Thermal Radiation, Heat Source/Sink and Work Done by Deformation Impacts on MHD Viscoelastic Fluid over a Nonlinear Stretching Sheet

Affiliation(s)

Mathematics Department, Faculty of Science, Assiut University, Assiut, Egypt.

Mathematics Department, Faculty of Science, South Valley University, Qena, Egypt.

Mathematics Department, Faculty of Science, Assiut University, Assiut, Egypt.

Mathematics Department, Faculty of Science, South Valley University, Qena, Egypt.

ABSTRACT

This work is focused on the effects of heat source/sink, viscous
dissipation, radiation and work done by deformation on flow and heat transfer
of a viscoelastic fluid over a nonlinear stretching sheet. The similarity
transformations have been used to convert the governing partial differential
equations into a set of nonlinear ordinary differential equations. These
equations are then solved numerically using a very efficient implicit finite
difference method. Favorable comparison with previously published work is
performed and it is found to be in excellent agreement. The results of this
parametric study are shown in several plots and tables and the physical aspects
of the problem are highlighted and discussed.

KEYWORDS

Flow and Heat Transfer; Second Grade Fluid; Nonlinear Stretching Sheet; Heat Source; Radiation

Flow and Heat Transfer; Second Grade Fluid; Nonlinear Stretching Sheet; Heat Source; Radiation

Cite this paper

F. Hady, R. Mohamed and H. ElShehabey, "Thermal Radiation, Heat Source/Sink and Work Done by Deformation Impacts on MHD Viscoelastic Fluid over a Nonlinear Stretching Sheet,"*World Journal of Mechanics*, Vol. 3 No. 4, 2013, pp. 203-214. doi: 10.4236/wjm.2013.34020.

F. Hady, R. Mohamed and H. ElShehabey, "Thermal Radiation, Heat Source/Sink and Work Done by Deformation Impacts on MHD Viscoelastic Fluid over a Nonlinear Stretching Sheet,"

References

[1] A. C. Truesdell and W. Noll, “The Non-Linear Field Theories of Mechanics,” In: S. Flugge, Ed., Encyclopedia of Physics, Springer, Berlin, 1965, pp. 1-591.

[2] K. R. Rajagopal, “On Boundary Conditions for Fluids of the Differential Type,” In: A. Sequeira, Ed., Navier— Stokes Equations and Related Non-linear Problems, Plenum Press, New York, 1995, pp. 273-278.

[3] K. R. Rajagopal and P. N. Kaloni, “Some Remarks on Boundary Conditions for Fluids of the Differential Type,” In: G. A. C. Graham and S. K. Malik , Eds., Continuum Mechanics and Its Applications, Hemisphere, New York, 1989, pp. 935-942.

[4] K. R. Rajagopal and A. S. Gupta, “An Exact Solution for the Flow of a Non-Newtonian Fluid past an Infinite Plate,” Meccanica, Vol. 19, No. 2, 1984, pp. 158-160. doi:10.1007/BF01560464

[5] D. W. Beard and K. Walters, “Elastico-Viscous Boundary Layer Flows,” Proceedings of the Cambridge Philosophi cal Society, Vol. 60, 1964, pp. 667-674. doi:10.1017/S0305004100038147

[6] J. E. Danberg and K. S. Fansler, “A Non Similar Moving Wall Boundary-Layer Problem,” Quarterly of Applied Mathematics, Vol. 34, 1976, pp. 305-309.

