Dynamics of Unsteady MHD Convective Flow with Thermophoresis of Particles and Variable Thermo-Physical Properties past a Vertical Surface Moving through Binary Mixture
Author(s)
Isaac Lare Animasaun
Affiliation(s)
Department of Mathematical Sciences, Federal University of Technology, Akure, Nigeria.
Department of Mathematical Sciences, Federal University of Technology, Akure, Nigeria.
ABSTRACT
The dynamics of unsteady magnetohydrodynamic convective fluid flow with radiation and thermophoresis of particles past a vertical porous plate moving through a binary mixture in an optically thin environment is investigated. The approximate form of the radiative heat flux is considered as the fourth power of temperature. Chemical reaction that occurs as the chemically reacting fluid flow through binary mixture is accounted for in energy and species concentration equations. Exponential space dependent heat source is introduced to generate additional heat energy across the fluid domain. The corresponding influence of heat energy is properly accounted for. It is assumed that viscosity and thermal conductivity vary as a linear function of temperature. The governing boundary layer equations are converted to nonlinear ordinary differential equations using similarity variables. A novel method of obtaining root finding starting with three guesses in shooting techniques is presented. The corresponding nonlinear coupled ordinary differential equations is solved numerically by shooting technique along with quadratic interpolation scheme. Graphical results of the dimensionless velocity, temperature and concentration distributions are shown for certain pertinent parameters controlling the fluid flow. The quadratic interpolation method is found to produce better estimated values of
, which satisfy the degree of accuracy and proportional to the physical quantities.
The dynamics of unsteady magnetohydrodynamic convective fluid flow with radiation and thermophoresis of particles past a vertical porous plate moving through a binary mixture in an optically thin environment is investigated. The approximate form of the radiative heat flux is considered as the fourth power of temperature. Chemical reaction that occurs as the chemically reacting fluid flow through binary mixture is accounted for in energy and species concentration equations. Exponential space dependent heat source is introduced to generate additional heat energy across the fluid domain. The corresponding influence of heat energy is properly accounted for. It is assumed that viscosity and thermal conductivity vary as a linear function of temperature. The governing boundary layer equations are converted to nonlinear ordinary differential equations using similarity variables. A novel method of obtaining root finding starting with three guesses in shooting techniques is presented. The corresponding nonlinear coupled ordinary differential equations is solved numerically by shooting technique along with quadratic interpolation scheme. Graphical results of the dimensionless velocity, temperature and concentration distributions are shown for certain pertinent parameters controlling the fluid flow. The quadratic interpolation method is found to produce better estimated values of
, which satisfy the degree of accuracy and proportional to the physical quantities.
KEYWORDS
Variable Viscosity, Variable Thermal Conductivity, Binary Mixture, Quadratic Interpolation, Unsteady Flow, Thermophoresis, Exponential Heat Source, Shooting Technique
Variable Viscosity, Variable Thermal Conductivity, Binary Mixture, Quadratic Interpolation, Unsteady Flow, Thermophoresis, Exponential Heat Source, Shooting Technique
Cite this paper
Animasaun, I. (2015) Dynamics of Unsteady MHD Convective Flow with Thermophoresis of Particles and Variable Thermo-Physical Properties past a Vertical Surface Moving through Binary Mixture. Open Journal of Fluid Dynamics, 5, 106-120. doi: 10.4236/ojfd.2015.52013.
Animasaun, I. (2015) Dynamics of Unsteady MHD Convective Flow with Thermophoresis of Particles and Variable Thermo-Physical Properties past a Vertical Surface Moving through Binary Mixture. Open Journal of Fluid Dynamics, 5, 106-120. doi: 10.4236/ojfd.2015.52013.
References
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http://dx.doi.org/10.1115/1.2920985 [2] Wang, S.H., Al-Sharif, A., Rajagopal, K.R. and Szeri, A.Z. (1993) Lubrication with Binary Mixtures: Liquid-Liquid Emulsion in an EHL Conjunction. Journal of Tribology, 115, 515-522.
http://dx.doi.org/10.1115/1.2921668 [3] Makinde, O.D., Olanrewaju, P.O. and Charles, W.M. (2011) Unsteady Convection with Chemical Reaction and Radiative Heat Transfer past a Flat Porous Plate Moving through a Binary Mixture. Afrika Matematika, 21, 1-17. [4] Makinde, O.D. and Olanrewaju, P.O. (2011) Unsteady Mixed Convection with Soret and Dufour Effects past a Porous Plate Moving through a Binary Mixture of Chemically Reacting Fluid. Chemical Engineering Communications, 22, 65-78.
