Semi Analytical Solution of Boundary-Layer Flow of a Micropolar Fluid through a Porous Channel

Mohammad Mehdi RASHIDI; Najib LARAQI; Seyed Majid SADRI

Authors

Mohammad Mehdi RASHIDI Mechanical Engineering Department, Engineering Faculty of Bu-Ali Sina University, Hamedan
Najib LARAQI Universite Paris 10, Laboratoire TIE, EA 4415, Departement GTE, 50 Rue de Sevres, F92410 Ville d'Avray
Seyed Majid SADRI Mechanical Engineering Department, Engineering Faculty of Bu-Ali Sina University, Hamedan

Keywords:

Micropolar fluid, differential transform method (DTM), boundary-layer, similarity transformation

Abstract

In this study, the problem of boundary-layer flow of a micropolar fluid through a porous channel is dealt with. The flow occurs due to suction and injection at the walls of the channel. The micropolar fluid fills the space inside the channel. The similarity transformations are applied to reduce governing partial differential equations (PDEs) to a set of nonlinear coupled ordinary differential equations (ODEs) in dimensionless form. An efficient mathematical technique, called the differential transform method (DTM), is used to solve the nonlinear differential equations governing the problem in the form of series with easily computable terms. Graphical results are presented to investigate the influence of the pertinent parameters on the velocity and micro-rotation. The results illustrate the reliability and the performance of the DTM in comparison with the numerical method (forth-order Runge-Kutta) in solving this problem.

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Author Biography

Mohammad Mehdi RASHIDI, Mechanical Engineering Department, Engineering Faculty of Bu-Ali Sina University, Hamedan