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

[8] B. S. Dandapat 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. doi:10.1016/0020-7462(89)90040-1

[9] F. M. Hady and R. S. R. Gorla, “Heat Transfer from a Continuous Surface in a Parallel Free Stream of Viscoe lastic Fluid,” Acta Mechanica, Vol. 128, No. 3, 1998, pp. 201-208. doi:10.1007/BF01251890

[10] C. I. Cookey, A. Ogulu and V. B. Omubo-Pepple, “Influence of Viscous Dissipation and Radiation on Un steady MHD Free-Convection Flow past an Infinite Heated Vertical Plate in a Porous Medium with Time-Dependent Suction,” International Journal of Heat and Mass Trans fer, Vol. 46, No. 13, 2003, pp. 2305-2311. doi:10.1016/S0017-9310(02)00544-6

[11] M. Kumari and G. Nath, “Radiation Effect on Mixed Convection from a Horizontal Surface in a Porous Me dium,” Mechanics Research Communications, Vol. 31, No. 4, 2004, pp. 483-491. doi:10.1016/j.mechrescom.2003.11.006

[12] M. A. Abd El-Naby, E. M. E. Elbarbary and N. Y. Abde lazem, “Finite Difference Solution of Radiation Effects on MHD Unsteady Free-Convection Flow over Vertical Porous Plate,” Applied Mathematics and Computation, Vol. 151, No. 2, 2004, pp. 327-346. doi:10.1016/S0096-3003(03)00344-8

[13] S. Abel, K. V. Prasad and A. Mahaboob, “Bouyancy Force and Thermal Radiation Effects in MHD Boundary Layer Viscoelastic Fluid Flow over Continuously Moving Stretching Surface,” International Journal of Thermal Sciences, Vol. 44, No. 5, 2005, pp. 465-476. doi:10.1016/j.ijthermalsci.2004.08.005

[14] S. K. Khan, “Heat Transfer in a Viscoelastic Fluid Flow over a Stretching Surface with Heat Source/Sink, Suction/ Blowing and Radiation,” International Journal of Heat and Mass Transfer, Vol. 49, No. 3-4, 2006, pp. 628-639. doi:10.1016/j.ijheatmasstransfer.2005.07.049

[15] Mahdy and H. M. ElShehabey, “Uncertainties in Physical Property Effects on Viscous Flow and Heat Transfer over a Nonlinearly Stretching Sheet with Nanofluids,” International Journal of Heat and Mass Transfer, Vol. 39, No. 5, 2012, pp. 713-719. doi:10.1016/j.icheatmasstransfer.2012.03.019

[16] R. Cortell, “Effects of Viscous Dissipation and Radiation on the Thermal Boundary Layer over a Nonlinearly Stretching Sheet,” Physics Letters A, Vol. 372, No. 13, 2008, pp. 631-636. doi:10.1016/j.physleta.2007.08.005

[17] K. Vajravelu, “Viscous Flow over a Nonlinearly Stretch ing Sheet,” Applied Mathematics and Computation, Vol. 124, No. 3, 2001, pp. 281-288. doi:10.1016/S0096-3003(00)00062-X

[18] R. Cortell, “Viscous Flow and Heat Transfer over a Non linearly Stretching Sheet,” Applied Mathematics and Computation, Vol. 184, No. 2, 2007, pp. 864-873. doi:10.1016/j.amc.2006.06.077

[19] S. Awang Kechil and I Hashim, “Series Solution of Flow over Nonlinearly Stretching Sheet with Chemical Reaction and Magnetic Field,” Physics Letters A, Vol. 372, No. 13, 2008, pp. 2258-2263. doi:10.1016/j.physleta.2007.11.027

[20] Z. Ziabakhsh, G. Domairry, H. Bararnia and H. Baba zadeh, “Analytical Solution of Flow and Diffusion of Chemically Reactive Species over a Nonlinearly Stretching Sheet Immersed in a Porous Medium,” Journal of the Taiwan Institute of Chemical Engineers, Vol. 41, No. 1, 2010, pp. 22-28. doi:10.1016/j.jtice.2009.04.011

[21] Muhaimina, R. Kandasamya and I. Hashimb, “Effect of Chemical Reaction, Heat and Mass Transfer on Nonlinear Boundary Layer past a Porous Shrinking Sheet in the Presence of Suction,” Nuclear Engineering and Design, Vol. 240, No. 5, 2010, pp. 933-939. doi:10.1016/j.nucengdes.2009.12.024