http://dx.doi.org/10.1080/00986445.2011.545296 [5] Sastry, D.R.V.S.R.K. and Murti, A.S.N. (2013) A Double Diffusive Unsteady MHD Convective Flow past a Flat Porous Plate Moving through a Binary Mixture with Suction or Injection. Journal of Fluids, 2013, Article ID: 935156. [6] Somasundaran, P. (2006) Encyclopaedia of Surface and Colloid Science. CRC Press, Boca Raton, 711. [7] Giddings, J.C., Yang, F.J. and Myers, M.N. (1976) Flow Field-Flow Fractionation: A Versatile New Separation Method. Science, 193, 1244-1245.
http://dx.doi.org/10.1126/science.959835 [8] Talbot, L., Cheng, R.K., Scheffer, R.W. and Wills, D.P. (1980) Thermophoresis of Particles in a Heated Boundary Layer. Journal Fluid Mechanics, 101, 737-758.
http://dx.doi.org/10.1017/S0022112080001905 [9] Chamkha, A.J. and Isaa, C. (2000) Effects of Heat Generation/Absorption and Thermophoresis on Hydromagnetic Flow with Heat and Mass Transfer over a Flat Surface. International Journal of Numerical Methods for Heat and Fluid flow, 4, 432-448.
http://dx.doi.org/10.1108/09615530010327404 [10] Animasaun, I.L (2015) Effects of Thermophoresis, Variable Viscosity and Thermal Conductivity on Free Convective Heat and Mass Transfer of Non-Darcian MHD Dissipative Casson Fluid Flow with Suction and nth Order of Chemical Reaction. Journal of the Nigerian Mathematical Society, 34, 11-31.
http://dx.doi.org/10.1016/j.jnnms.2014.10.008 [11] Fagbade, A.I., Falodun, B.O. and Boneze, C.U. (2015) Influence of Magnetic Field, Viscous Dissipation and Thermophoresis on Darcy-Forcheimer Mixed Convection Flow in Fluid Saturated Porous Media. American Journal of Computational Mathematics, 5, 18-40.
http://dx.doi.org/10.4236/ajcm.2015.51002 [12] Singh, A.K. and Dikshit, C.K. (1988) Hydromagnetic Flow past a Continuously Moving Semi-Infinite Plate for Large Suction. Astrophysics and Space Science, 148, 249-256.
http://dx.doi.org/10.1007/BF00645964 [13] Vajravelu, K. and Hadjinicolaou, A. (1993) Heat Transfer in a Viscous Fluid over a Stretching Sheet with Viscous Dissipation and Internal Heat Generation. International Communications in Heat and Mass Transfer, 20, 417-430.
http://dx.doi.org/10.1016/0735-1933(93)90026-R [14] Takhar, H.S., Gorla, R.S.R. and Soundalgekar, V.M. (1996) Radiation Effects on MHD Free Convection Flow of a Gas past a Semi-Infinite Vertical Plate. International Journal of Numerical Methods for Heat and Fluid Flow, 6, 77-83.
http://dx.doi.org/10.1108/09615539610113118 [15] Kim, Y.J. (2001) Unsteady MHD Convection Flow of Polar Fluids past a Vertical Moving Porous Plate in a Porous Medium. International Journal of Heat and Mass Transfer, 44, 2791-2799.
http://dx.doi.org/10.1016/S0017-9310(00)00332-X [16] Raptis, A., Perdikis, C. and Leontitsis, A. (2003) Effects of Radiation in an Optically Thin Gray Gas Flowing past a Vertical Infinite Plate in the Presence of a Magnetic Field. Heat and Mass Transfer, 39, 771-773.
http://dx.doi.org/10.1007/s00231-002-0317-8 [17] Israel-Cookey, C., Ogulu, A. and Omubo-Pepple, V.B. (2003) Influence of Viscous Dissipation and Radiation on Unsteady MHD Free-Convection Flow past an Infinite Heated Vertical Plate in a Porous Medium with Time-Dependent Suction. International Journal of Heat and Mass Transfer, 46, 2305-2311.
http://dx.doi.org/10.1016/S0017-9310(02)00544-6 [18] Chamkha, A.J. (2004) Unsteady MHD Convective Heat and Mass Transfer past a Semi-Infinite Vertical Permeable Moving Plate with Heat Absorption. International Journal of Engineering Science, 42, 217-230.
