow over a Vertical Flat Plate with Radiative Heat Transfer,"

References

[1] E. M. Sparrow, R. Eichorn and J. L. Gregg, “Combined Forced and Free Convection in Boundary Layer Flow,” Physics of Fluids, Vol. 2, No. 3, 1959 , pp. 319-328. doi:10.1063/1.1705928

[2] J. H. Merkin, “The Effects of Buoyancy Forces on the Boundary Layer Flow over Semi-Infinite Vertical Flat Plate in a Uniform Free Stream,” Journal of Fluid Mechanics, Vol. 35, No. 3, 1969, pp. 4398-4450. doi:10.1017/S0022112069001212

[3] J. R. Lloyd and E. M. Sparrow, “Combined Forced and Free Convection Flow on Vertical Surfaces,” International Journal of Heat and Mass Transfer, Vol. 13, No. 2, 1970, pp. 434-438. doi:10.1016/0017-9310(70)90119-5

[4] G. Wilks, “Combined Forced and Free Convective Flow on Vertical Surfaces,” International Journal of Heat and Mass Transfer, Vol. 16, No. 10, 1973, pp. 1958-1964. doi:10.1016/0017-9310(73)90216-0

[5] G. Tingwei, R. Bachrum and M. Dagguent, “Influence de la Convective Natural le Sur la Convection Force Andessus D’Une Surface Plane Vertical Voumise a un Flux de Rayonnement,” International Journal of Heat and Mass Transfer, Vol. 25, No. 7, 1982, pp. 1061-1065. doi:10.1016/0017-9310(82)90081-3

[6] M. S. Raju, X. R. Liu and C. K. Law, “A Formulation of Combined Forced and Free Convection past Horizontal and Vertical Surfaces,” International Journal of Heat and Mass Transfer, Vol. 27, No. 12, 1984, pp. 2215-2224. doi:10.1016/0017-9310(84)90080-2

[7] B. Gebhart, Y. Jaluria, R. L. Mahajan and B. Sammakia, “Buoyancy-Induced Flows Transport,” Hemisphere, New York, 1988.

[8] M. N. Ozisik, “Thermal Radiation Transfer and Interactions with Conduction and Convection,” John Wiley & Sons, New York, 1973.

[9] E. M. Sparrow and R. D. Cess, “Radiation Heat Transfer, Augmented Edition, Hemishpere Media,” International Journal of Heat and Mass Transfer, Vol. 5, 1962, pp. 179-806.

[10] R. D. Cess, “The Interaction of Thermal Radiation in Boundary Layer Heat Transfer,” Proceedings of the 3rd International Heat Transfer Conference, Vol. 5, 1966, pp. 154-163.

[11] V. S. Arpaci, “Effect of Thermal Radiation on the Laminar Free Convection from a Heated Vertical Plate,” International Journal of Heat and Mass Transfer, Vol. 11, No. 5, 1968, pp. 871-881. doi:10.1016/0017-9310(68)90130-0

[12] M. A. Hossain, M. Kutubuddin and I. Pop, “Effect of Conduction-Radiation Interaction on the Mixed Convection Flow from a Horizaontal Cylinder,” Heat Mass Transfer, Vol. 35, 1999, pp. 307-314. doi:10.1007/s002310050329

[13] M. A. Hossain, M. Anghel and I. Pop, “Thermal Radiation Effects on Free Convection over a Rotating Axisymmetric Body with Application to a Rotating Hemisphere,” Archives of Mechanics, Vol. 54, 2002, pp. 55-74.

[14] M. A. Hossain and M. S. Munir, “Natural Convection Flow of a Viscous Fluid about a Truncated Cone with Temperature Dependent Viscosity and Thermal Conductivity,” International Journal of Numerical Methods Heat & Fluid Flow, Vol. 11, No. 6, 2001, pp. 494-510. doi:10.1108/09615530110399459

[15] M. A. Hossain and D. A. S. Rees, “Radiation Conduction Interaction on Mixed Convection Flow along a Slender Vertical Cylinder,” AIAA Journal of Thermophysics and Heat Transfer, Vol. 12, 1998, pp. 611-614.

[16] M. M. Molla and M. A. Hossain, “Radiation Effect on Mixed Convection Laminar Flow along a Vertical Wavy Surface,” International Journal of Thermal Sciences, Vol. 46, No. 9, 2007, pp. 926-935. doi:10.1016/j.ijthermalsci.2006.10.010

[17] S. Siddiqa, S. Asghar and M. A. Hossain, “Radiation Effects on Natural Convection Flow over an Inclined Flat Plate with Temperature-Dependent Viscosity,” Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, Vol. 225, No. 2, 2011, pp. 407-419. doi:10.1243/09544062JMES2205

[18] S. Siddiqa, S. Asghar and M. A. Hossain, “Radiation Effect on Mixed Convection Flow of Viscous Fluid Having Temperature Dependent Density along Permeable Vertical Plate,” Journal of Engineering Physics and Thermophysics, Vol. 85, No. 2, 2012, pp. 339-348. doi:10.1007/s10891-012-0658-1

[19] O. G. Martynenko, Yu. A. Sokovishin and M. V. Shapiro, “Free-Convection and Radiative Heat Transfer on a Vertical Plane Surface,” Inzhenerno-Fizicheskii Zhurnal, Vol. 32, 1977, pp. 1071-1079.

