Magnetohydrodynamic Flow of Viscous Fluid over a Non-Linearly Stretching Sheet

Yohannes Yirga^*, Daniel Tesfay

Department Mathematics, Mekelle University, Mekelle, Ethiopia.

DOI: 10.4236/oalib.1101030                

Abstract

In this paper, the Magnetohydrodynamic (MHD) Flow of Viscous Fluid over a Nonlinear Stretching Sheet is investigated numerically. The partial differential equations governing the flow are reduced to a non linear ordinary differential equations by using similarity transformations. The resulting transformed equations are numerically solved by an explicit finite difference scheme known as the Keller Box Method. The velocity profiles are determined and the effects of the magnetic parameter and non linear stretching parameter on the flow characteristics are investigated. In addition to this the numerical results for the local skin friction coefficients are computed. Comparison with the exact solution and previously reported analytic solutions is made and excellent agreement is noted. Moreover, the velocity profile obtained by Keller box method is in a better agreement to the exact solution than by the Homotopy Analysis Method. It is also found that, an increase in the magnetic parameter or non-linearity parameter causes a decrease in the velocity profile and velocity distribution.

Keywords

Magneto-Hydrodynamics, Non-Linear Stretching, Homotopy Analysis Method, Keller Box Method

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Sakiadis, B.C. (1961) Boundary Layer Behavior on Continuous Solid Surfaces. AIChE Journal, 7, 26-28, http://dx.doi.org/10.1002/aic.690070108
[2]	Crane, L.J. (1970) Flow past a Stretching Plate. Zeitschrift für angewandte Mathematik und Physik ZAMP, 21, 645-647. http://dx.doi.org/10.1007/BF01587695
[3]	Brady, J.F. and Acrivos, A. (1981) Steady Flow in a Channel or Tube with Accelerating Surface Velocity. An exact solution to the Navier-Stokes Equation with Reverse Flow. Journal of Fluid Mechanics, 112, 127-150. http://dx.doi.org/10.1017/S0022112081000323
[4]	Wang, C.Y. (1984) Fluid Flow Due to Stretching Cylinder. Physics of Fluids, 31, 466-468. http://dx.doi.org/10.1063/1.866827
[5]	Nadeem, S. (2009) Anwar Hussain: MHD Flow of a Viscous Fluid on a Nonlinear Porous Shrinking Sheet with Homotopy Analysis Method. Applied Mathematics and Mechanics (Engl. Ed.), 30, 1569-1578.
[6]	Sajid, M. (2009) Homotopy Analysis of Stretching Flows with Partial Slip. International Journal of Nonlinear Science, 8, 284-290.
[7]	Nadeem, S. and Lee, C. (2012) Boundary Layer Flow of Nanofluid over an Exponentially Stretching Surface. Nanoscale Research Letters, 7, 94. http://dx.doi.org/10.1186/1556-276X-7-94
[8]	Sharidan, S., Mahmood, T. and Pop, I. (2006) Similarity Solutions for the Unsteady Boundary Layer Flow and Heat Transfer Due to a Stretching Sheet. International Journal of Applied Mechanics and Engineering, 11, 647-654.
[9]	Cortell, R. (2007) Viscous Flow and Heat Transfer over a Nonlinearly Stretching Sheet. Applied Mathematics and Computation (Elsevier), 184, 864-873. http://dx.doi.org/10.1016/j.amc.2006.06.077
[10]	Alinejad, J. and Samarbakhsh, S. (2012) Viscous Flow over Nonlinearly Stretching Sheet with Effects of Viscous Dissipation. Journal of Applied Mathematics (Hindawi Publishing Corporation), 2012, Article ID: 587834.
[11]	Devi, C.D.S., Takhar, H.S. and Nath, G. (1991) Unsteady Mixed Convection Flow in Stagnation Region Adjacent to a Vertical Surface. Heat and Mass Transfer, 26, 71-79.
[12]	Andersson, H.I., Aarseth, J.B. and Dandapat, B.S. (2000) Heat Transfer in a Liquid Film on an Unsteady Stretching Surface. International Journal of Heat and Mass Transfer, 43, 69-74. http://dx.doi.org/10.1016/S0017-9310(99)00123-4
[13]	Nazar, R., Amin, N. and Pop, I. (2004) Unsteady Boundary Layer Flow Due to Stretching Surface in a Rotating Fluid. Mechanics Research Communications, 31, 121-128. http://dx.doi.org/10.1016/j.mechrescom.2003.09.004
[14]	Elbashbeshy, E.M.A. and Bazid, M.A.A. (2004) Heat Transfer over an Unsteady Stretching Surface. Heat and Mass Transfer, 41, 1-4. http://dx.doi.org/10.1007/s00231-004-0520-x
[15]	Ishak, A., Nazar, R. and Pop, I. (2009) Heat Transfer over an Unsteady Stretching Permeable Surface with Prescribed Wall Temperature. Nonlinear Analysis: Real World Applications, 10, 2909-2913. http://dx.doi.org/10.1016/j.nonrwa.2008.09.010
[16]	Wubshet, I., Shankar, B. and Nandeppanavar, M.M. (2013) MHD Stagnation Point Flow and Heat Transfer Due to Nanofluid towards Astretching Sheet. International Journal of Heat and Mass Transfer, 56, 1-9.
[17]	Turkyilmazoglu, M. and Pop, I. (2013) Heat and Mass Transfer of Unsteady Natural Convection Flow of Some Nano-fluids Past Avertical Infinite Flat Plate with Radiation Effect. International Journal of Heat and Mass Transfer, 59, 167-171. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.12.009
[18]	Xu, H., Pop, I. and You, X.C. (2013) Flow and Heat Transfer in a Nano-Liquid Film over an Unsteady Stretching Surface. International Journal of Heat and Mass Transfer, 60, 646-652. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.01.046
[19]	Lyapunov, A.M. (1992) General Problem on Stability of Motion (English Translation). Taylor and Francis, London.
[20]	Karmishin, A.V., Zhukov, A.I. and Kolosov, V.G. (1990) Methods of Dynamics Calculation and Testing for Thin-Walled Structures. Mashinostroyenie, Moscow.
[21]	Adomian, G. (1994) Solving Frontier Problems of Physics: The Decomposition Method. Kluwer Academic Publishers, Boston and London.
[22]	Liao, S.J. (1992) The Proposed Homotopy Analysis Technique for the Solution of Nonlinear Problems. PhD Thesis, Shanghai Jiao Tong University, Shanghai.
[23]	Keller, H.B. (1971) A New Difference Scheme for Parabolic Problems. In: Hubbard, B., Ed., Numerical Solutions of Partial Differential Equations, II, Academic Press, New York, 327-350.
[24]	Hayat, T., Hussain, Q. and Javed, T. (2007) The Modified Decomposition Method and Pade Approximation for the MHD Flow over a Non-Linear Stretching Sheet. Nonlinear Analysis: Real World Applications, 10, 966-973.
[25]	Ghotbi, A.R. (2009) Homotopy Analysis Method for Solving the MHD Flow over a Non-Linear Stretching Sheet. Communications in Nonlinear Science and Numerical Simulation, 14, 2653-2663.
[26]	Cebeci, T. and Pradshaw, P. (1988) Physical and Computational Aspects of Convective Heat Transfer. Springer, New York. http://dx.doi.org/10.1007/978-1-4612-3918-5