Springer Nature is making Coronavirus research free. View research | View latest news | Sign up for updates

Thermal Analysis of Natural Convection in Porous Fins with Homotopy Perturbation Method (HPM)

  • 963 Accesses

  • 6 Citations


In this paper, porous fin has been studied and its nonlinear ordinary differential equation has been solved through homotopy perturbation method. In this method, a homotopy is introduced to be constructed for the equation. The initial approximations can be freely chosen with possible unknown constants, which can be determined by imposing the boundary and initial conditions. To study the thermal performance, one type case is considered: finite-length fin with insulated tip.



Thermal conductivity


Darcy number, K/t2


Thermal conductivity ratio, (k eff /k f )


Permeability of porous fin




convection parameter


Heat transfer rateα


Rayleigh number, Gr × Pr


Porous parameter


Temperature at any point


Temperature at fin base


Thickness of the fin

ν W(x) :

Velocity of fluid passing through the fin anypoint


Width of the fin


Axial coordinate


Dimensionless axial coordinate, (x/L)

θ :

Dimensionless temperature


Base temperature difference, (Tb–T∞)


Solid properties


Fluid properties


Porous properties


  1. 1

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

  2. 2

    Hamel G.: Spiralförmige Bewgungen Zäher Flüssigkeiten. Jahresber Deutsch Math- Verein. 25, 34–60 (1916)

  3. 3

    Rosenhead L.: The steady two-dimensional radial flow of viscous fluid between two inclined plane walls. Proc. R. Soc. A. 175, 436–467 (1940)

  4. 4

    Batchelor K.: An Introduction to Fluid Dynamics. Cambridge University Press, London (1967)

  5. 5

    Sobey I.J., Drazin P.G.: Bifurcations of two-dimensional channel flows. J. Fluid Mech. 171, 263–287 (1986)

  6. 6

    Hamadiche M., Scott J., Jeandel D.: Temporal stability of Jeffery-Hamel flow. J. Fluid Mech. 268, 71–88 (1994)

  7. 7

    Fraenkel L.E.: Laminar flow in symmetrical channels with slightly curved walls.on the Jeffery-Hamel solutions for flow between plane walls. Proc. R. Soc. Lond. A. 267, 119–138 (1962)

  8. 8

    Makinde O.D., Mhone P.Y.: Hermite-Padé approximation approach to MHD Jeffery- Hamel flows. Appl. Math. Comput. 181, 966–972 (2006)

  9. 9

    Schlichting H.: Boundary-layer theory. McGraw-Hill Press, New York (2000)

  10. 10

    Rathy R.K.: An introduction to fluid dynamics. Oxford and IBH Pl, New Delhi (1976)

  11. 11

    McAlpine A., Drazin P.G.: On the spatio-temporal development of small perturbations of Jeffery-Hamel flows. Fluid Dynam. Res. 22, 123–138 (1998)

  12. 12

    Tolou N., Ganji D.D., Hosseini M.J., Ganji Z.Z.: Application of homotopy perturbation method in nonlinear heat diffusionconvection- reaction equations. Open Mech. J. 1, 20–25 (2007)

  13. 13

    Jamshidi N., Ganji D.D., Ganji Z.Z.: HPM and VIM methods for finding the exact solutions of the nonlinear dispersive equations and seventh-order Sawada-Kotara equation. Int. J. Modern Phys. B 23(1), 39–52 (2009)

  14. 14

    He J.H.: Homotopy perturbation method: a new nonlinear analytical technique. J. Appl. Math. Comput. 135, 73–79 (2003)

  15. 15

    He J.H.: The Homotopy perturbation method for nonlinear oscillators with discontinuities. J. Appl. Math. Comput. 151, 287–292 (2004)

  16. 16

    Ganji D.D., Sadighi A.: Application of homotopy-perturbation and variational iteration methods to nonlinear heat transfer and porous media equations. J. Comput. Appl. Math. 207, 24–34 (2007)

