Application of He s homotopy perturbation method to boundary layer flow and convection heat transfer over a flat plate

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1 Physics Letters A ) Application of He s homotopy perturbation method to boundary layer flow and convection heat transfer over a flat plate M. Esmaeilpour, D.D. Ganji Department of Mechanical Engineering, Mazandaran University, PO Box 484, Babol, Iran Received 4 April 007; accepted July 007 Available online 5 July 007 Communicated by A.R. Bishop Abstract In this Letter, the problem of forced convection over a horizontal flat plate is presented and the homotopy perturbation method HPM) is employed to compute an approximation to the solution of the system of nonlinear differential equations governing on the problem. It has been attempted to show the capabilities and wide-range applications of the homotopy perturbation method in comparison with the previous ones in solving heat transfer problems. The obtained solutions, in comparison with the exact solutions admit a remarkable accuracy. A clear conclusion can be drawn from the numerical results that the HPM provides highly accurate numerical solutions for nonlinear differential equations. 007 Elsevier B.V. All rights reserved. Keywords: Convection heat transfer; Nonlinear equations; Homotopy perturbation method HPM); Numerical method NM) 1. Introduction Most scientific problems such as heat transfer are inherently of nonlinearity. We know that except a limited number of these problems, most of them do not have analytical solution. Therefore, these nonlinear equations should be solved using other methods. Some of them are solved using numerical techniques and some are solved using the analytical method of perturbation. In the numerical method, stability and convergence should be considered so as to avoid divergence or inappropriate results. In the analytical perturbation method, we should exert the small parameter in the equation. Therefore, finding the small parameter and exerting it into the equation are difficulties of this method. Since there are some limitations with the common perturbation method, and also because the basis of the common perturbation method is upon the existence of a small parameter, developing the method for different applications is very difficult. Therefore, many different methods have recently introduced some ways to eliminate the small parameter, such as artificial parameter method introduced by He [1,], the homotopy perturbation method by He [3,4], the variational iteration method by He [5 7]. One of the semi-exact methods is the homotopy perturbation method [8 0]. The applications of this method in different fields of nonlinear equations, integro-differential equations, Laplace transform, fluid mechanics and heat transfer have been studied by Cai [1], Cveticanin [], El-Shahed [3], Abbasbandy [4], Siddiqui [19,0] and Ganji [5 7]. In this Letter, we will apply He s homotopy perturbation method to the problem of forced convection over a horizontal flat plate for finding the approximate solution. * Corresponding author. Tel./fax: addresses: ddg_davood@yahoo.com, mirgang@nit.ac.ir D.D. Ganji) /$ see front matter 007 Elsevier B.V. All rights reserved. doi: /j.physleta

2 34 M. Esmaeilpour, D.D. Ganji / Physics Letters A ) Basic idea of homotopy perturbation method The homotopy perturbation method is a combination of the classical perturbation technique and homotopy technique. To explain the basic idea of the HPM for solving nonlinear differential equations, we consider the following nonlinear differential equation: Au) fr)= 0, r Ω, subject to boundary condition 1) Bu, u/ n) = 0, r Γ, where A is a general differential operator, B a boundary operator, fr)is a known analytical function, Γ is the boundary of domain Ω and u/ n denotes differentiation along the normal drawn outwards from Ω. The operator A can, generally speaking, be divided into two parts: a linear part L and a nonlinear part N.Eq.) therefore can be rewritten as follows: ) Lu) + Nu) fr)= 0. In case the nonlinear equation 1) has no small parameter, we can construct the following homotopy, 3) Hv,p)= Lv) Lu 0 ) + plu 0 ) + p Nv) fr) ) = 0 where p is called homotopy parameter. According to the homotopy perturbation method, the approximation solution of Eq. 4) can be expressed as a series of the power of p, i.e., v = v 0 + pv 1 + p v +, u = lim v p 1 = v 0 + v 1 + v +. When Eq. 4) corresponds to Eq. 1), and Eq. 5b) becomes the approximate solution of Eq. 1). Some interesting results have been attained using this method [11,1,,3,6]. 3. Governing equations Boundary layer flow over a flat plate is governed by the continuity and the Navier Stokes equations. Under the boundary layer assumptions and a constant property assumption, the continuity and Navier Stokes equations become [8]: u x + v y = 0, u u x + v u y = 1 dp ρ dx + ν u y + gβt T ). Under a boundary layer assumption, the energy transport equation is also simplified. u T x + v T y = α T y. From Eqs. 7) and 8), the solutions of the energy and momentum equations are coup led. However, the buoyancy force may be neglected if there is a pressure gradient perpendicular to the gravitational force. Thus, in the case of the forced convection over a horizontal flat plate, the solution to the momentum equation is decoupled from the energy solution. However, the solution of the energy equation is still linked to the momentum solution. The following dimensionless variables are introduced in the transformation: η = y x Re 0.5 x, θη)= T T 10) T W T where θ is nondimensional form of the temperature and the Reynolds number is defined as: Re = u x 11) v. Using Eqs. 6) through 10), the governing equations can be reduced to two equations where f is a function of the similarity variable η): 4) 5a) 5b) 6) 7) 8) 9) f + 1 ff = 0, εθ + 1 fθ = 0 1)

