Modified Differential Transform Method (DTM) Simulation of Hydromagnetic Multi-Physical Flow Phenomena from a Rotating Disk

ABSTRACT

A similarity solution for the steady hydromagnetic convective heat and mass transfer with slip flow from a spinning disk with viscous dissipation and Ohmic heating yields a system of non-linear, coupled, ordinary differential equations. These equations are analytically solved by applying a newly developed method namely the**DTM-Padé** technique which is a combination of the **D**ifferential **T**ransform **M**ethod (**DTM**) and the Padé approximation. A full analytical solution is presented, as a benchmark for alternative numerical solutions. **DTM-Padé** is implemented without requiring linearization, discretization, or perturbation, and holds significant potential for solving strongly nonlinear differential equations which arise frequently in fluid dynamics. The regime studied is shown to be controlled by the slip parameter (*γ*), magnetohydrodynamic body force parameter (*M*), Eckert (viscous heating) number (*Ec*), Schmidt number (Sc), Soret number (Sr), Dufour number (*Du*) and Prandtl number (*Pr*). The influence of selected parameters on the evolution of dimensionless velocity, temperature and concentration distributions is studied graphically. Increasing magnetic field (*M*) is found to significantly inhibit the radial (*f*) and tangential (*g*) velocities, but to accentuate the axial velocity field (*h*); furthermore temperature (*θ*) and concentration (*φ*) are both enhanced with increasing *M*. Increasing Soret number (*Sr*) acts to boost the dimensionless concentration (*φ*). Temperatures are significantly elevated in the boundary layer regime with a rise in Eckert number (*Ec*). Excellent correlation between the **DTM-Padé** technique and numerical (shooting) solutions is achieved. The model has important applications in industrial energy systems, process mechanical engineering, electromagnetic materials processing and electro-conductive chemical transport processes.

A similarity solution for the steady hydromagnetic convective heat and mass transfer with slip flow from a spinning disk with viscous dissipation and Ohmic heating yields a system of non-linear, coupled, ordinary differential equations. These equations are analytically solved by applying a newly developed method namely the

KEYWORDS

Differential Transform Method, Padé Approximants, Thermal-Diffusion, Heat Transfer, Soret Effect, Boundary-Layers, Hydromagnetics, Slip, Dissipation, Electromagnetic Processing of Materials

Differential Transform Method, Padé Approximants, Thermal-Diffusion, Heat Transfer, Soret Effect, Boundary-Layers, Hydromagnetics, Slip, Dissipation, Electromagnetic Processing of Materials

Cite this paper

nullM. Rashidi, E. Erfani, O. Bég and S. Ghosh, "Modified Differential Transform Method (DTM) Simulation of Hydromagnetic Multi-Physical Flow Phenomena from a Rotating Disk,"*World Journal of Mechanics*, Vol. 1 No. 5, 2011, pp. 217-230. doi: 10.4236/wjm.2011.15028.

nullM. Rashidi, E. Erfani, O. Bég and S. Ghosh, "Modified Differential Transform Method (DTM) Simulation of Hydromagnetic Multi-Physical Flow Phenomena from a Rotating Disk,"

References

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[2] A. Arikoglu, I. Ozkol and G. Komurgoz, “Effect of Slip on Entropy Generation in a Single Rotating Disk in MHD Flow,” Applied Energy, Vol. 85, 2008, pp. 1225-1236.

[3] E. A. Salem, M. F. Khalil and S. A. Hakim, “Thermohydrodynamic Lubrication of Rotating Magnetohydrodynamic Thrust Bearings,” Wear, Vol. 62, 1980, pp. 337- 348.

[4] J. Zueco and O. A. Bég, “Network Numerical Analysis of Hydromagnetic Squeeze Film Flow Dynamics between Two Parallel Rotating Disks with Induced Magnetic Field Effects,” Tribology International, Vol. 43, 2010, pp. 532-543.