[20] V. V. Salomatov and E. M. Puzyrev, “Influence of Thermal Radiation on the Laminar Boundary Layer of a Nonabsorbing Fluid,” Inzhenerno-Fizicheskii Zhurnal, Vol. 20, 1971, pp. 1008-1014.

[21] Yu. Ao Sokovishin and M. V. Shapiro, “Heat Transfer of a Vertical Cylinder by Free Convection and Radiation,” Inzhenerno-Fizicheskii Zhurnal, Vol. 33, 1977, pp. 694-699.

[22] S. Siddiqa, S. Asghar and M. A. Hossain, “Natural Convection Flow over an Inclined Flat Plate with Internal Heat Generation and Variable Viscosity,” Mathematical and Computer Modelling, Vol. 52, No. 9-10, 2010, pp. 1739-1751. doi:10.1016/j.mcm.2010.07.001

[23] M. M. Molla, M. A. Hossain and R. S. R. Gorla, “Radiation Effect on Natural Convection Boundary Layer Flow over a Vertical Wavy Frustum of a Cone,” Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, Vol. 223, No. 7, 2009, pp. 1605-1614. doi:10.1243/09544062JMES1323

[24] Al. M. Morega and A. Bejan, “Heatline Visualization of Forced Convection Laminar Boundary Layers,” International Journal of Heat and Mass Transfer, Vol. 36, No. 16, 1993, pp. 3957-3966. doi:10.1016/0017-9310(93)90146-W

[25] S. Kimura and A. Bejan, “The “Heatline” Visualization of Convective Heat Transfer,” Journal of Heat Transfer, Vol. 105, No. 4, 1983, pp. 916-919. doi:10.1115/1.3245684

[26] E. M. Sparrow and R. D. Cess, “The Effect of Magnetic Field on Free Convection Heat Transfer,” International Journal of Heat and Mass Transfer, Vol. 3, No. 4, 1961, pp. 267-274. doi:10.1016/0017-9310(61)90042-4

[1] E. M. Sparrow, R. Eichorn and J. L. Gregg, “Combined Forced and Free Convection in Boundary Layer Flow,” Physics of Fluids, Vol. 2, No. 3, 1959 , pp. 319-328. doi:10.1063/1.1705928

[2] J. H. Merkin, “The Effects of Buoyancy Forces on the Boundary Layer Flow over Semi-Infinite Vertical Flat Plate in a Uniform Free Stream,” Journal of Fluid Mechanics, Vol. 35, No. 3, 1969, pp. 4398-4450. doi:10.1017/S0022112069001212

[3] J. R. Lloyd and E. M. Sparrow, “Combined Forced and Free Convection Flow on Vertical Surfaces,” International Journal of Heat and Mass Transfer, Vol. 13, No. 2, 1970, pp. 434-438. doi:10.1016/0017-9310(70)90119-5

[4] G. Wilks, “Combined Forced and Free Convective Flow on Vertical Surfaces,” International Journal of Heat and Mass Transfer, Vol. 16, No. 10, 1973, pp. 1958-1964. doi:10.1016/0017-9310(73)90216-0

[5] G. Tingwei, R. Bachrum and M. Dagguent, “Influence de la Convective Natural le Sur la Convection Force Andessus D’Une Surface Plane Vertical Voumise a un Flux de Rayonnement,” International Journal of Heat and Mass Transfer, Vol. 25, No. 7, 1982, pp. 1061-1065. doi:10.1016/0017-9310(82)90081-3

[6] M. S. Raju, X. R. Liu and C. K. Law, “A Formulation of Combined Forced and Free Convection past Horizontal and Vertical Surfaces,” International Journal of Heat and Mass Transfer, Vol. 27, No. 12, 1984, pp. 2215-2224. doi:10.1016/0017-9310(84)90080-2

[7] B. Gebhart, Y. Jaluria, R. L. Mahajan and B. Sammakia, “Buoyancy-Induced Flows Transport,” Hemisphere, New York, 1988.

[8] M. N. Ozisik, “Thermal Radiation Transfer and Interactions with Conduction and Convection,” John Wiley & Sons, New York, 1973.

[9] E. M. Sparrow and R. D. Cess, “Radiation Heat Transfer, Augmented Edition, Hemishpere Media,” International Journal of Heat and Mass Transfer, Vol. 5, 1962, pp. 179-806.