[22] R. Gorder, “High-Order Nonlinear Boundary Value Problems Admitting Multiple Exact Solutions with Application to the Fluid Flow over a Sheet,” Applied Mathematics and Computation, Vol. 216, No. 7, 2010, pp. 2177-2182. doi:10.1016/j.amc.2010.03.053

[23] R. Cortell, “Heat and Fluid Flow Due to Non-Linearly Stretching Surfaces,” Applied Mathematics and Computation, Vol. 271, 2011, pp. 7564-7564. doi:10.1016/j.amc.2011.02.029

[24] R. Gorder, K. Vajravelu and F. T. Akyildiz, “Existence and Uniqueness Results for a Nonlinear Differential Equation Arising in Viscous Flow over a Nonlinearly Stretching Sheet,” Applied Mathematics Letters, Vol. 24, No. 2, 2011, pp. 238-242. doi:10.1016/j.aml.2010.09.011

[25] K. Vajravelu, K. V. Prasad and N. S. Prasanna, “Diffusion of a Chemically Reactive Species of a Power-Law Fluid past a Stretching Surface,” Computers & Mathematics with Applications, Vol. 62, No. 1, 2011, pp. 93-108. doi:10.1016/j.camwa.2011.04.055

[26] R. Cortell, “Effects of Heat Source/Sink, Radiation and Work Done by Deformation on Flow and Heat Transfer of a Viscoelastic Fluid over a Stretching Sheet,” Computers & Mathematics with Applications, Vol. 53, No. 2, 2007 pp. 305-316. doi:10.1016/j.camwa.2006.02.041

[27] P. S. Datti, K. V. Prasad, M. Subhas Abel and A. Joshi, “MHD Viscoelastic Fluid Flow over a Non-Isothermal Stretching Sheet,” International Journal of Engineering Science, Vol. 42, No. 8-9, 2004, pp. 935-946. doi:10.1016/j.ijengsci.2003.09.008

[28] M. M. Rahman and T. Sultana, “Radiative Heat Transfer Flow of Micropolar Fluid with Variable Heat Flux in a Porous Medium,” Nonlinear Analysis: Modelling and Control, Vol. 13, No. 1, 2008, pp. 71-87

[29] F. M. Hady, F. S. Ibrahim, H. M. El-Hawary and A. M. AbdElhady, “Forced Convection Flow of Nanofluids past Power Law Stretching Horizontal Plates,” Applied Mathematics, Vol. 3, No. 2, 2012, pp. 121-126. doi:10.4236/am.2012.32019

[30] L. F. Shampine, M. W. Reichelt and J. Kierzenka, “Solving Boundary Value Problems for Ordinary Differential Equations in MATLAB with bvp4c.” http://www.mathworks.com/bvp_tutorial

[31] L. F. Shampine, I. Gladwell and S. Thompson, “Solving ODEs with MATLAB,” Cambridge University Press, Cambridge, 2003. doi:10.1017/CBO9780511615542

[1] A. C. Truesdell and W. Noll, “The Non-Linear Field Theories of Mechanics,” In: S. Flugge, Ed., Encyclopedia of Physics, Springer, Berlin, 1965, pp. 1-591.

[2] K. R. Rajagopal, “On Boundary Conditions for Fluids of the Differential Type,” In: A. Sequeira, Ed., Navier— Stokes Equations and Related Non-linear Problems, Plenum Press, New York, 1995, pp. 273-278.

[3] K. R. Rajagopal and P. N. Kaloni, “Some Remarks on Boundary Conditions for Fluids of the Differential Type,” In: G. A. C. Graham and S. K. Malik , Eds., Continuum Mechanics and Its Applications, Hemisphere, New York, 1989, pp. 935-942.