3 M. Esmaeilpour, D.D. Ganji / Physics Letters A ) where ε = Pr 1 and f is related to the u velocity by f = u. 13) u The reference velocity is the free stream velocity of forced convection. The boundary conditions for are obtained from the similarity variables. For the forced convection case [9]: f0) = 0, f 0) = 0, θ0) = 1, f ) = 1, θ ) = HPM solution for flow over a flat plate In this section, we will apply the HPM to nonlinear ordinary differential system 1). According to the HPM, we can construct a homotopy of system 1) as follows: 1 p) f f 0 ) + p f + 1 ) ff We consider f and θ as following: = 0, 1 p) εθ εθ 0 ) + p εθ + 1 ) fθ = 0. f = f 0 + pf 1 + p f + p 3 f 3 +, θ = θ 0 + pθ 1 + p θ + p 3 θ 3 +. Assuming f 0 = θ 0 = 0 and substituting f and θ from Eq. 16) into Eq. 15) and some simplification and rearranging based on powers of p-terms, we have: p 0 : f = 0, εθ = 0, f 0 0) = 0, f 0 0) = 0, f 0 ) = 1, θ 00) = 1, θ 0 ) = 0, p 1 : f ) f 0f 0 = 0, εθ f 0θ 0 = 0, f 1 0) = 0, f 1 0) = 0, f 1 ) = 0, θ 10) = 0, θ 1 ) = 0, 0) 14) 15) 16) 17) 18) p : f + 1 f0 f 1 + f 1f 0 ) = 0, εθ + 1 f0 θ 1 + f 1θ 0) = 0, f 0) = 0, f 0) = 0, f ) = 0, θ 0) = 0, θ ) = 0, p 3 : f f0 f + f 1f 1 + f f 0 ) = 0, εθ f0 θ + f 1θ 1 + f θ 0) = 0, f 3 0) = 0, f 3 0) = 0, f 3 ) = 0, θ 30) = 0, θ 3 ) = 0. Solving Eqs. 17) 3) with boundary conditions 18) 4), wehave: 1) ) 3) 4) f 0 = 1 10 η, f 1 = η η, 11 f = η η η, 1 f 3 = η11 + θ 0 = 1 5 η + 1, η η η, θ 1 = Pr 100 η4 5 Pr 48 η, 1 1 θ = Pr 4 0Pr )η7 + 1 ) 65Pr + 65)η Pr 4 0Pr + ) Pr ) ) η, θ 3 = Pr Pr + 80Pr ) η Pr 36750Pr 3500 ) η Pr Pr ) ) η Pr ) 6) 7) 8) 9) 30) 31) Pr Pr )η. 36 3)

4 36 M. Esmaeilpour, D.D. Ganji / Physics Letters A ) Table 1 The results of HPM and NM methods for fη)and θη) fη) f η) θη) η HPM NM HPM NM HPM NM E According to Eq. 16) and the assumption p = 1, we get: fη)= η η η η11, θη)= 1 5 η Pr η Pr η Pr Pr Pr ) η Pr 36750Pr 3500 ) η Pr Pr )η Pr + 65)η Pr Pr + 80Pr ) η Pr Pr ) ) η 4 Pr Pr )η. 36 ) 33) 34) Since, Eqs. 1) cannot be easily solved by the analytical method; Eqs. 1) is, therefore, solved by the numerical method using the software MAPLE whose results are given in Table 1, and also the consequent results of the two different methods of homotopy perturbation and numerical are compared in Figs. 1,, 3, 4 and 5. As you can see in ε 1, the HPM has a high accuracy. 5. Conclusions In this Letter we have studied heat transfer problem with a small parameter with the homotopy perturbation method. The results show that this perturbation scheme provides excellent approximations to the solution of this nonlinear system with high accuracy. This new method accelerated the convergence to the solutions. As shown in Eq. 5b), the homotopy perturbation method does not need a small parameter. Finally, it has been attempted to show the capabilities and wide-range applications of the homotopy perturbation method in comparison with the previous ones in solving heat transfer problems.