[5] G. Hinds, F. E. Spada, J. M. D. Coey, T. R. N. Mhíocháin and M. E. G. Lyons, “Magnetic Field Effects on Copper Electrolysis,” Journal of Physical Chemistry B, Vol. 105, 2001, pp. 9487-9502.

[6] R. Bessaih, P. Marty and M. Kadja, “Numerical Study of Disk Driven Rotating MHD Flow of a Liquid Metal in a Cylindrical Enclosure,” Acta Mechanica, Vol. 135, 1999, pp. 153-167.

[7] N. Harada, H. Yamasaki and S. Shioda, “High Enthalpy Extraction from a Helium-Driven Disk Magnetohydrodynamic Generator,” American Institute of Aeronautics and Astronautics (AIAA) Journal of Propulsion and Power, Vol. 5, 1989, pp. 353-357.

[8] A. Liberati and Y. Okuno, “Influence of Anode-Region Boundary-Layer Separation on Disk MHD-Generator Performance,” IEEE Transactions on Plasma Science, Vol. 35, 2007, pp. 1588-1597.

[9] P. Intani, T. Sasaki, T. Kikuchi and N. Harada, “Analysis of Disk AC MHD Generator Performance by Finite Element Method,” Journal of Plasma Fusion Research (Japan), Vol. 9, 2010, pp. 580-585.

[10] A. L. Aboul-Hassan and H. A. Attia, “Flow Due to a Rotating Disk with Hall Effect,” Physics Letters A, Vol. 228, 1997, pp. 286-290.

[11] M. Turkyilmazoglu, “Heat and Mass Transfer on the Unsteady Magnetohydrodynamic Flow Due to a Porous Rotating Disk Subject to a Uniform Outer Radial Flow,” American Society of Mechanical Engineers (ASME) Journal of Heat Transfer, Vol. 132, 2010, pp. 1-6.

[12] A. Postelnicu, “Influence of a Magnetic Field on Heat and Mass Transfer by Natural Convection from Vertical Surfaces in Porous Media Considering Soret and Dufour Effects,” International Journal of Heat and Mass Transfer, Vol. 47, 2004, pp. 1467-1472.

[13] R. Bhargava, R. Sharma and O. A. Bég, “Oscillatory Chemically-Reacting MHD Free Convection Heat and Mass Transfer in a Porous Medium with Soret and Dufour Effects: Finite Element Modeling,” Inernational Journal of Applied Mathematics and Mechanics, Vol. 5, 2009, pp. 15-37.

[14] O. A. Bég, A. Y. Bakier and V. R. Prasad, “Numerical Study of Free Convection Magnetohydrodynamic Heat and Mass Transfer from a Stretching Surface to a Saturated Porous Medium with Soret and Dufour Effects,” Computational Materials Science, Vol. 46, 2009, pp. 57-65.

[15] K. A. Maleque and M. A. Sattar, “Similarity Solution of MHD Free-Convective and Mass Transfer Flow over an Vertical Porous Plate with Thermal Diffusion Effects,” Arabian Journal of Science Engineering, Vol. 1, 2002, pp. 44-55.

[16] H. A. Attia, “On the Effectivness of Ion Slip and Uniform Suction or Injection on Steady MHD Flow Due to a Rotating Disk with Heat Transfer and Ohmic Heating,” Chemical Engineering Communications, Vol. 194, 2007, pp. 1396-1407.

[17] J. Zueco, O. A. Bég, H. S. Takhar and V. R. Prasad, “Thermophoretic Hydromagnetic Dissipative Heat and Mass Transfer with Lateral Mass Flux, Heat Source, Ohmic Heating and Thermal Conductivity Effects: Network Simulation Numerical Study,” Applied Thermal Engineering, Vol. 29, 2009, pp. 2808-2815.

[18] O. A. Bég, J. Zueco and H. S. Takhar, “Unsteady Magnetohydrodynamic Hartmann-Couette Flow and Heat Transfer in a Darcian Channel with Hall Current, Ionslip, Viscous and Joule Heating Effects: Network Numerical Solutions,” Comunication Nonlinear Science Numerical Simulation, Vol. 14, 2009, pp. 1082-1097.