[4] K. R. Rajagopal and A. S. Gupta, “An Exact Solution for the Flow of a Non-Newtonian Fluid past an Infinite Plate,” Meccanica, Vol. 19, No. 2, 1984, pp. 158-160. doi:10.1007/BF01560464

[5] D. W. Beard and K. Walters, “Elastico-Viscous Boundary Layer Flows,” Proceedings of the Cambridge Philosophi cal Society, Vol. 60, 1964, pp. 667-674. doi:10.1017/S0305004100038147

[6] J. E. Danberg and K. S. Fansler, “A Non Similar Moving Wall Boundary-Layer Problem,” Quarterly of Applied Mathematics, Vol. 34, 1976, pp. 305-309.

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

[8] B. S. Dandapat 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. doi:10.1016/0020-7462(89)90040-1

[9] F. M. Hady and R. S. R. Gorla, “Heat Transfer from a Continuous Surface in a Parallel Free Stream of Viscoe lastic Fluid,” Acta Mechanica, Vol. 128, No. 3, 1998, pp. 201-208. doi:10.1007/BF01251890

[10] C. I. Cookey, A. Ogulu and V. B. Omubo-Pepple, “Influence of Viscous Dissipation and Radiation on Un steady MHD Free-Convection Flow past an Infinite Heated Vertical Plate in a Porous Medium with Time-Dependent Suction,” International Journal of Heat and Mass Trans fer, Vol. 46, No. 13, 2003, pp. 2305-2311. doi:10.1016/S0017-9310(02)00544-6

[11] M. Kumari and G. Nath, “Radiation Effect on Mixed Convection from a Horizontal Surface in a Porous Me dium,” Mechanics Research Communications, Vol. 31, No. 4, 2004, pp. 483-491. doi:10.1016/j.mechrescom.2003.11.006

[12] M. A. Abd El-Naby, E. M. E. Elbarbary and N. Y. Abde lazem, “Finite Difference Solution of Radiation Effects on MHD Unsteady Free-Convection Flow over Vertical Porous Plate,” Applied Mathematics and Computation, Vol. 151, No. 2, 2004, pp. 327-346. doi:10.1016/S0096-3003(03)00344-8

[13] S. Abel, K. V. Prasad and A. Mahaboob, “Bouyancy Force and Thermal Radiation Effects in MHD Boundary Layer Viscoelastic Fluid Flow over Continuously Moving Stretching Surface,” International Journal of Thermal Sciences, Vol. 44, No. 5, 2005, pp. 465-476. doi:10.1016/j.ijthermalsci.2004.08.005

[14] S. K. Khan, “Heat Transfer in a Viscoelastic Fluid Flow over a Stretching Surface with Heat Source/Sink, Suction/ Blowing and Radiation,” International Journal of Heat and Mass Transfer, Vol. 49, No. 3-4, 2006, pp. 628-639. doi:10.1016/j.ijheatmasstransfer.2005.07.049

[15] Mahdy and H. M. ElShehabey, “Uncertainties in Physical Property Effects on Viscous Flow and Heat Transfer over a Nonlinearly Stretching Sheet with Nanofluids,” International Journal of Heat and Mass Transfer, Vol. 39, No. 5, 2012, pp. 713-719. doi:10.1016/j.icheatmasstransfer.2012.03.019

[16] R. Cortell, “Effects of Viscous Dissipation and Radiation on the Thermal Boundary Layer over a Nonlinearly Stretching Sheet,” Physics Letters A, Vol. 372, No. 13, 2008, pp. 631-636. doi:10.1016/j.physleta.2007.08.005

[17] K. Vajravelu, “Viscous Flow over a Nonlinearly Stretch ing Sheet,” Applied Mathematics and Computation, Vol. 124, No. 3, 2001, pp. 281-288. doi:10.1016/S0096-3003(00)00062-X

[18] R. Cortell, “Viscous Flow and Heat Transfer over a Non linearly Stretching Sheet,” Applied Mathematics and Computation, Vol. 184, No. 2, 2007, pp. 864-873. doi:10.1016/j.amc.2006.06.077

