Non-Newtonian Power-Law Fluid Flow and Heat Transfer over a Non-Linearly Stretching Surface

Affiliation(s)

Department of Mathematics, Central College Campus, Bangalore University, Bangalore, India.

Tata Institute of Fundamental Research, Centre for Applicable Mathematics, Bangalore, India.

Department of Mathematics, Central College Campus, Bangalore University, Bangalore, India.

Tata Institute of Fundamental Research, Centre for Applicable Mathematics, Bangalore, India.

ABSTRACT

The problem of magneto-hydrodynamic flow and heat transfer of an electrically conducting non-Newtonian power-law fluid past a non-linearly stretching surface in the presence of a transverse magnetic field is considered. The stretching velocity, the temperature and the transverse magnetic field are assumed to vary in a power-law with the distance from the origin. The flow is induced due to an infinite elastic sheet which is stretched in its own plane. The governing equations are reduced to non-linear ordinary differential equations by means of similarity transformations. These equations are then solved numerically by an implicit finite-difference scheme known as Keller-Box method. The numerical solution is found to be dependent on several governing parameters, including the magnetic field parameter, power-law index, velocity exponent parameter, temperature exponent parameter, Modified Prandtl number and heat source/sink parameter. A systematic study is carried out to illustrate the effects of these parameters on the fluid velocity and the temperature distribution in the boundary layer. The results for the local skin-friction coefficient and the local Nusselt number are tabulated and discussed. The results obtained reveal many interesting behaviors that warrant further study on the equations related to non-Newtonian fluid phenomena.

The problem of magneto-hydrodynamic flow and heat transfer of an electrically conducting non-Newtonian power-law fluid past a non-linearly stretching surface in the presence of a transverse magnetic field is considered. The stretching velocity, the temperature and the transverse magnetic field are assumed to vary in a power-law with the distance from the origin. The flow is induced due to an infinite elastic sheet which is stretched in its own plane. The governing equations are reduced to non-linear ordinary differential equations by means of similarity transformations. These equations are then solved numerically by an implicit finite-difference scheme known as Keller-Box method. The numerical solution is found to be dependent on several governing parameters, including the magnetic field parameter, power-law index, velocity exponent parameter, temperature exponent parameter, Modified Prandtl number and heat source/sink parameter. A systematic study is carried out to illustrate the effects of these parameters on the fluid velocity and the temperature distribution in the boundary layer. The results for the local skin-friction coefficient and the local Nusselt number are tabulated and discussed. The results obtained reveal many interesting behaviors that warrant further study on the equations related to non-Newtonian fluid phenomena.

Cite this paper

K. Prasad, S. Santhi and P. Datti, "Non-Newtonian Power-Law Fluid Flow and Heat Transfer over a Non-Linearly Stretching Surface,"*Applied Mathematics*, Vol. 3 No. 5, 2012, pp. 425-435. doi: 10.4236/am.2012.35065.

K. Prasad, S. Santhi and P. Datti, "Non-Newtonian Power-Law Fluid Flow and Heat Transfer over a Non-Linearly Stretching Surface,"

References

[1] B. Vujannovic, A. M. Status and D. J. Djukiv, “A Variational Solution of the Rayleigh Problem for Power-Law Non-Newtonian Conducting Fluid,” Archive of Applied Mechanics, Vol. 41, No. 6, 1971, pp. 381-386.

[2] B. C. Sakiadis, “Boundary Layer Behavior on Continuous Solid Surfaces,” AIChE Journal, Vol. 7, No. 1, 1961, pp. 26-28. doi:10.1002/aic.690070108

[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] P. S. Gupta and A. S. Gupta, “Heat and Mass Transfer on a Stretching Sheet with Suction or Blowing,” The Canadian Journal of Chemical Engineering, Vol. 55, No. 6, 1977, pp. 744-746. doi:10.1002/cjce.5450550619

[5] J. P. Jadhav and B. B. Waghmode, “Heat Transfer to Non-Newtonian Power-Law Fluid past a Continuously Moving Porous Flat Plate with Heat Flux,” Heat and Mass Transfer, Vol. 25, No. 6, 1990, pp. 377-380. doi:10.1007/BF01811562

[6] T. Sarpakaya, “Flow on Non-Newtonian Fluids in a Magnetic Field,” AIChE Journal, Vol. 7, 1961, pp. 26-28.

