Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2013, Article ID 685454, 8 pages
http://dx.doi.org/10.1155/2013/685454
Research Article

Solution of the Boundary Layer Equation of the Power-Law Pseudoplastic Fluid Using Differential Transform Method

Department of Mechanical Engineering, Islamic Azad University, Najafabad Branch, Isfahan, Iran

Received 26 July 2013; Revised 26 September 2013; Accepted 22 October 2013

Academic Editor: Metin O. Kaya

Copyright © 2013 Sobhan Mosayebidorcheh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. S.-J. Liao, “A general approach to get series solution of non-similarity boundary-layer flows,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 5, pp. 2144–2159, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  2. A. R. Ghotbi, H. Bararnia, G. Domairry, and A. Barari, “Investigation of a powerful analytical method into natural convection boundary layer flow,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 5, pp. 2222–2228, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  3. Z. Ziabakhsh and G. Domairry, “Analytic solution of natural convection flow of a non-Newtonian fluid between two vertical flat plates using homotopy analysis method,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 5, pp. 1868–1880, 2009. View at Publisher · View at Google Scholar · View at Scopus
  4. S. Abbasbandy and T. Hayat, “Solution of the MHD Falkner-Skan flow by homotopy analysis method,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 9-10, pp. 3591–3598, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  5. J. Cheng, S. Liao, R. N. Mohapatra, and K. Vajravelu, “Series solutions of nano boundary layer flows by means of the homotopy analysis method,” Journal of Mathematical Analysis and Applications, vol. 343, no. 1, pp. 233–245, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  6. S.-J. Liao and I. Pop, “Explicit analytic solution for similarity boundary layer equations,” International Journal of Heat and Mass Transfer, vol. 47, no. 1, pp. 75–85, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  7. S. Liao, “A new branch of solutions of boundary-layer flows over an impermeable stretched plate,” International Journal of Heat and Mass Transfer, vol. 48, no. 12, pp. 2529–2539, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  8. T. Hayat and M. Sajid, “Homotopy analysis of MHD boundary layer flow of an upper-convected Maxwell fluid,” International Journal of Engineering Science, vol. 45, no. 2–8, pp. 393–401, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  9. A. R. Ghotbi, “Homotopy analysis method for solving the MHD flow over a non-linear stretching sheet,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 6, pp. 2653–2663, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  10. B. Raftari and A. Yildirim, “The application of homotopy perturbation method for MHD flows of UCM fluids above porous stretching sheets,” Computers and Mathematics with Applications, vol. 59, no. 10, pp. 3328–3337, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  11. M. Esmaeilpour and D. D. Ganji, “Application of He's homotopy perturbation method to boundary layer flow and convection heat transfer over a flat plate,” Physics Letters A, vol. 372, no. 1, pp. 33–38, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  12. L. Xu, “He's homotopy perturbation method for a boundary layer equation in unbounded domain,” Computers and Mathematics with Applications, vol. 54, no. 7-8, pp. 1067–1070, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  13. M. Fathizadeh and F. Rashidi, “Boundary layer convective heat transfer with pressure gradient using Homotopy Perturbation Method (HPM) over a flat plate,” Chaos, Solitons and Fractals, vol. 42, no. 4, pp. 2413–2419, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  14. H. Bararnia, E. Ghasemi, S. Soleimani, A. R. Ghotbi, and D. D. Ganji, “Solution of the Falkner-Skan wedge flow by HPM-Pade' method,” Advances in Engineering Software, vol. 43, no. 1, pp. 44–52, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  15. H. Mirgolbabaei, D. D. Ganji, M. M. Etghani, and A. Sobati, “Adapted variational iteration method and axisymmetric flow over a stretching sheet,” World Journal of Modelling and Simulation, vol. 5, no. 4, pp. 307–314, 2009. View at Google Scholar · View at Scopus
  16. S. T. Mohyud-Din, A. Yildirim, S. Anl Sezer, and M. Usman, “Modified variational iteration method for free-convective boundary-layer equation using padé approximation,” Mathematical Problems in Engineering, vol. 2010, Article ID 318298, 11 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  17. A.-M. Wazwaz, “The variational iteration method for solving two forms of Blasius equation on a half-infinite domain,” Applied Mathematics and Computation, vol. 188, no. 1, pp. 485–491, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  18. A.-M. Wazwaz, “The modified decomposition method and Padé approximants for a boundary layer equation in unbounded domain,” Applied Mathematics and Computation, vol. 177, no. 2, pp. 737–744, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  19. E. Alizadeh, K. Sedighi, M. Farhadi, and H. R. Ebrahimi-Kebria, “Analytical approximate solution of the cooling problem by Adomian decomposition method,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 2, pp. 462–472, 2009. View at Publisher · View at Google Scholar · View at Scopus
