Unsteady Boundary Layer Flow past a Stretching Plate and Heat Transfer with Variable Thermal Conductivity

Affiliation(s)

Depatment of Applied Sciences, NSIT, New Delhi, India.

Department of Mathematics Jamia Millia Islamia, New Delhi, India.

Department of Mathematics & Statistics, Faculty of Science, University of Maiduguri, Maiduguri, Nigeria.

Depatment of Applied Sciences, NSIT, New Delhi, India.

Department of Mathematics Jamia Millia Islamia, New Delhi, India.

Department of Mathematics & Statistics, Faculty of Science, University of Maiduguri, Maiduguri, Nigeria.

ABSTRACT

An unsteady boundary layer flow of viscous incompressible fluid over a stretching plate has been considered to solve heat flow problem with variable thermal conductivity. First, using similarity transformation, the velocity components have been obtained, and then the heat flow problem has been attempted in the following two ways: 1) prescribed stretching surface temperature (PST), and 2) prescribed stretching surface heat flux (PHF) Flow and temperature fields have been analyzed through graphs. The expressions for skin friction and coefficient of convective heat transfer Nusselt number in PST and PHF cases have been derived.

An unsteady boundary layer flow of viscous incompressible fluid over a stretching plate has been considered to solve heat flow problem with variable thermal conductivity. First, using similarity transformation, the velocity components have been obtained, and then the heat flow problem has been attempted in the following two ways: 1) prescribed stretching surface temperature (PST), and 2) prescribed stretching surface heat flux (PHF) Flow and temperature fields have been analyzed through graphs. The expressions for skin friction and coefficient of convective heat transfer Nusselt number in PST and PHF cases have been derived.

Cite this paper

nullM. Misra, N. Ahmad and Z. Siddiqui, "Unsteady Boundary Layer Flow past a Stretching Plate and Heat Transfer with Variable Thermal Conductivity,"*World Journal of Mechanics*, Vol. 2 No. 1, 2012, pp. 35-41. doi: 10.4236/wjm.2012.21005.

nullM. Misra, N. Ahmad and Z. Siddiqui, "Unsteady Boundary Layer Flow past a Stretching Plate and Heat Transfer with Variable Thermal Conductivity,"

References

[1] B. C. Sakiadis, “Boundary Layer Behavior on Continuous Solid Surfaces: II Boundary Layer on a Continuous Flat Surface,” AIChE Journal, Vol. 7, No. 2, 1961, pp. 221- 225. doi:10.1002/aic.690070211

[2] F. K. Tsou, E. M. Sparrow and R. J. Goldstein, “Flow and Heat Transfer in the Boundary Layer on a Continuous Moving Surfaces,” International Journal of Heat and Mass Transfer, Vol. 10, No. 2, 1967, pp. 219-235. doi:10.1016/0017-9310(67)90100-7

[3] L. J. Crane, “Flow past a Stretching Plate,” Zeitschrift für Angewandte Mathematik und Physik, Vol. 21, No. 4, 1970, pp. 645-647. doi:10.1007/BF01587695

[4] K. R. Rajagopal, T. Y. Na and A. S. Gupta, “Flow of a Visco-Elastic Fluid over a Stretching Sheet,” Rheologica Acta, Vol. 23, No. 2, 1984, pp. 213-215. doi:10.1007/BF01332078

[5] K. R. Rajagopal, T. Y. Na and A. S. Gupta, “A Non Similar Boundary Layer on a Stretching Sheet in a Non-New- tonian Fluid with Uniform Free Stream,” Journal of Ma- thematical Physics, Vol. 21, No. 2, 1987, pp. 189-200.

