NEW ANALYTICAL SOLUTION FOR NATURAL CONVECTION OF DARCIAN FLUID IN POROUS MEDIA PRESCRIBED SURFACE HEAT FLUX

Abstract

A new analytical method called He's Variational Iteration Method is introduced to be applied to solve nonlinear equations. In this method, general Lagrange multipliers are introduced to construct correction functional for the problems. It is strongly and simply capable of solving a large class of linear or nonlinear differential equations without the tangible restriction of sensitivity to the degree of the nonlinear term and also is very user friend because it reduces the size of calculations besides; its iterations are direct and straightforward. In this paper the powerful method called Variational Iteration Method is used to obtain the solution for a nonlinear Ordinary Differential Equations that often appear in boundary layers problems arising in heat and mass transfer which these kinds of the equations contain infinity boundary condition. The boundary layer approximations of fluid flow and heat transfer of vertical full cone embedded in porous media give us the similarity solution for full cone subjected to surface heat flux boundary conditions. The obtained Variational Iteration Method solution in comparison with the numerical ones represents a remarkable accuracy.

Keywords

Dates

  • Submission Date2010-04-24
  • Revision Date2010-06-08
  • Acceptance Date2011-01-06

DOI Reference

10.2298/TSCI100424001V

References

  1. A.H. Nayfeh, Introduction to Perturbation Techniques, John Wiley and Sons, New York (1981).
  2. D.A. Nield, A. Bejan, Convection in Porous Media, third ed., Springer-Verlag, New York, (2006).
  3. I. Pop, T. Y. Na, Natural convection over a frustum of a wavy cone in a porous medium, Mech. Res. Comm. 22 (1995), pp. 181-190.
  4. I. Pop, D. B. Ingham: Convective Heat Transfer: Mathematical and Computational Modeling of Viscous Fluids and Porous Media, Pergamon, Oxford (2001).
  5. P. Cheng, T.T. Le, I. Pop, Natural convection of a Darcian fluid about a cone, International Communications Heat Mass Transfer 12 (1985). pp. 705-717.
  6. J.D. Cole, Perturbation Methods in Applied Mathematics, Blaisdell Waltham, MA, (1968).
  7. A.H. Nayfeh, Perturbation Methods, Wiley, New York, (2000).
  8. A.M. Lyapunov, General Problem on Stability of Motion, Taylor & Francis, London, (English translation). (1992) .
  9. A.V. Karmishin, A.I. Zhukov, V.G. Kolosov, Methods of Dynamics Calculation and Testing for Thin-Walled Structures, Mashinostroyenie, Moscow, (1990).
  10. G. Adomian, Solving Frontier Problems on Physics: The Decomposition Method, Kluwer Academic, Dordrecht, (1994).
  11. J.H. He, Homotopy Perturbation Method for Bifurcation of Nonlinear Problems, International Journal of Nonlinear Sciences and Numerical Simulation. 6 (2005). pp. 207-211.
  12. J.H. He, The homotopy perturbation method for nonlinear oscillators with discontinuities Applied Mathematics and Computation, 151 (2004). pp. 287-292.
  13. D.D. Ganji, A. Rajabi, Assessment of homotopy-perturbation and perturbation methods in heat radiation equations International Communications in Heat and Mass Transfer, 33 (2006), pp. 391-400.
  14. J.H. He, Homotopy perturbation method for solving boundary value problems, Physics Letters A, 350 (2006), pp. 87-88.
  15. J.H. He, Some asymptotic methods for strongly nonlinear equations, International Journal of Modern Physics B 20 (2006), pp. 1141-1199.
  16. Ganji, D.D., H. Mirgolbabaei, Me.miansari, Mo. Miansari, "Application of homotopy perturbation method to Solve linear and non-linear systems of ordinary differential equations and differential equation of order three", Journal of Applied Sciences 8 (2008), pp. 1256-1261.
  17. H. Mirgolbabaei, D.D. Ganji, H. Taherian, "Soliton sulotion of the Kadomtse-Petviashvili equation by homotopy perturbation method", World Journal of Modeling and Simulation 5 (2009), pp.38-44 .
  18. H. Mirgolbabaei, D.D. Ganji, "Application of homotopy perturbation method to the combined KdV-MKdV equation", Journal of Applied Sciences, 9 (2009), pp. 3587-3592.
  19. M. Omidvar, A. Barari, H. Mirgolbabaei, D.D. Ganji, "Solution of Diffusion Equations using Homotopy Perturbation and Variational Iteration Methods", MAEJO International Journal of Science and Technology(in press).
  20. A. Barari, H. Mirgolbabaei, D.D. Ganji, "Application of He's Variational Iteration Method to Nonlinear Helmholtz and Fifth-order KDV Equations", SDU Journal of Science(in press).
  21. He JH. A coupling method of homotopy and perturbation technique for nonlinear problems. International Journal of Nonlinear Mech. 35 (2000), pp. 37-43.
  22. He JH. Homotopy perturbation method, a new nonlinear analytic technique. Applied Mathematics and computations. 135 (2003), pp. 73-79.
  23. He JH. A new approach to nonlinear partial differential equations. Communication in Nonlinear Sciences and Numerical Simulation. 2 (1997), pp. 230-235.
  24. He JH. Comparison of homotopy6 perturbation and homotopy analysis method. Applied Mathematics and Computations, 156 (2004), pp. 527-539.
  25. He JH. An approximation solution technique depending upon an artificial parameter Communication in Nonlinear Sciences and Numerical Simulation. 3(1998), pp. 92-97.
  26. He JH. Application of homotopy perturbation method to nonlinear wave equations. Chaos Solutions and Fractals. 26 (2005), pp. 695-700.
  27. He JH. The homotopy perturbation method for nonlinear oscillators with discontinuities. Applied Mathematics and Computations. 151(2004), pp. 287-292.
  28. He JH. A simple perturbation approach to Blasius equation. Applied Mathematics and Computations. 140 (2003), pp. 217-222 .
  29. Öziş T., A. Yildrim., Comparison between Adomian's method and He's homotopy perturbation method. Computers and Mathematics with Applications. 56 (2008), pp. 1216-1224.
  30. Konuralp, A., the steady temperature distributions with different types of nonlinearities. ( in press) (2009).
  31. J.H. He, Variational iteration method for autonomous ordinary differential systems, Applied Mathematics and Computation, 114 (2000), pp. 115-123.
  32. J.H. He, Approximate solution of nonlinear differential equations with convolution product nonlinearities, Computer Methods in Applied Mechanics and Engineering, 167 (1998), pp. 69-73.
  33. D.D. Ganji, A. Sadighi, Application of homotopy-perturbation and variational iteration methods to nonlinear heat transfer and porous media equations, Journal of Computational and Applied Mathematics . 207 (2007), pp. 24-34.
  34. MATLAB 6 - the language of technical computing; 2006.
Volume 15, Issue 12, Pages221 -227