  20. S. A. Kechil and I. Hashim, “Non-perturbative solution of free-convective boundary-layer equation by Adomian decomposition method,” Physics Letters A, vol. 363, no. 1-2, pp. 110–114, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  21. E. Alizadeh, M. Farhadi, K. Sedighi, H. R. Ebrahimi-Kebria, and A. Ghafourian, “Solution of the Falkner-Skan equation for wedge by Adomian Decomposition Method,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 3, pp. 724–733, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  22. J. K. Zhou, Differential Transformation and Its Applications for Electrical CircuIts, Huazhong University Press, Wuhan, China, 1986, (Chinese).
  23. F. Ayaz, “Applications of differential transform method to differential-algebraic equations,” Applied Mathematics and Computation, vol. 152, no. 3, pp. 649–657, 2004. View at Publisher · View at Google Scholar · View at Scopus
  24. H. Liu and Y. Song, “Differential transform method applied to high index differential-algebraic equations,” Applied Mathematics and Computation, vol. 184, no. 2, pp. 748–753, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  25. F. Ayaz, “Solutions of the system of differential equations by differential transform method,” Applied Mathematics and Computation, vol. 147, no. 2, pp. 547–567, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  26. A. S. V. Ravi Kanth and K. Aruna, “Differential transform method for solving linear and non-linear systems of partial differential equations,” Physics Letters A, vol. 372, no. 46, pp. 6896–6898, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  27. F. Ayaz, “On the two-dimensional differential transform method,” Applied Mathematics and Computation, vol. 143, no. 2-3, pp. 361–374, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  28. X. Yang, Y. Liu, and S. Bai, “A numerical solution of second-order linear partial differential equations by differential transform,” Applied Mathematics and Computation, vol. 173, no. 2, pp. 792–802, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  29. S.-H. Chang and I.-L. Chang, “A new algorithm for calculating two-dimensional differential transform of nonlinear functions,” Applied Mathematics and Computation, vol. 215, no. 7, pp. 2486–2494, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  30. B. Jang, “Solving linear and nonlinear initial value problems by the projected differential transform method,” Computer Physics Communications, vol. 181, no. 5, pp. 848–854, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  31. Z. M. Odibat, “Differential transform method for solving Volterra integral equation with separable kernels,” Mathematical and Computer Modelling, vol. 48, no. 7-8, pp. 1144–1149, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  32. A. Arikoglu and I. Ozkol, “Solutions of integral and integro-differential equation systems by using differential transform method,” Computers and Mathematics with Applications, vol. 56, no. 9, pp. 2411–2417, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  33. A. Arikoglu and I. Ozkol, “Solution of boundary value problems for integro-differential equations by using differential transform method,” Applied Mathematics and Computation, vol. 168, no. 2, pp. 1145–1158, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  34. C.-L. Chen and Y.-C. Liu, “Differential transformation technique for steady nonlinear heat conduction problems,” Applied Mathematics and Computation, vol. 95, no. 2-3, pp. 155–164, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  35. S. Mosayebidorcheh and T. Mosayebidorcheh, “Series solution of convective radiative conduction equation of the nonlinear fin with temperature dependent thermal conductivity,” In press. View at Publisher · View at Google Scholar
  36. S.-U. Islam, S. Haq, and J. Ali, “Numerical solution of special 12th-order boundary value problems using differential transform method,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 4, pp. 1132–1138, 2009. View at Publisher · View at Google Scholar · View at Scopus
  37. M. Torabi and A. Aziz, “Convective-radiative fins with simultaneous variation of thermal conductivity, heat transfer coefficient, and surface emissivity with temperature,” Heat Transfer, vol. 41, no. 2, pp. 99–113, 2012. View at Publisher · View at Google Scholar · View at Scopus
