Numerical Algorithms

, Volume 61, Issue 3, pp 499–514

On a new analytical method for flow between two inclined walls

  • Sandile S. Motsa
  • Precious Sibanda
  • Gerald T. Marewo
Original Paper


Efficient analytical methods for solving highly nonlinear boundary value problems are rare in nonlinear mechanics. The purpose of this study is to introduce a new algorithm that leads to exact analytical solutions of nonlinear boundary value problems and performs more efficiently compared to other semi-analytical techniques currently in use. The classical two-dimensional flow problem into or out of a wedge-shaped channel is used as a numerical example for testing the new method. Numerical comparisons with other analytical methods of solution such as the Adomian decomposition method (ADM) and the improved homotopy analysis method (IHAM) are carried out to verify and validate the accuracy of the method. We show further that with a slight modification, the algorithm can, under certain conditions, give better performance with enhanced accuracy and faster convergence.


Nonlinear equations Exact analytical solutions Successive linearisation method Channel flow 


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  1. 1.
    Akulenko, L.D., Georgievskii, D.V., Kumakshev, S.A.: Solutions of the Jeffery–Hamel problem regularly extendable in the Reynolds number. Fluid Dyn. 39, 12–28 (2004)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Rivkind, L., Solonnikov, V.A.: Jeffery–Hamel asymptotics for steady state Navier–Stokes flow in domains with sector-like outlets to infinity. J. Math. Fluid Mech. 2, 324–352 (2000)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Fraenkel, L.E.: Laminar flow in symmetrical channels with slightly curved walls- I: on the Jeffery–Hamel solutions for flow between plane walls. Proc. R. Soc. A 267, 119–138 (1962)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Jeffery, G.B.: The two-dimensional steady motion of a viscous fluid. Phil. Mag. 6(29), 455–465 (1915)Google Scholar
  5. 5.
    Riley, N.: Heat transfer in Jeffery–Hamel flow. Q. J. Mech. Appl. Math. 42, 203–211 (1989)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Rosenhead, L.: The steady two-dimensional radial flow of viscous fluid between two inclined plane walls. Proc. R. Soc. A 175(963), 436–467 (1940)CrossRefGoogle Scholar
  7. 7.
    Makinde, O.D.: Effect of arbitrary magnetic Reynolds number on MHD flows in convergent-divergent channels. Int. J. Numer. Meth. Heat Fluid Flow 18(6), 697–707 (2008)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Reza, M.S.: Channel entrance flow. PhD thesis. Department of Mechanical Engineering, University of Western Ontario (1997)Google Scholar
  9. 9.
    Millsaps, K., Pohlhausen, K.: Thermal distribution in Jeffrey–Hamel flows between non-parallel plane walls. J. Aeronaut. Sci. 20, 187–196 (1953)MathSciNetMATHGoogle Scholar
  10. 10.
    Marshall, R.S.: Symmetrical velocity profiles for Jeffery–Hamel flow. ASME Transactions: J. Appl. Mech. 46, 214–215 (1979)MATHCrossRefGoogle Scholar
  11. 11.
    Esmaili, Q., Ramiar, A., Alizadeh, E., Ganji, D.D.: An approximation of the analytical solution of the Jeffery-Hamel flow by decomposition method. Phys. Lett. A 372, 3434–3439 (2008)MATHCrossRefGoogle Scholar
  12. 12.
    Ganji, Z.Z., Ganji, D.D., Esmaeilpour, M.: Study on nonlinear Jeffery–Hamel flow by He’s semi-analytical methods and comparison with numerical results. Comput. Math. Appl. (2009). doi:10.1016/j.camwa.2009.03.044 MathSciNetGoogle Scholar
  13. 13.
    Domairry, G., Mohsenzadeh, A., Famouri, M.: The application of homotopy analysis method to solve nonlinear differential equation governing Jeffery–Hamel flow. Commun. Nonlinear Sci. Numer. Simulat. 14, 85–95 (2008)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Esmaeilpour, M., Ganji, D.D.: Solution of the Jeffery–Hamel flow problem by optimal homotopy asymptotic method. Comput. Math. Appl. 59, 3405–3411 (2010)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Joneidi, A.A., Domairry, G., Babaelahi, M.: Three analytical methods applied to Jeffery–Hamel flow. Commun. Nonlinear Sci. Numer. Simulat. 15, 3423–3434 (2010)CrossRefGoogle Scholar