http://dx.doi.org/10.1016/S0020-7225(03)00285-4 [19] Koriko, O.K., Omowaye, A.J. and Animasaun, I.L. (2015) Effects of Some Thermo-Physical Parameters on Free Convective Heat and Transfer over Vertical Stretching Surface at Absolute Zero. Journal of Heat and Mass Transfer Research, 1, in Press. [20] Na, T.Y. (1979) Computational Methods in Engineering Boundary Value Problems. Academic Press, New York. [21] Cheng, P. (1964) Two Dimensional Radiation Gas Flow by Moment Method. American Institute of Aeronautics and Astronautics Journal, 2, 1662-1664.
http://dx.doi.org/10.2514/3.2645 [22] Incropera, F. (1985) Fundamentals of Heat and Mass Transfer. John Wiley and Sons, New York. [23] Chamkha, A.J., Al-Mudhaf, A.F. and Pop, I. (2006) Effect of Heat Generation or Absorption on Thermophoretic Free Convection Boundary Layer from a Vertical Flat Plate Embedded in a Porous Medium. International Communications in Heat and Mass Transfer, 33, 1096-1102.
http://dx.doi.org/10.1016/j.icheatmasstransfer.2006.04.009 [24] Makinde, O.D. (2005) Free Convection Flow with Thermal Radiation and Mass Transfer past a Moving Vertical Porous Plate. International Communications in Heat and Mass Transfer, 32, 1411-1419.
http://dx.doi.org/10.1016/j.icheatmasstransfer.2005.07.005 [25] Crepeau, J.C. and Clarksean, R. (1997) Similarity Solutions of Natural Convection with Internal Heat Generation. Transactions of ASME—Journal of Heat Transfer, 119, 184-185.
http://dx.doi.org/10.1115/1.2824086 [26] Salem, A.M. and El-Aziz, M.A. (2007) MHD-Mixed Convection and Mass Transfer from a Vertical Stretching Sheet with Diffusion of Chemically Reactive Species and Space- or Temperature-Dependent Heat Source. Canadian Journal of Physics, 85, 359-373.
http://dx.doi.org/10.1139/P07-048 [27] Salem, A.M. and El-Aziz, M.A. (2008) Effect of Hall Currents and Chemical Reaction on Hydromagnetic Flow of a Stretching Vertical Surface with Internal Heat Generation/Absorption. Applied Mathematical Modelling, 32, 1236-1254.
http://dx.doi.org/10.1016/j.apm.2007.03.008 [28] Animasaun, I.L., Adebile, E.A. and Fagbade, A.I. (2015) Casson Fluid Flow with Variable Thermo-Physical Property along Exponentially Stretching Sheet with Suction and Exponentially Decaying Internal Heat Generation Using the Homotopy Analysis Method. Journal of the Nigerian Mathematical Society, in Press.
http://dx.doi.org/10.1016/j.jnnms.2015.02.001 [29] Animasaun, I.L. (2015) Casson Fluid Flow of Variable Viscosity and Thermal Conductivity along Exponentially Stretching Sheet Embedded in a Thermally Stratified Medium with Exponentially Heat Generation. Journal of Heat and Mass Transfer Research, 1, in Press. [30] Batchelor, G.K. (1987) An Introduction to Fluid Dynamics. Cambridge University Press, London. [31] Sivagnana-Prabhu, K.K., Kandasamy, R. and Saravanan, R. (2009) Lie Group Analysis for the Effect of Viscosity and Thermophoresis Particle Deposition on Free Convective Heat and Mass Transfer in the Presence of Suction/Injection. Theoretical and Applied Mechanics, 36, 275-298.
http://dx.doi.org/10.2298/TAM0904275S [32] Animasaun, I.L. and Oyem, A.O. (2014) Effect of Variable Viscosity, Dufour, Soret and Thermal Conductivity on Free Convective Heat and Mass Transfer of Non-Darcian Flow past Porous Flat Surface. American Journal of Computational Mathematics, 4, 357-365.
http://dx.doi.org/10.4236/ajcm.2014.44030 [33] Charraudeau, J. (1975) Influence de gradients de properties physiques en convection force application au cas du tube. International Journal of Heat and Mass Transfer, 18, 87-95.