[19] S. Awang Kechil and I Hashim, “Series Solution of Flow over Nonlinearly Stretching Sheet with Chemical Reaction and Magnetic Field,” Physics Letters A, Vol. 372, No. 13, 2008, pp. 2258-2263. doi:10.1016/j.physleta.2007.11.027

[20] Z. Ziabakhsh, G. Domairry, H. Bararnia and H. Baba zadeh, “Analytical Solution of Flow and Diffusion of Chemically Reactive Species over a Nonlinearly Stretching Sheet Immersed in a Porous Medium,” Journal of the Taiwan Institute of Chemical Engineers, Vol. 41, No. 1, 2010, pp. 22-28. doi:10.1016/j.jtice.2009.04.011

[21] Muhaimina, R. Kandasamya and I. Hashimb, “Effect of Chemical Reaction, Heat and Mass Transfer on Nonlinear Boundary Layer past a Porous Shrinking Sheet in the Presence of Suction,” Nuclear Engineering and Design, Vol. 240, No. 5, 2010, pp. 933-939. doi:10.1016/j.nucengdes.2009.12.024

[22] R. Gorder, “High-Order Nonlinear Boundary Value Problems Admitting Multiple Exact Solutions with Application to the Fluid Flow over a Sheet,” Applied Mathematics and Computation, Vol. 216, No. 7, 2010, pp. 2177-2182. doi:10.1016/j.amc.2010.03.053

[23] R. Cortell, “Heat and Fluid Flow Due to Non-Linearly Stretching Surfaces,” Applied Mathematics and Computation, Vol. 271, 2011, pp. 7564-7564. doi:10.1016/j.amc.2011.02.029

[24] R. Gorder, K. Vajravelu and F. T. Akyildiz, “Existence and Uniqueness Results for a Nonlinear Differential Equation Arising in Viscous Flow over a Nonlinearly Stretching Sheet,” Applied Mathematics Letters, Vol. 24, No. 2, 2011, pp. 238-242. doi:10.1016/j.aml.2010.09.011

[25] K. Vajravelu, K. V. Prasad and N. S. Prasanna, “Diffusion of a Chemically Reactive Species of a Power-Law Fluid past a Stretching Surface,” Computers & Mathematics with Applications, Vol. 62, No. 1, 2011, pp. 93-108. doi:10.1016/j.camwa.2011.04.055

[26] R. Cortell, “Effects of Heat Source/Sink, Radiation and Work Done by Deformation on Flow and Heat Transfer of a Viscoelastic Fluid over a Stretching Sheet,” Computers & Mathematics with Applications, Vol. 53, No. 2, 2007 pp. 305-316. doi:10.1016/j.camwa.2006.02.041

[27] P. S. Datti, K. V. Prasad, M. Subhas Abel and A. Joshi, “MHD Viscoelastic Fluid Flow over a Non-Isothermal Stretching Sheet,” International Journal of Engineering Science, Vol. 42, No. 8-9, 2004, pp. 935-946. doi:10.1016/j.ijengsci.2003.09.008

[28] M. M. Rahman and T. Sultana, “Radiative Heat Transfer Flow of Micropolar Fluid with Variable Heat Flux in a Porous Medium,” Nonlinear Analysis: Modelling and Control, Vol. 13, No. 1, 2008, pp. 71-87

[29] F. M. Hady, F. S. Ibrahim, H. M. El-Hawary and A. M. AbdElhady, “Forced Convection Flow of Nanofluids past Power Law Stretching Horizontal Plates,” Applied Mathematics, Vol. 3, No. 2, 2012, pp. 121-126. doi:10.4236/am.2012.32019

[30] L. F. Shampine, M. W. Reichelt and J. Kierzenka, “Solving Boundary Value Problems for Ordinary Differential Equations in MATLAB with bvp4c.” http://www.mathworks.com/bvp_tutorial

[31] L. F. Shampine, I. Gladwell and S. Thompson, “Solving ODEs with MATLAB,” Cambridge University Press, Cambridge, 2003. doi:10.1017/CBO9780511615542