  16. 16.
    Liao, S.J.: The proposed homotopy analysis technique for the solution of nonlinear problems. PhD thesis, Shanghai Jiao Tong University (1992)Google Scholar
  17. 17.
    Motsa, S.S., Sibanda, P., Awad, F.G., Shateyi, S.: A new spectral-homotopy analysis method for the MHD Jeffery–Hamel problem. Comput. Fluids 39, 1219–1225 (2010)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Motsa, S.S., Sibanda, P., Marewo, G.T., Shateyi, S.: A note on improved homotopy analysis method for solving the Jeffery–Hamel flow. Math. Probl. Eng. Article ID 359297, 11 (2010). doi:10.1155/2010/359297 Google Scholar
  19. 19.
    Awad, F.G., Sibanda, P., Motsa, S.S., Makinde, O.D.: Convection from an inverted cone in a porous medium with cross-diffusion effects. Comput. Math. Appl. 61, 1431–1441 (2011)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Makukula, Z.G., Sibanda, P., Motsa, S.S.: A novel numerical technique for two-dimensional laminar flow between two moving porous walls. Math. Probl. Eng. Article ID 528956, 15 (2010). doi:10.1155/2010/528956 Google Scholar
  21. 21.
    Makukula, Z.G., Sibanda, P., Motsa, S.S.: A note on the solution of the von Kármán equations using series and Chebyshev spectral methods. Bound Value Probl. Article ID 471793, 17 (2010). doi:10.1155/2010/471793
  22. 22.
    Makukula, Z.G., Sibanda, P., Motsa, S.S.: On new solutions for heat transfer in a visco-elastic fluid between parallel plates. Math. Model Meth. Appl. Sci. 4(4), 221–230 (2010)MathSciNetGoogle Scholar
  23. 23.
    Motsa, S.S., Shateyi, S.: A new approach for the solution of three-dimensional magnetohydrodynamic rotating flow over a shrinking sheet. Math. Probl. Eng. Article ID 586340, 15 (2010). doi:10.1155/2010/586340 Google Scholar
  24. 24.
    Motsa, S.S., Shateyi, S.: Successive linearisation solution of free convection non-Darcy flow with heat and mass transfer. In: El-Amin, M. (ed.) Advanced Topics in Mass Transfer. InTech Open Access Publishers, pp. 425–438 (2011)Google Scholar
  25. 25.
    Motsa, S.S., Sibanda, P., Shateyi, S.: On a new quasi-linearization method for systems of nonlinear boundary value problems. Math. Methods Appl. Sci. 34, 1406–1413 (2011)MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods in Fluid Dynamics. Springer, Berlin (1988)MATHCrossRefGoogle Scholar
  27. 27.
    Trefethen, L.N.: Spectral Methods in MATLAB. SIAM (2000)Google Scholar
  28. 28.
    Shateyi, S., Motsa, S.S.: Variable viscosity on magnetohydrodynamic fluid flow and heat transfer over an unsteady stretching surface with hall effect. Bound Value Probl. Article ID 257568, 20 (2010). doi:10.1155/2010/257568
  29. 29.
    Adomian, G.: A review of the decomposition method and some recent results for nonlinear equation. Math. Comput. Model. 13(7), 17–43 (1990)MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Adomian, G., Rach, R.: Noise terms in decomposition series solution. Comput. Math. Appl. 24(11), 61–64 (1992)MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Adomian, G.: Solving Frontier Problems of Physics: The Decomposition Method. Kluwer, Boston (1994)MATHGoogle Scholar
  32. 32.
    Kierzenka, J., Shampine, L.F.: A BVP solver based on residual control and the Matlab PSE. ACM Trans. Math. Softw. 27, 299–316 (2001)MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Shampine, L.F., Gladwell, I., Thompson, S.: Solving ODEs with MATLAB. Cambridge University Press, Cambridge (2003)CrossRefGoogle Scholar
  34. 34.
    Ganji, D.D., Sheikholeslami, M., Ashorynejad, H. R. : Analytical approximate solution of nonlinear differential equation governing Jeffery–Hamel flow with high magnetic field by adomian decomposition method. ISRN Mathematical Analysis. Article ID 937830, 16 (2011). doi:10.5402/2011/937830

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Sandile S. Motsa
    • 1
  • Precious Sibanda
    • 1
  • Gerald T. Marewo
    • 2
  1. 1.School of Mathematical SciencesUniversity of KwaZulu-NatalPietermaritzburgSouth Africa
  2. 2.Department of MathematicsUniversity of SwazilandKwaluseniSwaziland

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