[7] K. B. Pavlov, “Magnetohydrodynamic Flow of an Incompressible Viscous Fluid Caused by Deformation of a Plane Surface,” Magninaya Gidrodinamika (USSR), Vol. 4, 1974, pp. 146-147.

[8] H. I. Andersson, K. H. Bech and B. S. Dandapat, “Magnetohydrodynamic Flow of a Power-Law Fluid over a Stretching Sheet,” International Journal of Non-Linear Mechanics, Vol. 27, No. 6, 1992, pp. 929-936. doi:10.1016/0020-7462(92)90045-9

[9] M. I. Char, “Heat and Mass Transfer in a Hydromagnetic Flow of a 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

[10] R. Cortell, “A Note on Magneto Hydrodynamic Flow of a Power-Law Fluid over a Stretching Sheet,” Applied Mathematics and Computation, Vol. 168, No. 1, 2005, pp. 557566. doi:10.1016/j.amc.2004.09.046

[11] T. C. Chaim, “Hydromagnetic Flow over a Surface with a Power-Law Velocity,” International Journal of Engineering Science, Vol. 33, No. 3, 1995, pp. 429-435. doi:10.1016/0020-7225(94)00066-S

[12] A. Ishak, R. Nazar and I. Pop, “Magnetohydrodynamic Stagnation-Point Flow towards a Stretching Vertical Sheet,” Magnetohydrodynamics, Vol. 42, 2006, pp. 1730.

[13] S. P. Anjali Devi and M. Thiyagarajan, “Steady Nonlinear Hydromagnetic Flow and Heat Transfer over a Stretching Surface of Variable Temperature,” Heat and Mass Transfer, Vol. 42, No. 8, 2006, pp. 671-677. doi:10.1007/s00231-005-0640-y

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

[15] J. P. Denier and P. P. Dabrowski, “On the Boundary Layer Equations for Power-Law Fluids,” Proceedings of the Royal Society of London, Vol. A, 2004, pp. 31433158.

[16] A. Acrivos, M. J. Shah and E. E. Peterson, “Momentum and Heat Transfer in Laminar Boundary Layer Flows of Non-Newtonian Fluids past External Surfaces,” AIChE Journal, Vol. 6, No. 2, 1960, pp. 312-317. doi:10.1002/aic.690060227

[17] A. Acrivos, “A Theoretical Analysis of Laminar Natural Convection Heat Transfer to Non-Newtonian Fluids,” AIChE Journal, Vol. 6, No. 4, 1960, pp. 584-590. doi:10.1002/aic.690060416

[18] I. A. Hassanien, A. A. Abdullah and R. S. R. Gorla, “Flow and Heat Transfer in a Power-Law Fluid over a Non-Isothermal Stretching Sheet,” Mathematical and Computer Modelling, Vol. 28, No. 9, 1998, pp. 105-116. doi:10.1016/S0895-7177(98)00148-4

[19] H. I. Andersson and B. S. Dandapat, “Flow of a PowerLaw Fluid over a Stretching Sheet,” Stability and Applied Analysis of Continuous Media, Vol. 1, 1991, pp.339-347.

[20] L. G. Grubka and K. M. Bobba, “Heat Transfer Characteristics of a Continuous Stretching Surface with Variable Temperature,” Journal of Heat Transfer, Vol. 107, No. 1, 1985, pp. 248-250. doi:10.1115/1.3247387

[21] T. Cebeci and P. Bradshaw, “Physical and Computational Aspects of Convective Heat Transfer,” Springer-Verlag, New York, 1984.

[22] H. B. Keller, “Numerical Methods for Two-Point Boundary Value Problems,” Dover Publications, New York, 1992.

[1] B. Vujannovic, A. M. Status and D. J. Djukiv, “A Variational Solution of the Rayleigh Problem for Power-Law Non-Newtonian Conducting Fluid,” Archive of Applied Mechanics, Vol. 41, No. 6, 1971, pp. 381-386.