  38. A. A. Joneidi, D. D. Ganji, and M. Babaelahi, “Differential Transformation Method to determine fin efficiency of convective straight fins with temperature dependent thermal conductivity,” International Communications in Heat and Mass Transfer, vol. 36, no. 7, pp. 757–762, 2009. View at Publisher · View at Google Scholar · View at Scopus
  39. D. Nazari and S. Shahmorad, “Application of the fractional differential transform method to fractional-order integro-differential equations with nonlocal boundary conditions,” Journal of Computational and Applied Mathematics, vol. 234, no. 3, pp. 883–891, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  40. Z. Odibat, S. Momani, and V. S. Erturk, “Generalized differential transform method: application to differential equations of fractional order,” Applied Mathematics and Computation, vol. 197, no. 2, pp. 467–477, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  41. V. S. Erturk, S. Momani, and Z. Odibat, “Application of generalized differential transform method to multi-order fractional differential equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 13, no. 8, pp. 1642–1654, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  42. A. Arikoglu and I. Ozkol, “Solution of fractional differential equations by using differential transform method,” Chaos, Solitons and Fractals, vol. 34, no. 5, pp. 1473–1481, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  43. P. F. Lemieux, R. N. Dubey, and T. E. Unny, “Variational method for a pseudoplastic fluid in a laminar boundary layer over a flat plate,” Journal of Applied Mechanics, Transactions ASME, pp. 345–349, 1971. View at Google Scholar · View at Scopus
  44. G. A. Baker, Essentials of Pade Approximants, Academic Press, New York, NY, USA, 1975.
  45. T. G. Myers, “An approximate solution method for boundary layer flow of a power law fluid over a flat plate,” International Journal of Heat and Mass Transfer, vol. 53, no. 11-12, pp. 2337–2346, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  46. M. Benlahsen, M. Guedda, and R. Kersner, “The Generalized Blasius equation revisited,” Mathematical and Computer Modelling, vol. 47, no. 9-10, pp. 1063–1076, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  47. J. P. Denier and P. P. Dabrowski, “On the boundary-layer equations for power-law fluids,” Proceedings of the Royal Society A, vol. 460, no. 2051, pp. 3143–3158, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  48. C.-C. Hsu, “A simple solution for boundary layer flow of power law fluids past a semi-infinite flat plate,” AIChE Journal, vol. 15, no. 3, pp. 367–370, 1969. View at Publisher · View at Google Scholar
  49. H. Yaghoobi and M. Torabi, “The application of differential transformation method to nonlinear equations arising in heat transfer,” International Communications in Heat and Mass Transfer, vol. 38, no. 6, pp. 815–820, 2011. View at Publisher · View at Google Scholar · View at Scopus
  50. M. M. Rashidi, A. J. Chamkha, and 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, vol. 6, pp. 119–128, 2011. View at Google Scholar
  51. M. Keimanesh, M. M. Rashidi, A. J. Chamkha, and 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, vol. 62, no. 8, pp. 2871–2891, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  52. L.-T. Yu and C.-K. Chen, “The solution of the Blasius equation by the differential transformation method,” Mathematical and Computer Modelling, vol. 28, no. 1, pp. 101–111, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  53. B.-L. Kuo, “Application of the differential transformation method to the solutions of Falkner-Skan wedge flow,” Acta Mechanica, vol. 164, no. 3-4, pp. 161–174, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  54. Z. M. Odibat, C. Bertelle, M. A. Aziz-Alaoui, and G. H. E. Duchamp, “A multi-step differential transform method and application to non-chaotic or chaotic systems,” Computers and Mathematics with Applications, vol. 59, no. 4, pp. 1462–1472, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  55. H.-S. Peng and C.-L. Chen, “Hybrid differential transformation and finite difference method to annular fin with temperature-dependent thermal conductivity,” International Journal of Heat and Mass Transfer, vol. 54, no. 11-12, pp. 2427–2433, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  56. Y.-L. Yeh, C. C. Wang, and M.-J. Jang, “Using finite difference and differential transformation method to analyze of large deflections of orthotropic rectangular plate problem,” Applied Mathematics and Computation, vol. 190, no. 2, pp. 1146–1156, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  57. H. P. Chu and L. Ch. Chen, “Hybrid differential transform and finite difference method to solve the nonlinear heat conduction problem,” Communications in Nonlinear Science and Numerical Simulation, vol. 13, pp. 1605–1614, 2008. View at Publisher · View at Google Scholar