Mohammad Mehdi Rashidi, Ph.D. Associate Professor Bu-Ali Sina University Department of Mechanical Engineering P.O. Box 65175-4161 Hamedan, Iran mm.rashidi@usherbrooke.ca mm_rashidi@yahoo.com Tel.: +98 811 8257409; Fax: +98 811 8257400 http://profs.basu.ac.ir/rashidi/index.php?L=en&pc 1- M.M. Rashidi, H. Shahmohamadi, S. Dinarvand, Analytic Approximate Solutions for Unsteady Two-Dimensional and Axisymmetric Squeezing Flows between Parallel Plates, Mathematical Problems in Engineering, Volume 2008 (2008), Article ID 935095, 13 pages. 2- S. Dinarvand, M.M. Rashidi, A. Doosthoseini, Analytical approximate solutions for two-dimensional viscous flow through expanding or contracting gaps with permeable walls, Central European Journal of Physics 7 (4) (2009). 3- M.M. Rashidi, H. Shahmohamadi, Analytical solution of three-dimensional Navier–Stokes equations for the flow near an infinite rotating disk, Communications in Nonlinear Science and Numerical Simulation 14 (2009) 2999–3006. 4- M.M. Rashidi, G. Domairry, S. Dinarvand, Approximate solutions for the Burger and regularized long wave equations by means of the homotopy analysis method, Communications in Nonlinear Science and Numerical Simulation 14 (2009) 708–717. 5- M.M. Rashidi, D.D. Ganji, S. Dinarvand, Approximate Traveling Wave Solutions of Coupled Whitham-Broer-Kaup Shallow Water Equations by Homotopy Analysis Method, Differential Equations and Nonlinear Mechanics, Volume 2008, Article ID 243459, 8 pages. 6- M.M. Rashidi, D.D. Ganji, S. Dinarvand, Explicit Analytical Solutions of the Generalized Burger and Burger–Fisher Equations by Homotopy Perturbation Method, Numerical Methods for Partial Differential Equations 25 (2) (2009) 409-417. 7- M.M. Rashidi, G. Domairry, S. Dinarvand, The Homotopy Analysis Method for Explicit Analytical Solutions of Jaulent–Miodek Equations, Numerical Methods for Partial Differential Equations 25 (2) (2009) 430-439. 8- M.M. Rashidi, 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 10 (2009) 2346–2356. 9- S. Dinarvand, M.M. Rashidi, A reliable treatment of homotopy analysis method for two-dimensional viscous flow in a rectangular domain bounded by two moving porous walls, Nonlinear Analysis: Real World Applications 11 (2010) 1502-1512. 10- M.M. Rashidi, G. Domairry, New Analytical Solution of the Three-Dimensional Navier-Stokes Equations, Modern Physics Letters B 23 (26) (2009) 3147-3155. 11- S. Dinarvand, A. DoostHoseini, E. DoostHoseini, M.M. Rashidi, Series solutions for unsteady laminar MHD flow near forward stagnation point of an impulsively rotating and translating sphere in presence of buoyancy forces, Nonlinear Analysis: Real World Applications 11 (2010) 1159-1169. 12- M.M. Rashidi, E. Erfani, New analytical method for solving Burgers' and nonlinear heat transfer equations and comparison with HAM, Computer Physics Communications 180 (2009) 1539–1544. 13- M.M. Rashidi, The modified differential transform method for solving MHD boundary-layer equations, Computer Physics Communications180 (2009) 2210–2217. 14- M.M. Rashidi, G. Domairry, A. DoostHosseini, S. Dinarvand, Explicit Approximate Solution of the Coupled Kid Equations by using the Homotopy Analysis Method, Int. Journal of Math. Analysis 2 (12) (2008) 581–589. 15- M.M. Rashidi, H. Shahmohamadi, G. Domairry, Variational Iteration Method for Solving Three-Dimensional Navier–Stokes Equations of Flow Between Two Stretchable Disks, Numerical Methods for Partial Differential Equations 27 (2) (2011) 292–301. 16- S. Dinarvand, M.M. Rashidi, H. Shahmohamadi, Analytic approximate solution of three-dimensional Navier–Stokes equations of flow between two stretchable disks, Numerical Methods for Partial Differential Equations 26 (6) (2010) 1594–1607. 17- S. Dinarvand, S. Khosravi, A. Doosthoseini, M.M. Rashidi, The Homotopy Analysis Method for Solving the Sawada–Kotera and Lax’s Fifth-Order KdV Equations, Adv. Theor. Appl. Mech. 1 (7) (2008) 327 – 335. 18- S. Dinarvand, A. Doosthoseini, E. Doosthoseini, M.M. Rashidi, Comparison of HAM and HPM Methods for Berman’s Model of Two-Dimensional Viscous Flow in Porous Channel with Wall Suction or Injection, Adv. Theor. Appl. Mech. 1 (7) (2008) 337 – 347. 19- M.M. Rashidi, M.M. Bastani, NUMERICAL SIMULATION OF AXISYMMETRIC SUPERSONIC VISCOUS FLOW OVER BLUNT CONE BY USING DIAGONAL FOURTH ORDER FINITE DIFFERENCE METHOD, In Persian, Iranian Journal of Engineering Science 19 (10) (1387) (2009) 169-180. 20- M.M. Rashidi, D.D. Ganji, Homotopy Perturbation Combined with Pad´e Approximation for Solving Two Dimensional Viscous Flow in the Extrusion Process, International Journal of Nonlinear Science 7 (4) (2009) 387-394. 