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On spectral relaxation method approach for steady von Kármán flow of a Reiner-Rivlin fluid with Joule heating, viscous dissipation and suction/injection

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Central European Journal of Physics

Abstract

In this study we use the spectral relaxation method (SRM) for the solution of the steady von Kármán flow of a Reiner-Rivlin fluid with Joule heating and viscous dissipation. The spectral relaxation method is a new Chebyshev spectral collocation based iteration method that is developed from the Gauss-Seidel idea of decoupling systems of equations. In this work, we investigate the applicability of the method in solving strongly nonlinear boundary value problems of von Kármán flow type. The SRM results are validated against previous results present in the literature and with those obtained using the bvp4c, a MATLAB inbuilt routine for solving boundary value problems. The study highlights the accuracy and efficiency of the proposed SRM method in solving highly nonlinear boundary layer type equations.

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References

  1. T. von Kármán, ZAMM-Z. Angew. Math. Me. 1, 233 (1921)

    Article  MATH  Google Scholar 

  2. W. G. Cochran, Math. Proc. Cambridge 30, 365 (1934)

    Article  ADS  MATH  Google Scholar 

  3. E. R. Benton, J. Fluid Mech. 24, 781 (1966)

    Article  ADS  MATH  Google Scholar 

  4. H. I. Andersson, M. Rousselet, Int. J. Heat Fluid Fl. 27, 329 (2006)

    Article  Google Scholar 

  5. S. P. A. Devi, R. U. Devi, Journal of Applied Fluid Mechanics 5, 1 (2012)

    ADS  Google Scholar 

  6. M. Turkyilmazoglu, Appl. Therm. Eng. 35, 127 (2012)

    Article  Google Scholar 

  7. J.-H. He, Appl. Math. Comput. 143, 543 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. A. El-Nahhas, Proceedings of the Pakistan Academy of Sciences 44, 181 (2007)

    Google Scholar 

  9. M. Turkyilmazoglu, Phys. Fluids 21, 106104 (2009)

    Article  ADS  Google Scholar 

  10. M. Turkyilmazoglu, Comput. Fluids 39, 793 (2010)

    Article  MATH  Google Scholar 

  11. M. Turkyilmazoglu, Int. J. Mech. Sci. 52, 1735 (2010)

    Article  Google Scholar 

  12. C. Yang, S. J. Liao, Commun. Nonlinear Sci. 11, 83 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  13. M. A. Abdou, Acta. Appl. Math. 111, 7 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. A. Arikoglu, I. Ozkol, Int. J. Numer. Method. H. 16, 172 (2006)

    Article  MATH  Google Scholar 

  15. A. Arikoglu, I. Ozkol, G. Komurgoz, Appl. Energ. 85, 1225 (2008)

    Article  Google Scholar 

  16. M. M. Rashidi, S. A. M. Pour, African Journal of Mathematics and Computer Science Research 3, 93 (2010)

    Google Scholar 

  17. P. D. Ariel, ZAMM-Z. Angew. Math. Me. 82, 235 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  18. H. A. Attia, Turkish Journal of Engineering & Environmental Sciences 30, 231 (2006)

    Google Scholar 

  19. H. A. Attia, Kragujevac Journal of Science 30, 17 (2008)

    Google Scholar 

  20. H. A. Attia, Turkish Journal of Physics 30, 103 (2006)

    ADS  Google Scholar 

  21. H. A. Attia, Nonlinear Analysis: Modelling and Control 14, 21 (2009)

    MATH  Google Scholar 

  22. H. A. Attia, J. Braz. Soc. Mech. Sci. 29, 168 (2007)

    Google Scholar 

  23. P. Sibanda, O. D. Makinde, Int. J. Numer. Method. H. 20, 269 (2010)

    Google Scholar 

  24. E. Osalusi, J. Side, R. Harris, B. Johnston, Rom. Rep. Phys. 61, 71 (2009)

    Google Scholar 

  25. H. S. Takhar, A. K. Singh, G. Nath, Int. J. Therm. Sci. 41, 147 (2002)

    Article  Google Scholar 

  26. H. A. Attia, J. Mech. Sci. Technol. 21, 174 (2007)

    Article  Google Scholar 

  27. A. L. Aboul-Hassan, H. A. Attia, Phys. Lett. A 228, 286 (1997)

    Article  ADS  Google Scholar 

  28. P. Hatzikonstantinou, Astrophys. Space Sci. 161, 17 (1989)

    Article  ADS  Google Scholar 

  29. Z. G. Makukula, P. Sibanda, S. S. Motsa, Comput. Appl. Math. 31, 95 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  30. Z. G. Makukula, P. Sibanda, S. S. Motsa, DOI:10.1155/2010/471793

  31. Z. G. Makukula, P. Sibanda, S. S. Motsa, Lat. Am. Appl. Res. 42, 97 (2012)

    Google Scholar 

  32. B. Sahoo, Commun. Nonlinear Sci. 14, 2982 (2009)

    Article  Google Scholar 

  33. B. Sahoo, Bull. Braz. Math. Soc. 38, 595 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  34. B. Sahoo, DOI:10.1007/s12591-012-0117-7

  35. N. S. Akbar, S. Nadeem, Heat Mass Transfer 46, 531 (2010)

    Article  ADS  Google Scholar 

  36. H. A. Attia, M. E. S. Ahmed, ANZIAM J. 46, 237 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  37. E. Lelarasmee, A. Ruehli, A. Sangiovanni-Vincentelli, CAD Integrated Circuits and Systems 1, 132 (1982)

    Google Scholar 

  38. J. P. Boyd, Chebyshev and Fourier Spectral Methods (DOVER Publications Inc., New York, 2000)

    Google Scholar 

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

    Book  MATH  Google Scholar 

  40. L. N. Trefethen, Spectral Methods in MATLAB. (SIAM, Philadelphia, 2000)

    Book  Google Scholar 

  41. W. S. Don, A. Solomonoff, SIAM J. Sci. Comput. 16, 1253 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  42. J. A. C. Weideman, S. C. Reddy, ACM T. Math. Software 26, 465 (2000)

    Article  MathSciNet  Google Scholar 

  43. H. A. Attia, Mechanics and Mechanical Engineering 14, 119 (2010)

    Google Scholar 

  44. H. A. Attia, M. A. M. Abdeen, Kragujevac J. Sci. 34, 5 (2012)

    Google Scholar 

  45. H. A. Attia, Engineering Modelling 22, 57 (2009)

    Google Scholar 

  46. H. A. Attia, Commun. Nonlinear Sci. 13, 1571 (2008)

    Article  MATH  Google Scholar 

  47. H. A. Attia, Arab. J. Sci. Eng. 29, 165 (2004)

    Google Scholar 

  48. H. A. Attia, Tamkang Journal of Science and Engineering 9, 185 (2006)

    Google Scholar 

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Authors and Affiliations

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

    Sandile S. Motsa & Zodwa G. Makukula

Authors
  1. Sandile S. Motsa
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  2. Zodwa G. Makukula
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Correspondence to Sandile S. Motsa.

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Motsa, S.S., Makukula, Z.G. On spectral relaxation method approach for steady von Kármán flow of a Reiner-Rivlin fluid with Joule heating, viscous dissipation and suction/injection. centr.eur.j.phys. 11, 363–374 (2013). https://doi.org/10.2478/s11534-013-0182-8

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  • DOI: https://doi.org/10.2478/s11534-013-0182-8

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