[6] N. M. Bujurke, S. N. Biradar and P. S. Hiremath, “Second Order Fluid Flow Past a Stretching Sheet with Heat Transfer,” Zeitschrift für Angewandte Mathematik und Physik, Vol. 38, No. 4, 1987, pp. 890-892. doi:10.1007/BF00946345

[7] B. S. Dhanpat and A. S. Gupta, “Flow and Heat Transfer in a Visco-Elastic Fluid over a Stretching Sheet,” International Journal of Non-Linear Mechanics, Vol. 24, No. 3, 1989, pp. 215-219. doi:10.1016/0020-7462(89)90040-1

[8] N. Ahmad, G. S. Patel and B. Siddappa, “Visco-Elastic Boundary Layer Flow past a Stretching Plate and Heat Transfer,” Zeitschrift für Angewandte Mathematik und Physik, Vol. 41, No. 2, 1990, pp. 294-299. doi:10.1007/BF00945114

[9] D. Rollins and K. Vajravelu, “Heat Transfer in a Second Order Fluid over a Continuous Stretching Surface,” Acta Mechanica, Vol. 89, No. 1-4, 1991, pp. 167-178. doi:10.1007/BF01171253

[10] B. Siddappa and M. S. Abel, “Visco-Elastic Boundary Layer Flow past a Stretching Plate with Suction and Heat Transfer,” Rheologica Acta, Vol. 25, No. 3, 1986, pp. 319- 320.

[11] S. P. Lawrence and N. B. Rao, “Heat Transfer in the Flow of a Visco Elastic Fluid over a Stretching Sheet,” Acta Mechanica, Vol. 93, No. 1-4, 1992, pp. 53-61. doi:10.1007/BF01182572

[12] M. I. Char, “Heat and Mass Transfer in a Hydromagnetic Flow of Visco-Elastic Fluid over a Stretching Sheet,” Journal of Mathematical Analysis and Applications, Vol. 186, No. 3, 1994, pp. 674-689. doi:10.1006/jmaa.1994.1326

[13] A. Naseem, “On Temperature Distribution in a No-Parti- cipating Medium with Radiation Boundary Condition,” International Journal of Heat and Technology, Vol. 14, No. 1, 1995, pp. 19-28.

[14] N. Ahmad, “The Stretching Plate with Suction in Non-Participating Medium with Radiation Boundary Condition,” International Journal of Heat and Technology, Vol. 14, No. 1, 1996, pp. 59-66.

[15] D. Kelly, K. Vjravelu and L. Andrews, “Analysis of Heat Mass Transfer of a Visco-Elastic, Electrically Conducting Fluid past a Continuous Stretching Sheet,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 36, No. 6, 1999, pp. 767-784. doi:10.1016/S0362-546X(98)00128-X

[16] N. Ahmad and K. Marwah, “Visco Elastic Boundary Layer Flow past a Stretching Plate with Suction and Heat Transfer with Variable Conductivity,” International Journal on Multicultural Societies, Vol. 7, 2000, pp. 54- 56.

[17] A. M. Subhas, A. Joshi and R. M. Sonth, “Heat Transfer in MHD Visco-Elastic Fluid Flow over a Stretching Surface,” Zeitschrift für Angewandte Mathematik und Me- chanik, Vol. 81, No. 10, 2001, pp. 691-698. doi:10.1002/1521-4001(200110)81:10<691::AID-ZAMM691>3.0.CO;2-Z

[18] R. M. Sonth, S. K. Khan, A. M. Subhas and K. V. Prasad, “Heat and Mass Transfer in a Visco-Elastic Fluid Flow over an Accelerating Surface with Heat Source/Sink and Viscous Dissipation,” Heat and Mass Transfer, Vol. 38, No. 3, 2002, pp. 213-220. doi:10.1007/s002310100271

[19] P. G. Siddheshwar and U. S. Mahabaleswar, “Effects of Radiation and Heat Source on MHD Flow of Viscoelastic Liquid and Heat Transfer over Stretching Sheet,” International Journal of Non-Linear Mechanics, Vol. 40, No. 6, 2005, pp. 807-821. doi:10.1016/j.ijnonlinmec.2004.04.006

[20] M. S. Abel, P. G. Siddheshwar and M. M. Nandeppanavar, “Heat Transfer in a Visco-Elastic Boundary Layer Flow over a Stretching Sheet with Viscous Dissipation and Non-Uniform Heat Source,” International Journal of Mass and Heat Transfer, Vol. 50, No. 5-6, 2007, pp. 960-966.