21- M.M. Rashidi, S.A. Mohimanian Pour, Analytic Approximate Solutions for Unsteady Boundary-Layer Flow and Heat Transfer due to a Stretching Sheet by Homotopy Analysis Method, Nonlinear Analysis: Modelling and Control 15 (1) (2010) 83–95. …………. 22- M.M. Rashidi, S. Dinarvand, Explicit and Analytical Traveling Wave Solutions of Whitham–Broer–Kaup Shallow Water Equations by homotopy Perturbation Method, International Journal of Non-Linear dynamics and Engineering and Science, (2008). 23- M.M. Rashid, D.D. Ganji, Analytical Solution of Three-Dimensional Navier-Stokes Equations Using HPM-Padé, International Journal of Nonlinear Dynamics in Engineering and Science, In press. 24. M.M. Rashid, M. Keimanesh, Using differential transform method and Padé approximant for solving MHD flow in a laminar liquid film from a horizontal stretching surface, Mathematical Problems in Engineering, Volume 2010 (2010), Article ID 491319, 14 pages. 25. M.M. Rashidi, D.D. Ganji, Homotopy Perturbation Combined with Pad´e Approximation for Solving Two Dimensional Viscous Flow in the Extrusion, International Journal of Nonlinear Science 7 (4) (2009) 387-394. 26. M.M. Rashidi, S.M. Sadri, Solution of the Laminar Viscous Flow in a Semi-Porous Channel in the Presence of a Uniform Magnetic Field by Using the Differential Transform Method, Int. J. Contemp. Math. Sciences 5 (15) (2010) 711–720. 27. H. Shahmohamadi, M.M. Rashidi, A Novel Solution for the Glauert-jet problem by variational iteration method-Padé approximant, Mathematical Problems in Engineering Volume 2010 (2010), Article ID 501476, 7 pages. 28. M.M. Rashidi, E. Erfani, A NOVEL ANALYTICAL METHOD TO INVESTIGATE EFFECT OF RADIATION ON FLOW OF A MAGNETO-MICROPOLAR FLUID PAST A CONTINUOUSLY MOVING PLATE WITH SUCTION AND BLOWING, International Journal of Modeling, Simulation, and Scientific Computing 1 (2010) 219-238. 29. M.M. Rashidi, E. Erfani, Modified differential transform method for solving steady MHD convective and slip flow due to a rotating disk with viscous dissipation and Ohmic heating, International Journal for Computational Methods in Engineering Science & Mechanics, In press. 30. M.M. Rashidi, E. Erfani, Traveling Wave Solutions of WBK Shallow Water Equations by Differential Transform Method, Adv. Theor. Appl. Mech. 3(2010) 6 263 - 271. 31. M.M. Rashidi, S.M. Sadri, New Analytical Solution of Two-Dimensional Viscous Flow in a Rectangular Domain Bounded by Two Moving Porous Walls, International Journal for Computational Methods in Engineering Science & Mechanics 12 (2011) 26–33. 32. M.M. Rashidi, D.D. Ganji and S.M. Sadri, New Analytical Solution of Stagnation Point Flow in a Porous Medium, Journal of Porous Media 14 (12) (2011) 1125–1135. 33. M.M. Rashidi, S.A. Mohimanian Pour, Explicit Solution of Axisymmetric Stagnation Flow Towards a Shrinking Sheet by DTM-Padé, Applied Mathematical Sciences 4 (53) (2010) 2617-2632. 34. M.M. Rashidi, S.A. Mohimanian Pour, A Novel Analytical Solution of Heat Transfer of a Micropolar Fluid through a Porous Medium with Radiation by DTM-Padé, Heat Transfer-Asian Research 39 (8) (2010) 575-589. 35. M.M. Rashidi, N. Laraqi, S.M. Sadri, A Novel Analytical Solution of Mixed Convection about an Inclined Flat Plate Embedded in a Porous Medium Using the DTM-Padé, International Journal of Thermal Sciences 49 (2010) 2405-2412. 36. M.M. Rashidi, E. Erfani, The modified differential transform method for investigating nano boundary-layers over stretching surfaces, International Journal of Numerical Methods for Heat and Fluid Flow 21 (7) (2011) 864-883. 37. M.M. Rashidi and S.A. Mohimanian Pour, Analytic solution of steady three-dimensional problem of condensation film on inclined rotating disk by differential transform method, Mathematical Problems in Engineering, Volume 2010 (2010), Article ID 613230, 15 pages. 38. M.M. Rashidi, S.A. Mohimanian pour, S. Abbasbandy, Analytic Approximate Solutions for Heat Transfer of a Micropolar Fluid through a Porous Medium with Radiation, Communications in Nonlinear Science and Numerical Simulations 16 (2011) 1874–1889. 39. M.M. Rashidi, E. Erfani, A new analytical study of MHD stagnation–point flow in porous media with heat transfer, Computers & Fluids 40 (2011) 172–178. 40. M.M. Rashidi, S.A. Mohimanian Pour, N. Laraqi, A Semi-Analytical Solution of Micro Polar Flow in a Porous Channel with Mass Injection by using Differential Transform Method, NONLINEAR ANALYSIS: MODELLING AND CONTROL 15 (3) (2010) 341–350. 