5 M. Esmaeilpour, D.D. Ganji / Physics Letters A ) Fig. 1. The comparison of the answers resulted by HPM and NM for fη). Fig.. The comparison of the answers resulted by HPM and NM for f η). Fig. 3. The comparison of the answers resulted by HPM and NM for θη),at Pr = 1. Fig. 4. The comparison of the answers resulted by HPM and NM for θη),at Pr = 1.. Fig. 5. The comparison of the answers resulted by HPM and NM for θη),at Pr = 1andPr = 1..

6 38 M. Esmaeilpour, D.D. Ganji / Physics Letters A ) Appendix A. Nomenclature g gravitational force v velocity component in the y direction HPM homotopy perturbation method x dimensional vertical coordinate NM numerical method y dimensional horizontal coordinate P pressure p parameter of homotopy Greek symbols Pr Prandtl number ρ density Re Reynolds number β volumetric thermal expansion coefficient T temperature ε small parameter T W temperature imposed on the plate ν kinematic viscosity T local ambient temperature α thermal diffusivity u velocity component in the x direction θ dimensionless temperature References [1] J.H. He, Non-perturbative Methods for Strongly Nonlinear Problems, dissertation.de-verlag im Internet GmbH, Berlin, 006. [] J.H. He, Int. J. Mod. Phys. B 0 10) 006) [3] J.H. He, Phys. Lett. A ) 006) 87. [4] J.H. He, Chaos Solitons Fractals 6 3) 005) 695. [5] J.H. He, J. Comput. Math. Appl. Mech. Eng ) 57. [6] J.H. He, Comput. Math. Appl. Mech. Eng ) 69. [7] J.H. He, Int. J. Non-Linear Mech ) 699. [8] J.H. He, J. Comput. Math. Appl. Mech. Eng. 17 8) 1999) 57. [9] J.H. He, Int. J. Non-Linear Mech ) 37. [10] J.H. He, X.H. Wu, Chaos Solitons Fractals 9 1) 006) 108. [11] J.H. He, Phys. Lett. A ) 005) 8. [1] J.H. He, Chaos Solitons Fractals 6 3) 005) 87. [13] J.H. He, Int. J. Nonlinear Sci. Numer. Simul. 6 ) 005) 07. [14] P.D. Ariel, T. Hayat, S. Asghar, Int. J. Nonlinear Sci. Numer. Simul. 7 4) 006) 399. [15] D.D. Ganji, A. Sadighi, Int. J. Nonlinear Sci. Numer. Simul. 7 4) 006) 411. [16] M. Rafei, D.D. Ganji, Int. J. Nonlinear Sci. Numer. Simul. 7 3) 006) 31. [17] A.M. Siddiqui, R. Mahmood, Q.K. Ghori, Int. J. Nonlinear Sci. Numer. Simul. 7 1) 006) 7. [18] A.M. Siddiqui, M. Ahmed, Q.K. Ghori, Int. J. Nonlinear Sci. Numer. Simul. 7 1) 006) 15. [19] A. Beléndez, T. Hernández, et al., Int. J. Nonlinear Sci. Numer. Simul. 8 1) 007) 79. [0] J.H. He, Int. J. Mod. Phys. B 0 18) 0 July 006) 561. [1] X.C. Cai, W.Y. Wu, M.S. Li, Int. J. Nonlinear Sci. Numer. Simul. 7 1) 006) 109. [] L. Cveticanin, Chaos Solitons Fractals 30 5) 006) 11. [3] M. El-Shahed, Int. J. Nonlinear Sci. Numer. Simul. 6 ) 005) 163. [4] S. Abbasbandy, Chaos Solitons Fractals 30 5) 006) 106. [5] D.D. Ganji, M. Rafei, Phys. Lett. A 356 ) 006) 131. [6] D.D. Ganji, A. Rajabi, Int. Commun. Heat Mass Transfer 33 3) 006) 391. [7] D.D. Ganji, Phys. Lett. A ) 006) 337. [8] W.M. Kays, M.E. Crawford, Convective Heat and Mass Transfer, 3/e, McGraw Hill, New York, [9] A. Bejan, Convection Heat Transfer, Wiley, New York, 1995.

Homotopy Perturbation Method for Solving Systems of Nonlinear Coupled Equations

Applied Mathematical Sciences, Vol 6, 2012, no 96, 4787-4800 Homotopy Perturbation Method for Solving Systems of Nonlinear Coupled Equations A A Hemeda Department of Mathematics, Faculty of Science Tanta