[21] T. C. Chiam, “Heat Transfer with Variable Conductivity in Stagnation—Point Flow towards a Stretching Sheet,” International Communications in Heat and Mass Transfer, Vol. 23, No. 2, 1996, pp. 239-248. doi:10.1016/0735-1933(96)00009-7

[22] N. Ahmad and K. Marwah, “Visco-Elastic Boundary Layer Flow past a Stretching Plate and Heat Transfer with Variable Conductivity,” Proceedings of 44th Congress ISTAM, 22-25 December 1999, pp. 60-68.

[23] N. Ahmad and M. Mishra, “Unsteady Boundary Layer Flow and Heat Transfer over a Stretching Sheet,” Proceedings of 28th UIT Heat Transfer Congress, Brescia, 21-23 June 2010, pp. 37-40.

[1] B. C. Sakiadis, “Boundary Layer Behavior on Continuous Solid Surfaces: II Boundary Layer on a Continuous Flat Surface,” AIChE Journal, Vol. 7, No. 2, 1961, pp. 221- 225. doi:10.1002/aic.690070211

[2] F. K. Tsou, E. M. Sparrow and R. J. Goldstein, “Flow and Heat Transfer in the Boundary Layer on a Continuous Moving Surfaces,” International Journal of Heat and Mass Transfer, Vol. 10, No. 2, 1967, pp. 219-235. doi:10.1016/0017-9310(67)90100-7

[3] L. J. Crane, “Flow past a Stretching Plate,” Zeitschrift für Angewandte Mathematik und Physik, Vol. 21, No. 4, 1970, pp. 645-647. doi:10.1007/BF01587695

[4] K. R. Rajagopal, T. Y. Na and A. S. Gupta, “Flow of a Visco-Elastic Fluid over a Stretching Sheet,” Rheologica Acta, Vol. 23, No. 2, 1984, pp. 213-215. doi:10.1007/BF01332078

[5] K. R. Rajagopal, T. Y. Na and A. S. Gupta, “A Non Similar Boundary Layer on a Stretching Sheet in a Non-New- tonian Fluid with Uniform Free Stream,” Journal of Ma- thematical Physics, Vol. 21, No. 2, 1987, pp. 189-200.

[6] N. M. Bujurke, S. N. Biradar and P. S. Hiremath, “Second Order Fluid Flow Past a Stretching Sheet with Heat Transfer,” Zeitschrift für Angewandte Mathematik und Physik, Vol. 38, No. 4, 1987, pp. 890-892. doi:10.1007/BF00946345

[7] B. S. Dhanpat and A. S. Gupta, “Flow and Heat Transfer in a Visco-Elastic Fluid over a Stretching Sheet,” International Journal of Non-Linear Mechanics, Vol. 24, No. 3, 1989, pp. 215-219. doi:10.1016/0020-7462(89)90040-1

[8] N. Ahmad, G. S. Patel and B. Siddappa, “Visco-Elastic Boundary Layer Flow past a Stretching Plate and Heat Transfer,” Zeitschrift für Angewandte Mathematik und Physik, Vol. 41, No. 2, 1990, pp. 294-299. doi:10.1007/BF00945114

[9] D. Rollins and K. Vajravelu, “Heat Transfer in a Second Order Fluid over a Continuous Stretching Surface,” Acta Mechanica, Vol. 89, No. 1-4, 1991, pp. 167-178. doi:10.1007/BF01171253

[10] B. Siddappa and M. S. Abel, “Visco-Elastic Boundary Layer Flow past a Stretching Plate with Suction and Heat Transfer,” Rheologica Acta, Vol. 25, No. 3, 1986, pp. 319- 320.