[19] E. Osalusi, J. Side and R. Harris, “Thermal-Diffusion and Diffusion-Thermo Effects on Combined Heat and Mass Transfer of a Steady MHD Convective and Slip Flow due to a Rotating Disk with Viscous Dissipation and Ohmic Heating,” International Communications in Heat and Mass Transfer, Vol. 35, 2008, pp. 908-915.

[20] P. Sibanda and O. D. Makinde, “On Steady MHD Flow and Heat Transfer Past a Rotating Disk in a Porous Medium with Ohmic Heating and Viscous Dissipation,” International Journal of Numerical Methods Heat Fluid Flow, Vol. 20, 2010, pp. 269-285.

[21] M. M. Rashidi, M. Keimanesh, O. A. Bég and T. K. Hung, “Magnetohydrodynamic Biorheological Transport Phenomena in a Porous Medium: A Simulation of Mag- netic Blood Flow Control and Filtration,” International Journal for Numerical Methods in Biomedical Engineering, Vol. 27, 2011, pp. 805-821.

[22] J. K. Zhou, “Differential Transformation and Its Applications for Electrical Circuits,” Huazhong University Press, Wuhan, 1986.

[23] M. M. Rashidi and S. Dinarvand, “Purely Analytic Approximate Solutions for Steady Three-Dimensional Problem of Condensation Film on Inclined Rotating Disk by Homotopy Analysis Method,” Nonlinear Analysis Real World Applications, Vol. 10, 2009, pp. 2346-2356.

[24] A. Mehmood, A. Ali, H. S. Takhar, O. A. Bég, M. N. Islam and L. S. Wilson, “Unsteady Von Kármán Swirling Flow: Analytical Study Using the Homotopy Method,” International Journal of Applied Mathematics and Mechanics, Vol. 6, 2010, pp. 67-84.

[25] M. M. Rashidi, D. D. Ganji and S. Dinarvand, “Explicit Analytical Solutions of the Generalized Burger and Burger-Fisher Equations by Homotopy Perturbation Me- thod,” Numerical Methods for Partial Differential Equations, Vol. 25, 2009, pp. 409-417.

[26] A. Rajabi, D. D. Ganji and H. Taherian, “Application of Homotopy Perturbation Method in Nonlinear Heat Conduction and Convection Equations,” Physics Letters A, Vol. 360, 2007, pp. 570-573.

[27] M. M. Rashidi and E. Erfani, “New Analytical Method for Solving Burgers’ and Nonlinear Heat Transfer Equations and Comparison with HAM,” Computer Physics Communications, Vol. 180, 2009, pp. 1539-1544.

[28] C. K. Chen and S. H. Ho, “Applications of the Differential Transform to Eigenvalue Problems,” Applied Mathematics and Computation, Vol. 79, 1996, pp. 173-188.

[29] V. S. Erturk, “Differential Transformation Method for Solving Differential Equations of Lane-Emden Type,” Mathematical and Computational Applications, Vol. 12, 2007, pp. 135-139.

[30] V. S. Erturk, “Application of Differential Transformation Method to Linear Sixth-Order Boundary Value Problems,” Applied Mathematical Sciences, Vol. 1, 2007, pp. 51-58.

[31] M. M. Rashidi and E. Erfani, “Travelling Wave Solutions of WBK Shallow Water Equations by Differential Transform Method,” Advances in Theoretical and Applied Mechanics, Vol. 3, 2010, pp. 263-271.

[32] J. Biazar and F. Mohammadi, “Application of Differential Transform Method to the Sine-Gordon Equation,” International Journal of Nonlinear Science, Vol. 10, 2010, pp. 190-195.

