[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Springer Nature Link
Log in

On a new analytical method for flow between two inclined walls

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  4. Jeffery, G.B.: The two-dimensional steady motion of a viscous fluid. Phil. Mag. 6(29), 455–465 (1915)

    Google Scholar 

  5. Riley, N.: Heat transfer in Jeffery–Hamel flow. Q. J. Mech. Appl. Math. 42, 203–211 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  8. Reza, M.S.: Channel entrance flow. PhD thesis. Department of Mechanical Engineering, University of Western Ontario (1997)

  9. Millsaps, K., Pohlhausen, K.: Thermal distribution in Jeffrey–Hamel flows between non-parallel plane walls. J. Aeronaut. Sci. 20, 187–196 (1953)

    MathSciNet  MATH  Google Scholar 

  10. Marshall, R.S.: Symmetrical velocity profiles for Jeffery–Hamel flow. ASME Transactions: J. Appl. Mech. 46, 214–215 (1979)

    Article  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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

    MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  16. Liao, S.J.: The proposed homotopy analysis technique for the solution of nonlinear problems. PhD thesis, Shanghai Jiao Tong University (1992)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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. 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)

    MathSciNet  Google Scholar 

  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. 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)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  26. Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods in Fluid Dynamics. Springer, Berlin (1988)

    Book  MATH  Google Scholar 

  27. Trefethen, L.N.: Spectral Methods in MATLAB. SIAM (2000)

  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. Adomian, G.: A review of the decomposition method and some recent results for nonlinear equation. Math. Comput. Model. 13(7), 17–43 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  30. Adomian, G., Rach, R.: Noise terms in decomposition series solution. Comput. Math. Appl. 24(11), 61–64 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  31. Adomian, G.: Solving Frontier Problems of Physics: The Decomposition Method. Kluwer, Boston (1994)

    MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  33. Shampine, L.F., Gladwell, I., Thompson, S.: Solving ODEs with MATLAB. Cambridge University Press, Cambridge (2003)

    Book  Google Scholar 

  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

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, University of KwaZulu-Natal, Private Bag X01 Scottsville 3209, Pietermaritzburg, South Africa

    Sandile S. Motsa & Precious Sibanda

  2. Department of Mathematics, University of Swaziland, Private Bag 4, Kwaluseni, Swaziland

    Gerald T. Marewo

Authors
  1. Sandile S. Motsa
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Precious Sibanda
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Gerald T. Marewo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Precious Sibanda.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Motsa, S.S., Sibanda, P. & Marewo, G.T. On a new analytical method for flow between two inclined walls. Numer Algor 61, 499–514 (2012). https://doi.org/10.1007/s11075-012-9545-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-012-9545-2

Keywords

Search

Navigation