  17. 17

    Ganji D.D.: The application of Hes homotopy-perturbation method to nonlinear heat equations arising in heat transfer. Phys. Lett. A 355, 337–341 (2006)

  18. 18

    Ganji D.D., Jamshidi N.: Application of energy balance method and variational iteration method to an oscillation of a mass attached to a stretched elastic wire. Int. J. N. Sc. N. Sim. 7, 413–420 (2006)

  19. 19

    Sweilam N.H., Khader M.M.: Variational iteration method for one dimensional nonlinear thermoelasticity. Chaos Soliton Fract. 32, 145–149 (2007)

  20. 20

    He J.H.: Variational iteration method-a kind of non-linear analytical technique: Some examples. Int. J. Non-Linear Mech. 34(4), 699–708 (1999)

  21. 21

    Odibat Z.M., Momani S.: Application of variational iteration method to nonlinear differential equations of fractional order. Int. J. Nonlinear Sci. 7, 27–34 (2006)

  22. 22

    He J.H.: Variational iteration method for autonomous ordinary differential systems. A Math. Com. 114, 115–123 (2000)

  23. 23

    Ganji D.D., Jannatabadi M., Mohseni E.: Application of Hes variational iteration method to nonlinear Jaulent-Miodek equations and comparing it with ADM. J. Comput. Appl. Math 207(1), 35–45 (2006)

  24. 24

    Abbasbandy S.: Application of He’s homotopy perturbation method to functional integral equations. Chaos, Solitons Fract. 31, 1243–1247 (2007)

  25. 25

    Pakdemirli M., Karahan M.M.F., Boyaci H.: A new perturbation algorithm with better convergence properties: multiple scales lindstedt Poincare method. J. Comput. Appl. Math. 14, 31–44 (2009)

  26. 26

    Pakdemirli M., Karahan M.M.F.: A new perturbation solution for systems with strong quadratic and qubic nonlinearities. Math. Meth. Appl. Sci. 33, 704–712 (2010)

  27. 27

    Pakdemirli M., Karahan M.M.F., Boyaci H.: Forced vibrations of strongly nonlinear systems with multiple scales lindstedt Poincare method. J. Appl. Math. Comput. 16(4), 879–889 (2011)

  28. 28

    Pakdemirli M., Aksoy Y., Boyaci H.: A new perturbation iteration approach for first order differential equations. J. Appl. Math. Comput. 16(4), 890–899 (2011)

  29. 29

    Pakdemirli M., Sahin A.Z.: Group classification of fin equation with variable thermal properties. Int. J. Eng. Sci. 42, 1875–1889 (2004)

  30. 30

    Pakdemirli M., Sahin A.Z.: Similarity analysis of a nonlinear fin equation. Appl. Math. Lett. 19, 378–384 (2006)

  31. 31

    Kocak H., Yildirim A., Zhang D.H., Mohyud-Din S.T.: The comparative boubaker polynomials expansion scheme (BPES) and Homotopy Perturbation Method (HPM) for solving astandard nonlinear second order boundary value problem. J. Comput. Math . model. 54, 417–422 (2011)

  32. 32

    Raftari B., Yildirim A.: Series solution of a nonlinear ODE Arising in Magnetohydrodynamic by HPM-Pade Technique. J. Comput. Appl. Math. 61, 1676–1681 (2011)

  33. 33

    Khan Y., Faraz N., Yildirim A., Wu Q.: A series of the long porous slider. Tribol. Trans. 54(2), 187–191 (2011)

Download references

Author information

Correspondence to Majid Shahbabaei.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

Reprints and Permissions

About this article

Cite this article

Saedodin, S., Shahbabaei, M. Thermal Analysis of Natural Convection in Porous Fins with Homotopy Perturbation Method (HPM). Arab J Sci Eng 38, 2227–2231 (2013).

Download citation


  • Porous fin
  • Homotopy perturbation method (HPM)
  • Nonlinear ordinary differential equation
  • Finite-length fin with insulated tip