[11] S. P. Lawrence and N. B. Rao, “Heat Transfer in the Flow of a Visco Elastic Fluid over a Stretching Sheet,” Acta Mechanica, Vol. 93, No. 1-4, 1992, pp. 53-61. doi:10.1007/BF01182572

[12] M. I. Char, “Heat and Mass Transfer in a Hydromagnetic Flow of Visco-Elastic Fluid over a Stretching Sheet,” Journal of Mathematical Analysis and Applications, Vol. 186, No. 3, 1994, pp. 674-689. doi:10.1006/jmaa.1994.1326

[13] A. Naseem, “On Temperature Distribution in a No-Parti- cipating Medium with Radiation Boundary Condition,” International Journal of Heat and Technology, Vol. 14, No. 1, 1995, pp. 19-28.

[14] N. Ahmad, “The Stretching Plate with Suction in Non-Participating Medium with Radiation Boundary Condition,” International Journal of Heat and Technology, Vol. 14, No. 1, 1996, pp. 59-66.

[15] D. Kelly, K. Vjravelu and L. Andrews, “Analysis of Heat Mass Transfer of a Visco-Elastic, Electrically Conducting Fluid past a Continuous Stretching Sheet,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 36, No. 6, 1999, pp. 767-784. doi:10.1016/S0362-546X(98)00128-X

[16] N. Ahmad and K. Marwah, “Visco Elastic Boundary Layer Flow past a Stretching Plate with Suction and Heat Transfer with Variable Conductivity,” International Journal on Multicultural Societies, Vol. 7, 2000, pp. 54- 56.

[17] A. M. Subhas, A. Joshi and R. M. Sonth, “Heat Transfer in MHD Visco-Elastic Fluid Flow over a Stretching Surface,” Zeitschrift für Angewandte Mathematik und Me- chanik, Vol. 81, No. 10, 2001, pp. 691-698. doi:10.1002/1521-4001(200110)81:10<691::AID-ZAMM691>3.0.CO;2-Z

[18] R. M. Sonth, S. K. Khan, A. M. Subhas and K. V. Prasad, “Heat and Mass Transfer in a Visco-Elastic Fluid Flow over an Accelerating Surface with Heat Source/Sink and Viscous Dissipation,” Heat and Mass Transfer, Vol. 38, No. 3, 2002, pp. 213-220. doi:10.1007/s002310100271

[19] P. G. Siddheshwar and U. S. Mahabaleswar, “Effects of Radiation and Heat Source on MHD Flow of Viscoelastic Liquid and Heat Transfer over Stretching Sheet,” International Journal of Non-Linear Mechanics, Vol. 40, No. 6, 2005, pp. 807-821. doi:10.1016/j.ijnonlinmec.2004.04.006

[20] M. S. Abel, P. G. Siddheshwar and M. M. Nandeppanavar, “Heat Transfer in a Visco-Elastic Boundary Layer Flow over a Stretching Sheet with Viscous Dissipation and Non-Uniform Heat Source,” International Journal of Mass and Heat Transfer, Vol. 50, No. 5-6, 2007, pp. 960-966.

[21] T. C. Chiam, “Heat Transfer with Variable Conductivity in Stagnation—Point Flow towards a Stretching Sheet,” International Communications in Heat and Mass Transfer, Vol. 23, No. 2, 1996, pp. 239-248. doi:10.1016/0735-1933(96)00009-7

[22] N. Ahmad and K. Marwah, “Visco-Elastic Boundary Layer Flow past a Stretching Plate and Heat Transfer with Variable Conductivity,” Proceedings of 44th Congress ISTAM, 22-25 December 1999, pp. 60-68.

[23] N. Ahmad and M. Mishra, “Unsteady Boundary Layer Flow and Heat Transfer over a Stretching Sheet,” Proceedings of 28th UIT Heat Transfer Congress, Brescia, 21-23 June 2010, pp. 37-40.