Visco-Elastic Boundary Layer Flow past a Stretching Plate and Heat Transfer with Variable Thermal Conductivity

Author(s)
Naseem Ahmad

ABSTRACT

In the present paper, the boundary layer flow of Walters Liquid B Model over a stretching plate has been considered to solve heat flow problem with variable conductivity. First, using similarity transformation, the velocity components of velocity have been obtained. Then, the heat flow problem has been considered in two ways: 1) prescribed surface temperature (PST), and 2) prescribed stretching plate heat flux (PHF) in case of variable conductivity. Due to variable conductivity, temperature profile has its two part- one mean tempera-ture and other temperature profile induced due to variable conductivity. The related results have been dis-cussed with the help of graphs.

In the present paper, the boundary layer flow of Walters Liquid B Model over a stretching plate has been considered to solve heat flow problem with variable conductivity. First, using similarity transformation, the velocity components of velocity have been obtained. Then, the heat flow problem has been considered in two ways: 1) prescribed surface temperature (PST), and 2) prescribed stretching plate heat flux (PHF) in case of variable conductivity. Due to variable conductivity, temperature profile has its two part- one mean tempera-ture and other temperature profile induced due to variable conductivity. The related results have been dis-cussed with the help of graphs.

Cite this paper

nullN. Ahmad, "Visco-Elastic Boundary Layer Flow past a Stretching Plate and Heat Transfer with Variable Thermal Conductivity,"*World Journal of Mechanics*, Vol. 1 No. 2, 2011, pp. 15-20. doi: 10.4236/wjm.2011.12003.

nullN. Ahmad, "Visco-Elastic 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, February 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 (ZAMP), 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.

[5] K. R. Rajagopal, T. Y. Na and A. S. Gupta, “A Non Similar Boundary Layer on a Stretching Sheet in a Non- Newtonian Fluid with Uniform Free Stream,” Journal of Mathematical and Physical Sciences, 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 (ZAMP), 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 Viscoelastic Fluid over a Stretching Sheet,” International Journal of Non-Linear Mechanics, Vol. 24, No. 3, 1989, pp. 215-219.

[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 (ZAMP), Vol. 41, No. 2, 1990, pp. 294-298. 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] N. Ahmad, G. S. Patel and B. Siddappa, “Viscoelastic boundary layer flow past a stretching plate with suction and heat transfer,” Jnanabh, Vol. 21, 1991, pp. 47-58.

[11] S. P. Lawrence and N. B. Rao, “Heat Transfer in the Flow of a Viscoelastic 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 Viscoelastic Fluid over a Streatching 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-Participating 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, Vol. 36, No. 6, June 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 and Heat Transfer with Variable Conductivity,” Proceedings of 44th Congress ISTAM, 1999, pp. 60-68.

[17] N. Ahmad and K. Marwah, “Viscoelastic Boundary Layer Flow past a Stretching Plate with Suction and Heat Transfer with Variable Conductivity,” Indian Journal of Engineering and Material Sciences (IJMS), Vol. 7, 2000, pp. 54-56.

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

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

[20] 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, July 2005, pp. 807-821. doi:10.1016/j.ijnonlinmec.2004.04.006

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

[23] N. Ahmad, Z. U. Siddiqui and M. K. Mishra, “Boundary Layer Flow and Heat Transfer past a Stretching Plate with Variable Thermal Conductivity,” International Journal of Non-Linear Mechanics, Vol. 45, No. 3, April 2010, pp. 306-309. doi:10.1016/j.ijnonlinmec.2009.12.006

[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, February 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 (ZAMP), 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.

[5] K. R. Rajagopal, T. Y. Na and A. S. Gupta, “A Non Similar Boundary Layer on a Stretching Sheet in a Non- Newtonian Fluid with Uniform Free Stream,” Journal of Mathematical and Physical Sciences, 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 (ZAMP), 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 Viscoelastic Fluid over a Stretching Sheet,” International Journal of Non-Linear Mechanics, Vol. 24, No. 3, 1989, pp. 215-219.

[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 (ZAMP), Vol. 41, No. 2, 1990, pp. 294-298. 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] N. Ahmad, G. S. Patel and B. Siddappa, “Viscoelastic boundary layer flow past a stretching plate with suction and heat transfer,” Jnanabh, Vol. 21, 1991, pp. 47-58.

[11] S. P. Lawrence and N. B. Rao, “Heat Transfer in the Flow of a Viscoelastic 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 Viscoelastic Fluid over a Streatching 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-Participating 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, Vol. 36, No. 6, June 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 and Heat Transfer with Variable Conductivity,” Proceedings of 44th Congress ISTAM, 1999, pp. 60-68.

[17] N. Ahmad and K. Marwah, “Viscoelastic Boundary Layer Flow past a Stretching Plate with Suction and Heat Transfer with Variable Conductivity,” Indian Journal of Engineering and Material Sciences (IJMS), Vol. 7, 2000, pp. 54-56.

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

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

[20] 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, July 2005, pp. 807-821. doi:10.1016/j.ijnonlinmec.2004.04.006

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

[23] N. Ahmad, Z. U. Siddiqui and M. K. Mishra, “Boundary Layer Flow and Heat Transfer past a Stretching Plate with Variable Thermal Conductivity,” International Journal of Non-Linear Mechanics, Vol. 45, No. 3, April 2010, pp. 306-309. doi:10.1016/j.ijnonlinmec.2009.12.006