41. E. Erfani, M.M. Rashidi, A. Basiri parsa, The modified Differential Transform Method for solving off-centered stagnation flow towards a rotating disc, International Journal of Computational Methods 7 (4) (2010) 655-670. 42. M.M. Rashidi, S.A. Mohimanian Pour, A novel analytical solution of steady flow over a rotating disk in porous medium with heat transfer by DTM-PADÉ, African Journal of Mathematics and Computer Science Research 3(6) (2010) 93–100. 43. M.M. Rashidi, M. Keimanesh, O. Anwar Bég, T.K. Hung, Magnetohydrodynamic BioRheological Transport Phenomena in a Porous Medium: A simulation of magnetic blood flow control and filtration, International Journal for Numerical Methods in Biomedical Engineering 27 (6) (2011) 805–821. 44. M.M. Rashidi, O. Anwar Bég, A. Habibzadeh, First and second law analysis of an ejector expansion Joule-Thomson cryogenic refrigeration cycle, International Journal of Energy Research, Article first published online: 23 NOV 2010 DOI: 10.1002/er.1800. 45- M.M. Rashidi, N. Laraqi, A. Basiri Parsa, Analytical Modeling of heat convection in magnetized micropolar fluid by using modified differential transform method, Heat Transfer-Asian Research 40 (3) (2011) 187-204. 46- M.M. Rashidi, A.M. Siddiqui, M. Asadi, Application of homotopy analysis method to the unsteady squeezing flow of a second grade fluid between circular plates, Mathematical Problems in Engineering, Volume 2010, Article ID 706840, 18 pages, doi:10.1155/2010/706840. 47- M.M. Rashidi, D.D. Ganji, H. Shahmohamadi, Variational Iteration Method for two-dimensional steady slip flow in micro-channels, Archive of Applied Mechanics 81 (11) (2011) 1597-1605. 48- M. Hamraoui, T. Osman, A. Boucheffa, M.M. Rashidi, Analytical modelling of the threedimensional steady-state temperature in a bearing ring, Mecanique & Industrie 12 (1–4) (2011). 49- M.M. Rashidi, S.A. Mohimanian Pour, Analytic Approximate Solutions for Steady Flow over a Rotating Disk in Porous Medium by Homotopy Analysis Method with Two Auxiliary Parameters, International Journal of Applied Mathematics & Statistics, In press. 50- H. Shahmohamadi, M.M. Rashidi, New Solution for Steady Flow over a Rotating Disk in Porous Medium with Heat Transfer, Progress in Applied Mathematics 1(1) (2011). 51- M.M. Rashidi, N. Rahimzadeh, N. Laraqi, Evaluation of equations of state by using exergy for Air, Nitrogen and Oxygen on throttle reduction efficiency, International Journal of Exergy 9 (3) (2011) 297-318. 52- M.M. Rashidi, T. Hayat, E. Erfani, S.A. Mohimanian Pour, Awatif A-Hendi, Simultaneous effects of partial slip and thermal-diffusion and diffusion-thermo on steady MHD convective flow due to a rotating disk, Communications in Nonlinear Science and Numerical Simulations 16 (2011) 4303–4317. 53- M.M. Rashidi, O. Anwar Bég, M. Asadi, M.T. Rastegari, DTM- Padé Modeling of Natural Convective Boundary Layer Flow of a Nanofluid Past a Vertical Surface, International Journal of Thermal and Environmental Engineering 4 (1) (2012) 13-24. 54- M.M. Rashidi, M.M. Bastani, O. Anwar Bég, Numerical Simulation of Axisymmetric Supersonic Viscous Flow over a Blunt Cone with a Diagonal Fourth Order Finite Difference Method, Proceedings of the Institution of Mechanical Engineers, Part G, Journal of Aerospace Engineering, Published online before print November 11, 2011, doi:10.1177/0954410011408303. 55- M.M. Rashidi, E. Erfani, Analytical method for solving steady MHD convective and slip flow due to a rotating disk with viscous dissipation and Ohmic heating, Engineering Computations, In press. 56- M.M. Rashidi, O. Anwar Bég, A. Basiri Parsa, F. Nazari, Analysis and optimization of a transcritical power cycle with regenerator using artificial neural networks and genetic algorithms, PROC. IMECHE - PART A: J. POWER AND ENERGY 225 (6) (2011) 701-717. 57- M.M. Rashidi, M. Keimanesh, S.C. Rajvanshi, Study of Pulsatile flow in a porous annulus with the homotopy analysis method, International Journal of Numerical Methods for Heat and Fluid Flow, In press. 58- M.M. Rashidi, D.D. Ganji, Homotopy perturbation method for solving flow in the extrusion processes, Iranian Journal of Engineering Science 23 (3&4) (2010) 267-272. 59- N. Laraqi, M.M. Rashidi, J.M. Garcia de Maria, A. Baïri, Analytical model for the thermo-hydrodynamic behaviour of a thin lubricant film, Tribology International 44 (2011) 1083–1086. 60- M.M. Rashidi, T. Hayat, A. Basiri parsa, Solving of boundary-layer equations with transpiration effects, governance on a vertical permeable cylinder using modified differential transform method, Heat Transfer-Asian Research 40 (8) (2011) 677–692. 