[33] S. Catal, “Solution of Free Vibration Equations of Beams on Elastic Soil by Using Differential Transform Method,” Applied Mathematical Modelling, Vol. 32, 2008, pp. 1744-1757.

[34] Y. Keskin and G. Oturanc, “Numerical Solution of Regularized Long Wave Equation by Reduced Differential Transform Method,” Applied Mathematical Sciences, Vol. 4, 2010, pp. 1221-1231.

[35] L. T. Yu and C. K. Chen, “The Solution of the Blasius Equation by the Differential Transformation Method,” Mathematical and Computer Modelling, Vol. 28, 1998, pp. 101-111.

[36] S. H. Ho and C. K. Chen, “Free Transverse Vibration of An axially Loaded Non-Uniform Spinning Twisted Timoshenko Beam Using Differential Transform,” International Journal of Mechanical Sciences, Vol. 48, 2006, pp. 1323-1331.

[37] H. P. Chu and C. Y. Lo, “Application of the Hybrid Differential Transform-Finite Difference Method to Nonlinear Transient Heat Conduction Problems,” Numerical Heat Transfer, Part A, Vol. 53, 2008, pp. 295-307.

[38] G. A. Baker and P. Graves-Morris, “Padé Approximants,” Encyclopedia of Mathematics and Its Application 13, Parts I and II, Addison-Wesley Publishing Company, New York, 1981.

[39] G. A. Baker, “Essential of Padé Approximants,” Academic Press, London, 1975.

[40] M. M. Rashidi and G. Domairry, “New Analytical Solution of the Three-Dimensional Navier-Stokes Equations,” Modern Physics Letters B, Vol. 26, 2009, pp. 3147-3155.

[41] M. M. Rashidi, “The Modified Differential Transform Method for Solving MHD Boundary-Layer Equations,” Computer Physics Communications, Vol. 180, 2009, pp. 2210-2217.

[42] M. M. Rashidi and E. Erfani, “A Novel Analytical Solution of the Thermal Boundary-Layer over a Flat Plate with a Convective Surface Boundary Condition Using DTM-Padé,” International Conference on Applied Physics and Mathematics (ICAPM), Singapore, 2009.

[43] B. Gebhart and L. Pera, “The Nature of Vertical Natural Convection Flow from the Combined Buoyancy Effects on Thermal and Mass Diffusion,” International Journal of Heat and Mass Transfer, Vol. 14, 1971, pp. 2025- 2040.

[44] M. Gad-el-Hak, “The Fluid Mechanics of Microdevices: The Freeman Scholar Lecture,” ASME― Journal of Fluids Engineering, Vol. 121, 1999, pp. 5-33.

[45] O. A. Bég, J. Zueco and L. M. Lopez-Ochoa, “Network Numerical Analysis of Optically-Thick Hydromagnetic Slip Flow from a Porous Spinning Disk with Radiation Flux, Variable Thermophysical Properties, and Surface Injection Effects,” Chemical Engineering Communications, Vol. 198, 2010, pp. 360-384.

[46] H. Schlichting, “Boundary-Layer Theory,” 7th Edition, MacGraw-Hill, New York, 1979.

[47] M. M. Rashidi, O. Anwar Bég, M. Asadi and M. T. Rastegari, “DTM-Padé Modeling of Natural Convective Boundary Layer Flow of a Nanofluid Past a Vertical Surface,” International Journal of Thermal & Environmental Engineering, Vol. 4, 2012, pp. 13-24. (in press)

[1] [1] P. O. Tsatsin and V. P. Beskachko, “The Influence of Magnetic Field on the Inertial Deposition of a Particle on a Rotating Disk,” 13th International Conference on Liquid and Amorphous Metals: Special Volume, Journal of Physics: Conference Series, Vol. 98, 2008, pp. 1-4.

[2] A. Arikoglu, I. Ozkol and G. Komurgoz, “Effect of Slip on Entropy Generation in a Single Rotating Disk in MHD Flow,” Applied Energy, Vol. 85, 2008, pp. 1225-1236.