[10] R. D. Cess, “The Interaction of Thermal Radiation in Boundary Layer Heat Transfer,” Proceedings of the 3rd International Heat Transfer Conference, Vol. 5, 1966, pp. 154-163.

[11] V. S. Arpaci, “Effect of Thermal Radiation on the Laminar Free Convection from a Heated Vertical Plate,” International Journal of Heat and Mass Transfer, Vol. 11, No. 5, 1968, pp. 871-881. doi:10.1016/0017-9310(68)90130-0

[12] M. A. Hossain, M. Kutubuddin and I. Pop, “Effect of Conduction-Radiation Interaction on the Mixed Convection Flow from a Horizaontal Cylinder,” Heat Mass Transfer, Vol. 35, 1999, pp. 307-314. doi:10.1007/s002310050329

[13] M. A. Hossain, M. Anghel and I. Pop, “Thermal Radiation Effects on Free Convection over a Rotating Axisymmetric Body with Application to a Rotating Hemisphere,” Archives of Mechanics, Vol. 54, 2002, pp. 55-74.

[14] M. A. Hossain and M. S. Munir, “Natural Convection Flow of a Viscous Fluid about a Truncated Cone with Temperature Dependent Viscosity and Thermal Conductivity,” International Journal of Numerical Methods Heat & Fluid Flow, Vol. 11, No. 6, 2001, pp. 494-510. doi:10.1108/09615530110399459

[15] M. A. Hossain and D. A. S. Rees, “Radiation Conduction Interaction on Mixed Convection Flow along a Slender Vertical Cylinder,” AIAA Journal of Thermophysics and Heat Transfer, Vol. 12, 1998, pp. 611-614.

[16] M. M. Molla and M. A. Hossain, “Radiation Effect on Mixed Convection Laminar Flow along a Vertical Wavy Surface,” International Journal of Thermal Sciences, Vol. 46, No. 9, 2007, pp. 926-935. doi:10.1016/j.ijthermalsci.2006.10.010

[17] S. Siddiqa, S. Asghar and M. A. Hossain, “Radiation Effects on Natural Convection Flow over an Inclined Flat Plate with Temperature-Dependent Viscosity,” Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, Vol. 225, No. 2, 2011, pp. 407-419. doi:10.1243/09544062JMES2205

[18] S. Siddiqa, S. Asghar and M. A. Hossain, “Radiation Effect on Mixed Convection Flow of Viscous Fluid Having Temperature Dependent Density along Permeable Vertical Plate,” Journal of Engineering Physics and Thermophysics, Vol. 85, No. 2, 2012, pp. 339-348. doi:10.1007/s10891-012-0658-1

[19] O. G. Martynenko, Yu. A. Sokovishin and M. V. Shapiro, “Free-Convection and Radiative Heat Transfer on a Vertical Plane Surface,” Inzhenerno-Fizicheskii Zhurnal, Vol. 32, 1977, pp. 1071-1079.

[20] V. V. Salomatov and E. M. Puzyrev, “Influence of Thermal Radiation on the Laminar Boundary Layer of a Nonabsorbing Fluid,” Inzhenerno-Fizicheskii Zhurnal, Vol. 20, 1971, pp. 1008-1014.

[21] Yu. Ao Sokovishin and M. V. Shapiro, “Heat Transfer of a Vertical Cylinder by Free Convection and Radiation,” Inzhenerno-Fizicheskii Zhurnal, Vol. 33, 1977, pp. 694-699.

[22] S. Siddiqa, S. Asghar and M. A. Hossain, “Natural Convection Flow over an Inclined Flat Plate with Internal Heat Generation and Variable Viscosity,” Mathematical and Computer Modelling, Vol. 52, No. 9-10, 2010, pp. 1739-1751. doi:10.1016/j.mcm.2010.07.001

[23] M. M. Molla, M. A. Hossain and R. S. R. Gorla, “Radiation Effect on Natural Convection Boundary Layer Flow over a Vertical Wavy Frustum of a Cone,” Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, Vol. 223, No. 7, 2009, pp. 1605-1614. doi:10.1243/09544062JMES1323

[24] Al. M. Morega and A. Bejan, “Heatline Visualization of Forced Convection Laminar Boundary Layers,” International Journal of Heat and Mass Transfer, Vol. 36, No. 16, 1993, pp. 3957-3966. doi:10.1016/0017-9310(93)90146-W

[25] S. Kimura and A. Bejan, “The “Heatline” Visualization of Convective Heat Transfer,” Journal of Heat Transfer, Vol. 105, No. 4, 1983, pp. 916-919. doi:10.1115/1.3245684

[26] E. M. Sparrow and R. D. Cess, “The Effect of Magnetic Field on Free Convection Heat Transfer,” International Journal of Heat and Mass Transfer, Vol. 3, No. 4, 1961, pp. 267-274. doi:10.1016/0017-9310(61)90042-4