61- M.M. Rashidi, O. Anwar Bég, N. Rahimzadeh, A generalized Differential Transform Method for combined free and forced convection flow about inclined surfaces in porous media, Chemical Engineering Communications 199 (2) (2011) 257-282. 62- M.M. Rashidi, O. Anwar Bég, M.T. Rastegari, A study of non-Newtonian flow and heat transfer over a non-isothermal wedge using the Homotopy Analysis Method, Chemical Engineering Communications 199 (2) (2011) 231-256. 63- M.M. Rashidi, Ali J. Chamkha, M. Keimanesh, Application of multi-step differential transform method on flow of a second-grade fluid over a stretching or shrinking sheet, American Journal of Computational Mathematics 1 (2) (2011) 119-128. 64- A. Basiri parsa, M.M. Rashidi, O. Anwar Bég and M. Asadi, Approximate Explicit Solutions of Steady Similar Sakiadis Boundary-Layer Flows with the Homotopy Analysis Method, Advances in Applied Science Research, In press. 65- M.M. Rashidi, N. Galanis, F. Nazari, A. Basiri Parsa, L. Shamekhi, Parametric Analysis and Optimization of Regenerative Clausius and Organic Rankine Cycles with Two Feedwater Heaters Using Artificial Bees Colony and Artificial Neural Network, Energy 36 (2011) 5728-574. 66- O. Anwar Bég, M.M. Rashidi, T.A. Bég, M. Asadi, Homotopy Analysis of Transient Magneto-Bio-Fluid Dynamics of Micropolar Squeeze Film in a Porous Medium: A Model for Magneto-Bio-Rheological Lubrication, Journal of Mechanics in Medicine and Biology, DOI No: 10.1142/S0219519411004642 Accepted 2011-07-12. 67- M. Keimanesh, M.M. Rashidi, Ali J. Chamkha, R. Jafari, Study of a third grade non-Newtonian fluid flow between two parallel plates using the Multi-step differential transform method, Computers and Mathematics with Applications 62 (2011) 2871–2891. 68- M.M. Rashidi, S.A. Mohimanian Pour, T. Hayat, S. Obaidat, Analytic Approximate Solutions for Steady Flow over a Rotating Disk in Porous Medium with Heat Transfer by Homotopy Analysis Method, Computers and Fluids 54 (2012) 1–9. 69- M. Ferdows, Md. Jashim Uddin, M.M. Rashidi, N. Rahimzadeh, Numerical Analysis of Mixed Convection over Horizontal Moving Porous Flat Plate by the Method of One Parameter Continuous Group Theory, International Journal of Numerical Methods for Heat and Fluid Flow, In press. 70- M.M. Rashidi, T. Hayat, M. Keimanesh , A.A. Hendi, New analytical method for the study of natural convection flow of a non-Newtonian, International Journal of Numerical Methods for Heat and Fluid Flow, In press. 71- H. Shahmohamadi, M.M. Rashidi, Explicit Solutions for Steady Three-Dimensional Problem of Condensation Film on Inclined Rotating Disk, International Journal of Fluid Mechanics Research 38 (5.30) (2011) 424-436. 72- M.M. Rashidi, M. Keimanesh, S.C. Rajvanshi, S. Wasu, Pulsatile flow through annular space bounded by outer porous cylinder and an inner cylinder of permeable material, International Journal for Computational Methods in Engineering Science & Mechanics, In press. 73- M.M. Rashidi T. Hayat and A. Basiri parsa, MHD boundary-layer fow over a stretching surface with internal heat generation or absorption, International Journal of Applied Mathematics & Statistics, In press. 74- M.M. Rashidi, E. Erfani, O. Anwar Bég, S.K. Ghosh, Modified Differential Transform Method (DTM) Simulation of Hydromagnetic Multi-Physical Flow Phenomena from a Rotating Disk, World Journal of Mechanics 1 (2011) 217-230. 75- M.M. Rashidi, A.M. Siddiqui, M.T. Rastegari, Analytical Solution of Squeezing Flow between Two Circular Plates, International Journal for Computational Methods in Engineering Science & Mechanics, In press. 76- M.M. Rashidi, Group theory and differential transform analysis of mixed convective heat and mass transfer from a horizontal surface with chemical reaction effects, Chemical Engineering Communications, In press. 77- T. A. Bég, M.M. Rashidi, O. Anwar Bég, N. Rahimzadeh, Differential transform semi-numerical analysis of biofluid-particle suspension flow and heat transfer in non-Darcian porous mediad, Computer Methods in Biomechanics and Biomedical Engineering, In press. 78- O. Anwar Bég, M.M. Rashidi, A. Aziz and M. Keimanesh, DIFFERENTIAL TRANSFORM STUDY OF HYPERSONIC LAMINAR BOUNDARY LAYER AERODYNAMIC FLOW AND HEAT TRANSFER OVER SLENDER AXISYMMETRIC BODIES OF REVOLUTION, International Journal of Applied Mathematics and Mechanics, In press. 79- H. Shahmohamadi, M. M. Rashidi and O. Anwar Bég, A NEW TECHNIQUE FOR SOLVING STEADY FLOW AND HEAT TRANSFER FROM A ROTATING DISK IN HIGH PERMEABILITY MEDIA, International Journal of Applied Mathematics and Mechanics, In press.