http://dx.doi.org/10.1016/0017-9310(75)90011-3 [34] Tsai, R. (1999) A Simple Approach for Evaluating the Effect of Wall Suction and Thermophoresis on Aerosol Particle Deposition from a Laminar Flow over a Flat Plate. International Communications in Heat and Mass Transfer, 26, 249-257.
http://dx.doi.org/10.1016/S0735-1933(99)00011-1 [35] Batchelor, G.K. and Shen, C. (1985) Thermophoretic Deposition of Particles in Gas Flowing over Cold Surfaces. Journal of Colloid and Interface Science, 107, 21-37.
http://dx.doi.org/10.1016/0021-9797(85)90145-6 [36] Gill, S. (1951) A Process for the Step-by-Step Integration of Differential Equations in an Automatic Digital Computing Machine. Mathematical Proceedings of the Cambridge Philosophical Society, 47, 96-108.
http://dx.doi.org/10.1017/S0305004100026414 [37] Delin, T. and Zheng, C. (2012) On a General Formula of Fourth Order Runge Kutta Method. Journal of Mathematics Sciences and Mathematics Education, 7, 1-10. [38] Finlayson, B.A. (1980) Nonlinear Analysis in Chemical Engineering. McGrawHill, New York. [39] Animasaun, I.L. (2014) Effects of Some Thermo-Physical Parameters on Free Convective Heat and Mass Transfer over Vertical Surface. Master of Technology Thesis, Federal University of Technology, Akure. [40] Hoffman, J.D. (1992) Numerical Method’s for Engineers and Scientists. McGraw Hill, New York. [41] Alam, M.S. and Rahman, M.M. (2006) Dufour and Soret Effects on Mixed Convection Flow past a Vertical Porous Flat Plate with Variable Suction. Nonlinear Analysis: Modeling and Control, 11, 3-12.
[1] Al-Sharif, A., Chamniprasart, K., Rajagopal, K.R. and Szeri, A.Z. (1993) Lubrication with Binary Mixtures: Liquid-Liquid Emulsion. Journal of Tribology, 115, 46-55.
http://dx.doi.org/10.1115/1.2920985 [2] Wang, S.H., Al-Sharif, A., Rajagopal, K.R. and Szeri, A.Z. (1993) Lubrication with Binary Mixtures: Liquid-Liquid Emulsion in an EHL Conjunction. Journal of Tribology, 115, 515-522.
http://dx.doi.org/10.1115/1.2921668 [3] Makinde, O.D., Olanrewaju, P.O. and Charles, W.M. (2011) Unsteady Convection with Chemical Reaction and Radiative Heat Transfer past a Flat Porous Plate Moving through a Binary Mixture. Afrika Matematika, 21, 1-17. [4] Makinde, O.D. and Olanrewaju, P.O. (2011) Unsteady Mixed Convection with Soret and Dufour Effects past a Porous Plate Moving through a Binary Mixture of Chemically Reacting Fluid. Chemical Engineering Communications, 22, 65-78.
http://dx.doi.org/10.1080/00986445.2011.545296 [5] Sastry, D.R.V.S.R.K. and Murti, A.S.N. (2013) A Double Diffusive Unsteady MHD Convective Flow past a Flat Porous Plate Moving through a Binary Mixture with Suction or Injection. Journal of Fluids, 2013, Article ID: 935156. [6] Somasundaran, P. (2006) Encyclopaedia of Surface and Colloid Science. CRC Press, Boca Raton, 711. [7] Giddings, J.C., Yang, F.J. and Myers, M.N. (1976) Flow Field-Flow Fractionation: A Versatile New Separation Method. Science, 193, 1244-1245.
http://dx.doi.org/10.1126/science.959835 [8] Talbot, L., Cheng, R.K., Scheffer, R.W. and Wills, D.P. (1980) Thermophoresis of Particles in a Heated Boundary Layer. Journal Fluid Mechanics, 101, 737-758.
http://dx.doi.org/10.1017/S0022112080001905 [9] Chamkha, A.J. and Isaa, C. (2000) Effects of Heat Generation/Absorption and Thermophoresis on Hydromagnetic Flow with Heat and Mass Transfer over a Flat Surface. International Journal of Numerical Methods for Heat and Fluid flow, 4, 432-448.
http://dx.doi.org/10.1108/09615530010327404 [10] Animasaun, I.L (2015) Effects of Thermophoresis, Variable Viscosity and Thermal Conductivity on Free Convective Heat and Mass Transfer of Non-Darcian MHD Dissipative Casson Fluid Flow with Suction and nth Order of Chemical Reaction. Journal of the Nigerian Mathematical Society, 34, 11-31.