[3] E. A. Salem, M. F. Khalil and S. A. Hakim, “Thermohydrodynamic Lubrication of Rotating Magnetohydrodynamic Thrust Bearings,” Wear, Vol. 62, 1980, pp. 337- 348.

[4] J. Zueco and O. A. Bég, “Network Numerical Analysis of Hydromagnetic Squeeze Film Flow Dynamics between Two Parallel Rotating Disks with Induced Magnetic Field Effects,” Tribology International, Vol. 43, 2010, pp. 532-543.

[5] G. Hinds, F. E. Spada, J. M. D. Coey, T. R. N. Mhíocháin and M. E. G. Lyons, “Magnetic Field Effects on Copper Electrolysis,” Journal of Physical Chemistry B, Vol. 105, 2001, pp. 9487-9502.

[6] R. Bessaih, P. Marty and M. Kadja, “Numerical Study of Disk Driven Rotating MHD Flow of a Liquid Metal in a Cylindrical Enclosure,” Acta Mechanica, Vol. 135, 1999, pp. 153-167.

[7] N. Harada, H. Yamasaki and S. Shioda, “High Enthalpy Extraction from a Helium-Driven Disk Magnetohydrodynamic Generator,” American Institute of Aeronautics and Astronautics (AIAA) Journal of Propulsion and Power, Vol. 5, 1989, pp. 353-357.

[8] A. Liberati and Y. Okuno, “Influence of Anode-Region Boundary-Layer Separation on Disk MHD-Generator Performance,” IEEE Transactions on Plasma Science, Vol. 35, 2007, pp. 1588-1597.

[9] P. Intani, T. Sasaki, T. Kikuchi and N. Harada, “Analysis of Disk AC MHD Generator Performance by Finite Element Method,” Journal of Plasma Fusion Research (Japan), Vol. 9, 2010, pp. 580-585.

[10] A. L. Aboul-Hassan and H. A. Attia, “Flow Due to a Rotating Disk with Hall Effect,” Physics Letters A, Vol. 228, 1997, pp. 286-290.

[11] M. Turkyilmazoglu, “Heat and Mass Transfer on the Unsteady Magnetohydrodynamic Flow Due to a Porous Rotating Disk Subject to a Uniform Outer Radial Flow,” American Society of Mechanical Engineers (ASME) Journal of Heat Transfer, Vol. 132, 2010, pp. 1-6.

[12] A. Postelnicu, “Influence of a Magnetic Field on Heat and Mass Transfer by Natural Convection from Vertical Surfaces in Porous Media Considering Soret and Dufour Effects,” International Journal of Heat and Mass Transfer, Vol. 47, 2004, pp. 1467-1472.

[13] R. Bhargava, R. Sharma and O. A. Bég, “Oscillatory Chemically-Reacting MHD Free Convection Heat and Mass Transfer in a Porous Medium with Soret and Dufour Effects: Finite Element Modeling,” Inernational Journal of Applied Mathematics and Mechanics, Vol. 5, 2009, pp. 15-37.

[14] O. A. Bég, A. Y. Bakier and V. R. Prasad, “Numerical Study of Free Convection Magnetohydrodynamic Heat and Mass Transfer from a Stretching Surface to a Saturated Porous Medium with Soret and Dufour Effects,” Computational Materials Science, Vol. 46, 2009, pp. 57-65.

[15] K. A. Maleque and M. A. Sattar, “Similarity Solution of MHD Free-Convective and Mass Transfer Flow over an Vertical Porous Plate with Thermal Diffusion Effects,” Arabian Journal of Science Engineering, Vol. 1, 2002, pp. 44-55.

[16] H. A. Attia, “On the Effectivness of Ion Slip and Uniform Suction or Injection on Steady MHD Flow Due to a Rotating Disk with Heat Transfer and Ohmic Heating,” Chemical Engineering Communications, Vol. 194, 2007, pp. 1396-1407.