References

AC Eringen. Theory of micropolar fluids. Int. J. Math. Mech. 1966; 16, 1-18.

B Hadimoto and T Tokioka. Two-dimensional shear flows of linear micropolar fluids. Int. J. Eng. Sci. 1969; 7, 515-22.

FL Wood, M Benchaitra and S Friberg. Study of lyotropic liquid crystals in viscometric flow and clastohydrodynamic contact. ASLE Tri. Trans. 1987; 30, 539-48.

T Ariman, MA Turk and ND Sylvester. Microcontinuum fluid mechanics - a review. Int. J. Eng. Sci. 1974; 12, 273-93.

AC Eringen. Microcontinuum Field Theories II. Fluent Media, Springer, New York, 2001.

G Lukaszewicz. Micropolar Fluids. Theory and Applications, Basel. Birkhauser, 1999.

AS Berman. Laminar flow in a channel with porous walls. J. Appl. Phys. 1953; 24, 1232-5.

A Desseaus. Influence of a magnetic field over a laminar viscous flow in a semi-porous channel. Int. J. Eng. Sci. 1999; 37, 1781-94.

WA Robinson. The existence of multiple solutions for the laminar flow in a uniformly porous channel with suction at both walls. J. Eng. Math. 1976; 10, 23-40.

SW Yuan. Further investigation of laminar flow in channels with porous walls. J. Appl. Phys. 1956; 27, 267-9.

MB Zaturska, PG Drazin and WHH Banks. On the flow of a viscous fluid driven along a channel by suction at porous walls. Fluid Dyn. Resour. 1988; 4, 151-78.

