A Series Solution for Three-Dimensional Navier-Stokes Equations of Flow near an Infinite Rotating Disk


In this paper, homotopy analysis method (HAM) and Padé approximant will be considered for finding analytical solution of three-dimensional viscous flow near an infinite rotating disk which is a well-known classical problem in fluid mechanics. The solution is compared to the numerical (fourth-order Runge-Kutta) solution and the convergence of the obtained series solution is carefully analyzed. The results illustrate that HAM-Padé is an appropriate method in solving the systems of nonlinear equations.

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Shahmohamadi, H. and Mohammadpour, M. (2014) A Series Solution for Three-Dimensional Navier-Stokes Equations of Flow near an Infinite Rotating Disk. World Journal of Mechanics, 4, 117-127. doi: 10.4236/wjm.2014.45014.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] Von Karman, T. (1921) Uberlaminare und turbulence reibung. ZAMM, 1, 52-233.
[2] Rashidi, M.M. and Shahmohamadi, H. (2009) Analytical Solution of Three-Dimensional Navier-Stokes Equations for the Flow near an Infinite Rotating Disk. Communications in Nonlinear Science and Numerical Simulation, 14, 2999-3006.
[3] Ablowitz, M.J. and Clarkson, P.A. (1990) Solitons, Nonlinear Evolution Equations and Inverse Scattering Transform. Cambridge University Press, Cambridge.
[4] Fan, E. (2000) Extentedtanh-Function Method and Its Applications to Nonlinear Equations. Physics Letters A, 277, 212-218.
[5] Wazwaz, A.M. (2006) The Tanh and the Sine-Cosine Methods for a Reliable Treatment of the Modified Equal Width Equation and Its Variants. Communications in Nonlinear Science and Numerical Simulation, 11, 148-160.
[6] Fan, E. and Zhang, H. (1998) A Note on the Homogeneous Balance Method. Physical Letters A, 246, 403-406.
[7] He, J.H. and Wu, X.H. (2006) Exp-Function Method for Nonlinear Wave Equations. Chaos Solitons Fractals, 30, 700-708.
[8] Rashidi, M.M., Shahmohamadi, H. and Domairry, G. (2011) Variational Iteration Method for Solving Three-Dimensional Navier-Stokes Equations of Flow between Two Stretchable Disks. Numerical Methods for Partial Differential Equations, 27, 292-301.
[9] Rashidi, M.M., Ganji, D.D. and Shahmohamadi, H. (2011) Variational Iteration Method for Two-Dimensional Steady Slip Flow in Micro-Channels. Archive of Applied Mechanics, 81, 1597-1605.
[10] Shahmohamadi, H., Rashidi, M.M. and Anwar Beg, O. (2012) 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, 8, 1-17.
[11] Shahmohamadi, H. and Rashidi, M.M. (2010) A Novel Solution for the Glauert-Jet Problem by Variational Iteration Method-Pade Approximant. Mathematical Problems in Engineering, 2010, Article ID: 501476.
[12] Shahmohamadi, H. and Rashidi, M.M. (2011) Explicit Solutions for Steady Three-Dimensional Problem of Condensation Film on Inclined Rotating Disk. International Journal of Fluid Mechanics Research, 38, 353-365.
[13] Liao, S.J. (1992) The Proposed Homotopy Analysis Technique for the Solution of Nonlinear Problems. PhD Thesis, Shanghai Jiao Tong University, Shanghai.
[14] Liao, S.J. (2003) Beyond Perturbation: Introduction to Homotopy Analysis Method. Chapman & Hall/CRC, Boca Raton.
[15] He, J.H. (2000) A Coupling Method for Homotopy Technique and Perturbation Technique for Nonlinear Problem. International Journal of Non-Linear Mechanics, 35, 37-43.
[16] Liao, S.J. (2005) Comparison between the Homotopy Analysis Method and Homotopy Perturbation Method. Applied Mathematics and Computation, 169, 1186-1194.
[17] Sajid, M., Hayat, T. and Asghar, S. (2007) Comparison between the HAM and HPM Solutions of Tin Film Flows of Non-Newtonian Fluids on a Moving Belt. Nonlinear Dynamics, 50, 27-35.
[18] Abbasbandy, S. (2007) The Application of Homotopy Analysis Method to Solve a Generalized Hirota-Satsuma Coupled KdV Equation. Physics Letters A, 361, 478-483.
[19] Sajid, M. and Hayat, T. (2008) Comparison of HAM and HPM Methods in Nonlinear Heat Conduction and Convection Equations. Nonlinear Analysis: Real World Applications, 9, 2296-2301.
[20] Shahmohamadi, H. (2012) Analytic Study on Non-Newtonian Natural Convection Boundary Layer Flow with Variable Wall Temperature on a Horizontal Plate. Meccanica, 47, 1313-1323.
[21] Shahmohamadi, H. (2011) Reliable Treatment of a New Analytical Method for Solving MHD Boundary-Layer Equations. Meccanica, 46, 921-933.
[22] Rashidi, M.M., Shahmohamadi, H. and Dinarvand, S. (2008) Analytic Approximate Solutions for Unsteady Two-Dimensional and Axisymmetric Squeezing Flows between Parallel Plates. Mathematical Problems in Engineering, 2008, Article ID: 935095.
[23] Ismail, H.N.A. and Abde Rabboh, A.A. (2004) A Restrictive Pade Approximation for the Solution of the Generalized Fisher and Burger-Fisher Equations. Applied Mathematics and Computation, 154, 203-213.
[24] Liao, S.J. (2010) An Optimal Homotopy-Analysis Approach for Strongly Nonlinear Differential Equations. Communications in Nonlinear Science and Numerical Simulation, 15, 2003-2016. http://dx.doi.org/10.1016/j.cnsns.2009.09.002

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