[17] J. Zueco, O. A. Bég, H. S. Takhar and V. R. Prasad, “Thermophoretic Hydromagnetic Dissipative Heat and Mass Transfer with Lateral Mass Flux, Heat Source, Ohmic Heating and Thermal Conductivity Effects: Network Simulation Numerical Study,” Applied Thermal Engineering, Vol. 29, 2009, pp. 2808-2815.

[18] O. A. Bég, J. Zueco and H. S. Takhar, “Unsteady Magnetohydrodynamic Hartmann-Couette Flow and Heat Transfer in a Darcian Channel with Hall Current, Ionslip, Viscous and Joule Heating Effects: Network Numerical Solutions,” Comunication Nonlinear Science Numerical Simulation, Vol. 14, 2009, pp. 1082-1097.

[19] E. Osalusi, J. Side and R. Harris, “Thermal-Diffusion and Diffusion-Thermo Effects on Combined Heat and Mass Transfer of a Steady MHD Convective and Slip Flow due to a Rotating Disk with Viscous Dissipation and Ohmic Heating,” International Communications in Heat and Mass Transfer, Vol. 35, 2008, pp. 908-915.

[20] P. Sibanda and O. D. Makinde, “On Steady MHD Flow and Heat Transfer Past a Rotating Disk in a Porous Medium with Ohmic Heating and Viscous Dissipation,” International Journal of Numerical Methods Heat Fluid Flow, Vol. 20, 2010, pp. 269-285.

[21] M. M. Rashidi, M. Keimanesh, O. A. Bég and T. K. Hung, “Magnetohydrodynamic Biorheological Transport Phenomena in a Porous Medium: A Simulation of Mag- netic Blood Flow Control and Filtration,” International Journal for Numerical Methods in Biomedical Engineering, Vol. 27, 2011, pp. 805-821.

[22] J. K. Zhou, “Differential Transformation and Its Applications for Electrical Circuits,” Huazhong University Press, Wuhan, 1986.

[23] M. M. Rashidi and S. Dinarvand, “Purely Analytic Approximate Solutions for Steady Three-Dimensional Problem of Condensation Film on Inclined Rotating Disk by Homotopy Analysis Method,” Nonlinear Analysis Real World Applications, Vol. 10, 2009, pp. 2346-2356.

[24] A. Mehmood, A. Ali, H. S. Takhar, O. A. Bég, M. N. Islam and L. S. Wilson, “Unsteady Von Kármán Swirling Flow: Analytical Study Using the Homotopy Method,” International Journal of Applied Mathematics and Mechanics, Vol. 6, 2010, pp. 67-84.

[25] M. M. Rashidi, D. D. Ganji and S. Dinarvand, “Explicit Analytical Solutions of the Generalized Burger and Burger-Fisher Equations by Homotopy Perturbation Me- thod,” Numerical Methods for Partial Differential Equations, Vol. 25, 2009, pp. 409-417.

[26] A. Rajabi, D. D. Ganji and H. Taherian, “Application of Homotopy Perturbation Method in Nonlinear Heat Conduction and Convection Equations,” Physics Letters A, Vol. 360, 2007, pp. 570-573.

[27] M. M. Rashidi and E. Erfani, “New Analytical Method for Solving Burgers’ and Nonlinear Heat Transfer Equations and Comparison with HAM,” Computer Physics Communications, Vol. 180, 2009, pp. 1539-1544.

[28] C. K. Chen and S. H. Ho, “Applications of the Differential Transform to Eigenvalue Problems,” Applied Mathematics and Computation, Vol. 79, 1996, pp. 173-188.

[29] V. S. Erturk, “Differential Transformation Method for Solving Differential Equations of Lane-Emden Type,” Mathematical and Computational Applications, Vol. 12, 2007, pp. 135-139.

[30] V. S. Erturk, “Application of Differential Transformation Method to Linear Sixth-Order Boundary Value Problems,” Applied Mathematical Sciences, Vol. 1, 2007, pp. 51-58.