K Vafai and CL Tien. Boundary and inertia effects on flow and heat transfer in porous media. Int. J. Heat Mass Trans. 1981; 24, 195-203.

CL Tien and K Vafai. Convective and radiative heat transfer in porous media. Adv. Appl. Mech. 1989; 27, 225-81.

JJ Choi, Z Rusak and JA Tichy. Maxwell fluid suction flow in a channel. J. Non-Newtonian Fluid Mech. 1999; 85, 165-87.

G Ahmadi. Self-similar solution of incompressible micropolar boundary layer flow over a semi-infinite plate. Int. J. Eng. Sci. 1976; 14, 639-46.

DAS Rees and I Pop. Free convection boundary layer flow of micropolar fluid from a vertical flat plate. J. Appl. Math. 1998; 61, 179-97.

MI Char and CL Chang. Laminar free convection flow of micropolar fluids from a curved surface. J. Appl. Phys. 1995; 28, 1324-31.

D Lesnic, DB Ingham and I Pop. Free convection from a horizontal surface in a porous medium with Newtonian heating. J. Porous Media. 2000; 3, 227-35.

DAS Rees and I Pop. Vertical free convective boundary-layer flow in a porous medium using a thermal nonequilibrium model. J. Porous Media. 2000; 3, 31-44.

F Shahzad, M Sajid, T Hayat and M Ayub. Analytic solution for flow of a micropolar fluid. Acta Mechanica. 2007; 188, 93-102.

T Hayat, S Saif and Z Abbas. Mixed convection flow of a micropolar fluid over a non-linearly stretching sheet. Phys. Lett. A 2008; 372, 637-47.

T Hayat and N Ali. Effects of an endoscope on peristaltic flow of a micropolar fluid. Math. Comput. Modell. 2008; 48, 721-33.

M Sajid, N Ali and T Hayat. On exact solutions for thin film flows of micropolar fluid. Commun. Nonlinear Sci. Numer. Simul. 2009; 14, 451-61.

F Shahzad, T Hayat, M Ayub and S Asghar. Unsteady MHD flow due to noncoaxial rotations of micropolar fluid and an accelerated disk with partial slip condition. Numer. Meth. Part. Differ. Equat. 2010; 26, 176-87.

M Sajid, Z Abbas and T Hayat. Homotopy analysis for boundary layer flow of a micropolar fluid through a porous channel. J. Appl. Math. Modell. 2009; 33, 4120-5.

SK Jena and MN Mathur. Similarity solutions for laminar free convection flow of a thermo-micropolar fluid past a nonisothermal flat plate. Int. J. Eng. Sci. 1981; 19, 1431-9.

GS Guram and AC Smith. Stagnation flows of micropolar fluids with strong and weak interactions. J. Comput. Math. Appl. 1980; 6, 213-33.

J Biazar, E Babolian and R Islam. Solution of a system of Volterra integral equations of the first kind by Adomian method. J. Appl. Math. Comput. 2003; 139, 249-58.

HS Goghary, Sh Javadi and E Babolian. Restarted Adomian method for system of nonlinear Volterra integral equations. J. Appl. Math. Computa. 2005; 161, 745-51.

JH He. Homotopy perturbation technique. J. Applied Math. Comput. 1999; 178, 257-62.

JH He. A coupling method of a homotopy technique and a perturbation technique for non-linear problems. Int. J. Non-Linear Mech. 2000; 35, 37-43.

JH He. Homotopy perturbation method: A new nonlinear analytical technique. J. Appl. Math. Comput. 2003; 135, 73-9.

JH He. Comparison of homotopy perturbation method and homotopy analysis method. J. Appl. Math. Comput. 2004; 156, 527-39.

DD Ganji, M Esmaeilpour and E Mohseni. Application of the homotopy perturbation method to micropolar flow in a porous channel. J. Porous Media. 2009; 12, 451-9.

S Abbasbandy. The application of homotopy analysis method to nonlinear equations arising in heat transfer. Phys. Lett. A 2006; 360, 109-13.