http://dx.doi.org/10.1016/j.jnnms.2014.10.008 [11] Fagbade, A.I., Falodun, B.O. and Boneze, C.U. (2015) Influence of Magnetic Field, Viscous Dissipation and Thermophoresis on Darcy-Forcheimer Mixed Convection Flow in Fluid Saturated Porous Media. American Journal of Computational Mathematics, 5, 18-40.
http://dx.doi.org/10.4236/ajcm.2015.51002 [12] Singh, A.K. and Dikshit, C.K. (1988) Hydromagnetic Flow past a Continuously Moving Semi-Infinite Plate for Large Suction. Astrophysics and Space Science, 148, 249-256.
http://dx.doi.org/10.1007/BF00645964 [13] Vajravelu, K. and Hadjinicolaou, A. (1993) Heat Transfer in a Viscous Fluid over a Stretching Sheet with Viscous Dissipation and Internal Heat Generation. International Communications in Heat and Mass Transfer, 20, 417-430.
http://dx.doi.org/10.1016/0735-1933(93)90026-R [14] Takhar, H.S., Gorla, R.S.R. and Soundalgekar, V.M. (1996) Radiation Effects on MHD Free Convection Flow of a Gas past a Semi-Infinite Vertical Plate. International Journal of Numerical Methods for Heat and Fluid Flow, 6, 77-83.
http://dx.doi.org/10.1108/09615539610113118 [15] Kim, Y.J. (2001) Unsteady MHD Convection Flow of Polar Fluids past a Vertical Moving Porous Plate in a Porous Medium. International Journal of Heat and Mass Transfer, 44, 2791-2799.
http://dx.doi.org/10.1016/S0017-9310(00)00332-X [16] Raptis, A., Perdikis, C. and Leontitsis, A. (2003) Effects of Radiation in an Optically Thin Gray Gas Flowing past a Vertical Infinite Plate in the Presence of a Magnetic Field. Heat and Mass Transfer, 39, 771-773.
http://dx.doi.org/10.1007/s00231-002-0317-8 [17] Israel-Cookey, C., Ogulu, A. and Omubo-Pepple, V.B. (2003) Influence of Viscous Dissipation and Radiation on Unsteady MHD Free-Convection Flow past an Infinite Heated Vertical Plate in a Porous Medium with Time-Dependent Suction. International Journal of Heat and Mass Transfer, 46, 2305-2311.
http://dx.doi.org/10.1016/S0017-9310(02)00544-6 [18] Chamkha, A.J. (2004) Unsteady MHD Convective Heat and Mass Transfer past a Semi-Infinite Vertical Permeable Moving Plate with Heat Absorption. International Journal of Engineering Science, 42, 217-230.
http://dx.doi.org/10.1016/S0020-7225(03)00285-4 [19] Koriko, O.K., Omowaye, A.J. and Animasaun, I.L. (2015) Effects of Some Thermo-Physical Parameters on Free Convective Heat and Transfer over Vertical Stretching Surface at Absolute Zero. Journal of Heat and Mass Transfer Research, 1, in Press. [20] Na, T.Y. (1979) Computational Methods in Engineering Boundary Value Problems. Academic Press, New York. [21] Cheng, P. (1964) Two Dimensional Radiation Gas Flow by Moment Method. American Institute of Aeronautics and Astronautics Journal, 2, 1662-1664.
http://dx.doi.org/10.2514/3.2645 [22] Incropera, F. (1985) Fundamentals of Heat and Mass Transfer. John Wiley and Sons, New York. [23] Chamkha, A.J., Al-Mudhaf, A.F. and Pop, I. (2006) Effect of Heat Generation or Absorption on Thermophoretic Free Convection Boundary Layer from a Vertical Flat Plate Embedded in a Porous Medium. International Communications in Heat and Mass Transfer, 33, 1096-1102.
http://dx.doi.org/10.1016/j.icheatmasstransfer.2006.04.009 [24] Makinde, O.D. (2005) Free Convection Flow with Thermal Radiation and Mass Transfer past a Moving Vertical Porous Plate. International Communications in Heat and Mass Transfer, 32, 1411-1419.
http://dx.doi.org/10.1016/j.icheatmasstransfer.2005.07.005 [25] Crepeau, J.C. and Clarksean, R. (1997) Similarity Solutions of Natural Convection with Internal Heat Generation. Transactions of ASME—Journal of Heat Transfer, 119, 184-185.