[2] B. C. Sakiadis, “Boundary Layer Behavior on Continuous Solid Surfaces,” AIChE Journal, Vol. 7, No. 1, 1961, pp. 26-28. doi:10.1002/aic.690070108

[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] P. S. Gupta and A. S. Gupta, “Heat and Mass Transfer on a Stretching Sheet with Suction or Blowing,” The Canadian Journal of Chemical Engineering, Vol. 55, No. 6, 1977, pp. 744-746. doi:10.1002/cjce.5450550619

[5] J. P. Jadhav and B. B. Waghmode, “Heat Transfer to Non-Newtonian Power-Law Fluid past a Continuously Moving Porous Flat Plate with Heat Flux,” Heat and Mass Transfer, Vol. 25, No. 6, 1990, pp. 377-380. doi:10.1007/BF01811562

[6] T. Sarpakaya, “Flow on Non-Newtonian Fluids in a Magnetic Field,” AIChE Journal, Vol. 7, 1961, pp. 26-28.

[7] K. B. Pavlov, “Magnetohydrodynamic Flow of an Incompressible Viscous Fluid Caused by Deformation of a Plane Surface,” Magninaya Gidrodinamika (USSR), Vol. 4, 1974, pp. 146-147.

[8] H. I. Andersson, K. H. Bech and B. S. Dandapat, “Magnetohydrodynamic Flow of a Power-Law Fluid over a Stretching Sheet,” International Journal of Non-Linear Mechanics, Vol. 27, No. 6, 1992, pp. 929-936. doi:10.1016/0020-7462(92)90045-9

[9] M. I. Char, “Heat and Mass Transfer in a Hydromagnetic Flow of a 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

[10] R. Cortell, “A Note on Magneto Hydrodynamic Flow of a Power-Law Fluid over a Stretching Sheet,” Applied Mathematics and Computation, Vol. 168, No. 1, 2005, pp. 557566. doi:10.1016/j.amc.2004.09.046

[11] T. C. Chaim, “Hydromagnetic Flow over a Surface with a Power-Law Velocity,” International Journal of Engineering Science, Vol. 33, No. 3, 1995, pp. 429-435. doi:10.1016/0020-7225(94)00066-S

[12] A. Ishak, R. Nazar and I. Pop, “Magnetohydrodynamic Stagnation-Point Flow towards a Stretching Vertical Sheet,” Magnetohydrodynamics, Vol. 42, 2006, pp. 1730.

[13] S. P. Anjali Devi and M. Thiyagarajan, “Steady Nonlinear Hydromagnetic Flow and Heat Transfer over a Stretching Surface of Variable Temperature,” Heat and Mass Transfer, Vol. 42, No. 8, 2006, pp. 671-677. doi:10.1007/s00231-005-0640-y

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

[15] J. P. Denier and P. P. Dabrowski, “On the Boundary Layer Equations for Power-Law Fluids,” Proceedings of the Royal Society of London, Vol. A, 2004, pp. 31433158.

[16] A. Acrivos, M. J. Shah and E. E. Peterson, “Momentum and Heat Transfer in Laminar Boundary Layer Flows of Non-Newtonian Fluids past External Surfaces,” AIChE Journal, Vol. 6, No. 2, 1960, pp. 312-317. doi:10.1002/aic.690060227

[17] A. Acrivos, “A Theoretical Analysis of Laminar Natural Convection Heat Transfer to Non-Newtonian Fluids,” AIChE Journal, Vol. 6, No. 4, 1960, pp. 584-590. doi:10.1002/aic.690060416

[18] I. A. Hassanien, A. A. Abdullah and R. S. R. Gorla, “Flow and Heat Transfer in a Power-Law Fluid over a Non-Isothermal Stretching Sheet,” Mathematical and Computer Modelling, Vol. 28, No. 9, 1998, pp. 105-116. doi:10.1016/S0895-7177(98)00148-4

[19] H. I. Andersson and B. S. Dandapat, “Flow of a PowerLaw Fluid over a Stretching Sheet,” Stability and Applied Analysis of Continuous Media, Vol. 1, 1991, pp.339-347.

[20] L. G. Grubka and K. M. Bobba, “Heat Transfer Characteristics of a Continuous Stretching Surface with Variable Temperature,” Journal of Heat Transfer, Vol. 107, No. 1, 1985, pp. 248-250. doi:10.1115/1.3247387

[21] T. Cebeci and P. Bradshaw, “Physical and Computational Aspects of Convective Heat Transfer,” Springer-Verlag, New York, 1984.

[22] H. B. Keller, “Numerical Methods for Two-Point Boundary Value Problems,” Dover Publications, New York, 1992.