[31] M. M. Rashidi and E. Erfani, “Travelling Wave Solutions of WBK Shallow Water Equations by Differential Transform Method,” Advances in Theoretical and Applied Mechanics, Vol. 3, 2010, pp. 263-271.

[32] J. Biazar and F. Mohammadi, “Application of Differential Transform Method to the Sine-Gordon Equation,” International Journal of Nonlinear Science, Vol. 10, 2010, pp. 190-195.

[33] S. Catal, “Solution of Free Vibration Equations of Beams on Elastic Soil by Using Differential Transform Method,” Applied Mathematical Modelling, Vol. 32, 2008, pp. 1744-1757.

[34] Y. Keskin and G. Oturanc, “Numerical Solution of Regularized Long Wave Equation by Reduced Differential Transform Method,” Applied Mathematical Sciences, Vol. 4, 2010, pp. 1221-1231.

[35] L. T. Yu and C. K. Chen, “The Solution of the Blasius Equation by the Differential Transformation Method,” Mathematical and Computer Modelling, Vol. 28, 1998, pp. 101-111.

[36] S. H. Ho and C. K. Chen, “Free Transverse Vibration of An axially Loaded Non-Uniform Spinning Twisted Timoshenko Beam Using Differential Transform,” International Journal of Mechanical Sciences, Vol. 48, 2006, pp. 1323-1331.

[37] H. P. Chu and C. Y. Lo, “Application of the Hybrid Differential Transform-Finite Difference Method to Nonlinear Transient Heat Conduction Problems,” Numerical Heat Transfer, Part A, Vol. 53, 2008, pp. 295-307.

[38] G. A. Baker and P. Graves-Morris, “Padé Approximants,” Encyclopedia of Mathematics and Its Application 13, Parts I and II, Addison-Wesley Publishing Company, New York, 1981.

[39] G. A. Baker, “Essential of Padé Approximants,” Academic Press, London, 1975.

[40] M. M. Rashidi and G. Domairry, “New Analytical Solution of the Three-Dimensional Navier-Stokes Equations,” Modern Physics Letters B, Vol. 26, 2009, pp. 3147-3155.

[41] M. M. Rashidi, “The Modified Differential Transform Method for Solving MHD Boundary-Layer Equations,” Computer Physics Communications, Vol. 180, 2009, pp. 2210-2217.

[42] M. M. Rashidi and E. Erfani, “A Novel Analytical Solution of the Thermal Boundary-Layer over a Flat Plate with a Convective Surface Boundary Condition Using DTM-Padé,” International Conference on Applied Physics and Mathematics (ICAPM), Singapore, 2009.

[43] B. Gebhart and L. Pera, “The Nature of Vertical Natural Convection Flow from the Combined Buoyancy Effects on Thermal and Mass Diffusion,” International Journal of Heat and Mass Transfer, Vol. 14, 1971, pp. 2025- 2040.

[44] M. Gad-el-Hak, “The Fluid Mechanics of Microdevices: The Freeman Scholar Lecture,” ASME― Journal of Fluids Engineering, Vol. 121, 1999, pp. 5-33.

[45] O. A. Bég, J. Zueco and L. M. Lopez-Ochoa, “Network Numerical Analysis of Optically-Thick Hydromagnetic Slip Flow from a Porous Spinning Disk with Radiation Flux, Variable Thermophysical Properties, and Surface Injection Effects,” Chemical Engineering Communications, Vol. 198, 2010, pp. 360-384.

[46] H. Schlichting, “Boundary-Layer Theory,” 7th Edition, MacGraw-Hill, New York, 1979.

[47] M. M. Rashidi, O. Anwar Bég, M. Asadi and M. T. Rastegari, “DTM-Padé Modeling of Natural Convective Boundary Layer Flow of a Nanofluid Past a Vertical Surface,” International Journal of Thermal & Environmental Engineering, Vol. 4, 2012, pp. 13-24. (in press)