http://dx.doi.org/10.1115/1.2824086 [26] Salem, A.M. and El-Aziz, M.A. (2007) MHD-Mixed Convection and Mass Transfer from a Vertical Stretching Sheet with Diffusion of Chemically Reactive Species and Space- or Temperature-Dependent Heat Source. Canadian Journal of Physics, 85, 359-373.
http://dx.doi.org/10.1139/P07-048 [27] Salem, A.M. and El-Aziz, M.A. (2008) Effect of Hall Currents and Chemical Reaction on Hydromagnetic Flow of a Stretching Vertical Surface with Internal Heat Generation/Absorption. Applied Mathematical Modelling, 32, 1236-1254.
http://dx.doi.org/10.1016/j.apm.2007.03.008 [28] Animasaun, I.L., Adebile, E.A. and Fagbade, A.I. (2015) Casson Fluid Flow with Variable Thermo-Physical Property along Exponentially Stretching Sheet with Suction and Exponentially Decaying Internal Heat Generation Using the Homotopy Analysis Method. Journal of the Nigerian Mathematical Society, in Press.
http://dx.doi.org/10.1016/j.jnnms.2015.02.001 [29] Animasaun, I.L. (2015) Casson Fluid Flow of Variable Viscosity and Thermal Conductivity along Exponentially Stretching Sheet Embedded in a Thermally Stratified Medium with Exponentially Heat Generation. Journal of Heat and Mass Transfer Research, 1, in Press. [30] Batchelor, G.K. (1987) An Introduction to Fluid Dynamics. Cambridge University Press, London. [31] Sivagnana-Prabhu, K.K., Kandasamy, R. and Saravanan, R. (2009) Lie Group Analysis for the Effect of Viscosity and Thermophoresis Particle Deposition on Free Convective Heat and Mass Transfer in the Presence of Suction/Injection. Theoretical and Applied Mechanics, 36, 275-298.
http://dx.doi.org/10.2298/TAM0904275S [32] Animasaun, I.L. and Oyem, A.O. (2014) Effect of Variable Viscosity, Dufour, Soret and Thermal Conductivity on Free Convective Heat and Mass Transfer of Non-Darcian Flow past Porous Flat Surface. American Journal of Computational Mathematics, 4, 357-365.
http://dx.doi.org/10.4236/ajcm.2014.44030 [33] Charraudeau, J. (1975) Influence de gradients de properties physiques en convection force application au cas du tube. International Journal of Heat and Mass Transfer, 18, 87-95.
http://dx.doi.org/10.1016/0017-9310(75)90011-3 [34] Tsai, R. (1999) A Simple Approach for Evaluating the Effect of Wall Suction and Thermophoresis on Aerosol Particle Deposition from a Laminar Flow over a Flat Plate. International Communications in Heat and Mass Transfer, 26, 249-257.
http://dx.doi.org/10.1016/S0735-1933(99)00011-1 [35] Batchelor, G.K. and Shen, C. (1985) Thermophoretic Deposition of Particles in Gas Flowing over Cold Surfaces. Journal of Colloid and Interface Science, 107, 21-37.
http://dx.doi.org/10.1016/0021-9797(85)90145-6 [36] Gill, S. (1951) A Process for the Step-by-Step Integration of Differential Equations in an Automatic Digital Computing Machine. Mathematical Proceedings of the Cambridge Philosophical Society, 47, 96-108.
http://dx.doi.org/10.1017/S0305004100026414 [37] Delin, T. and Zheng, C. (2012) On a General Formula of Fourth Order Runge Kutta Method. Journal of Mathematics Sciences and Mathematics Education, 7, 1-10. [38] Finlayson, B.A. (1980) Nonlinear Analysis in Chemical Engineering. McGrawHill, New York. [39] Animasaun, I.L. (2014) Effects of Some Thermo-Physical Parameters on Free Convective Heat and Mass Transfer over Vertical Surface. Master of Technology Thesis, Federal University of Technology, Akure. [40] Hoffman, J.D. (1992) Numerical Method’s for Engineers and Scientists. McGraw Hill, New York. [41] Alam, M.S. and Rahman, M.M. (2006) Dufour and Soret Effects on Mixed Convection Flow past a Vertical Porous Flat Plate with Variable Suction. Nonlinear Analysis: Modeling and Control, 11, 3-12.