S Dinarvand, A Doosthoseini, E Doosthoseini and MM Rashidi. Series solutions for unsteady laminar MHD flow near forward stagnation point of an impulsively rotating and translating sphere in presence of buoyancy forces. Nonlinear Anal. Real World Appl. 2009; 11, 1159-69.

SJ Liao. 1992, The Proposed Homotopy Analysis Technique for the Solution of Nonlinear Problems, Ph.D. Thesis. Shanghai Jiao Tong University, Shanghai, China.

SJ Liao. Beyond Perturbation: Introduction to the Homotopy Analysis Method. Boca Raton. Chapman & Hall/CRC Press, 2003.

MM 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 Anal. Real World Appl. 2009; 10, 2346-56.

MM Rashidi and E Erfani. New analytical method for solving Burgers’ and nonlinear heat transfer equations and comparison with HAM. Comput. Phys. Commun. 2009; 180, 1539-44.

MM Rashidi, G Domairry and S Dinarvand. Approximate solutions for the Burger and regularized long wave equations by means of the Homotopy analysis method. Commun. Nonlinear. Numerical Simul. 2009; 14, 708-17.

T Hayat, N Ahmed, M Sajid and S Asghar. On the MHD flow of a second grade fluid in a porous channel. Comput. Math. Appl. 2007; 54, 407-14.

T Hayat, N Ahmed and M Sajid. Analytic solution for MHD flow of a third order fluid in a porous channel. J. Comput. Appl. Math. 2008; 214, 572-82.

MM Rashidi and H Shahmohamadi. Analytical solution of three-dimensional Navier–Stokes equations for the flow near an infinite rotating disk. Commun. Nonlinear. Numer. Simul. 2009; 14, 2999-3006.

SQ Wang and JH He. Variational iteration method for solving integro-differential equations. J. Phys. 2007; 367, 188-91.

JK Zhou. Differential Transformation and Its Applications for Electrical Circuits (in Chinese). Huazhong University Press, Wuhan, China, 1986.

CK Chen and SH Ho. Solving partial differential equations by two dimensional differential transform method. J. Appl. Math. Comput. 1999; 106, 171-9.

F Ayaz. Solutions of the systems of differential equations by differential transform method. J. Appl. Math. Comput. 2004; 147, 547-67.

IH Hassan Abdel-Halim. Comparison differential transformation technique with Adomian decomposition method for linear and nonlinear initial value problems. Chaos. Solitons Fractals. 2008; 36, 53-65.

MM Rashidi. Differential Transform Method for Solving Two Dimensional Viscous Flow. In: Proceedings of the International Conference on Applied Physics and Mathematics, Singapore. 2009.

MM Rashidi. Differential Transform Method for MHD Boundary-Layer Equations: Combination of the DTM and the Padé Approximant. In: Proceedings of the International Conference on Applied Physics and Mathematics, Singapore. 2009.

MM Rashidi. The modified differential transform method for solving MHD boundary-layer equations. J. Comput. Phys. Commun. 2009; 180, 2210-7.

MM 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. In: Proceedings of the International Conference on Applied Physics and Mathematics, Singapore. 2009.

MM Rashidi and SM Sadri. Solution of the laminar viscous flow in a semi-porous channel in the presence of a uniform magnetic field by using the differential transform method. Int. J. Contemp. Math. Sci. 2010; 5, 711-20.

MM Rashidi and SM Sadri. New analytical solution of two-dimensional viscous flow in a rectangular domain bounded by two moving porous walls. Int. J. Comput. Methods Eng. Sci. Mech. 2011; 12, 26-33.

MM Rashidi, DD Ganji and SM Sadri. New analytical solution of stagnation point flow in a porous medium. J. Porous Media. 2011; 14, 1125-35.

MM Rashidi, N Laraqi and SM Sadri. A novel analytical solution of mixed convection about an inclined flat plate embedded in a porous medium using the DTM-Padé. Int. J. Therm. Sci. 2010; 49, 2405-12.

A Arikoglu and I Ozkol. Solution of differential-difference equations by using differential transform method. Int. J. Appl. Math. Comput. 2006; 181, 153-62.