(Received 1 February 2012, accepted 29 June 2012) address: kamyar (K. Hosseini)

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1 ISSN (print), (online) International Journal of Nonlinear Science Vol.14(2012) No.2,pp Homotopy Analysis Method for a Fin with Temperature Dependent Internal Heat Generation and Thermal Conductivity K. Hosseini 1, B. Daneshian 1, N. Amanifard 2, R. Ansari 2 1 Department of Mathematics, Lahijan Branch, Islamic Azad University, P.O. Box 1616, Lahijan, Iran 2 Department of Mechanical Engineering, University of Guilan, P.O. Box 3756, Rasht, Iran (Received 1 February 2012, accepted 29 June 2012) Abstract: In this paper, an approximate but very accurate solution of a fin with temperature dependent internal heat generation and temperature dependent thermal conductivity is formally obtained using a powerful analytical method called the homotopy analysis method (HAM). A comparative study is also made between the results obtained by means of the HAM and a numerical procedure based on the finite difference method with Richardson extrapolation. It is shown that the freedom of choice of the auxiliary parameter h gives way to adjust and control the convergence of the solution series, which can be considered as a fundamental difference between the homotopy analysis method and other existing methods. Results confirm that the temperature distribution in the fin with temperature dependent internal heat generation and temperature dependent thermal conductivity can be accurately predicted with the HAM solution. Keywords: Fin; temperature dependent internal heat generation; temperature dependent thermal conductivity; homotopy analysis method; approximate analytical solution 1 Introduction Most of the problems arising in science and engineering can be expressed in terms of differential equations. However, it is not always easy and possible to obtain the exact solution for these equations, particularly for the nonlinear ones. This makes the development of approximate and numerical solutions clearly essential. In recent years, many analytical methods such as the homotopy analysis method [1-4], the homotopy perturbation method [5-10], the variational iteration method [11-16] and the Adomian decomposition method [17-22] have been utilized to solve linear and nonlinear problems approximately and exactly. Of these, the HAM was first proposed by Liao with the use of the concept of the homotopy in topology. Being different from perturbation techniques, the HAM is independent of any small/large physical parameters. Moreover, the HAM provides us with a simple way to ensure the convergence of solution series. Many researchers have applied this powerful method for their problems. To mention just some of applications of this well-established analytical technique, Moghimi and Ahmadian applied this method in studying dynamic pull-in instability of Microsystems [23]. Yucel employed the HAM for obtaining an approximate analytical solution of the sine-gordon equation with initial conditions [24]. Tan and Abbasbandy adopted the homotopy analysis method for solving quadratic Riccati differential equation [25]. Ziabakhsh and Domairry utilized the HAM for finding the solution of the laminar viscous flow in a semiporous channel in the presence of a uniform magnetic field [26]. Jafari et al. used the homotopy analysis method for solving multi-term linear and nonlinear diffusion-wave equations of fractional order [27]. The readers are also referred to see [28-38]. Therefore, in the present article, the homotopy analysis method will be applied for obtaining an approximate analytic but very accurate solution of a fin with temperature dependent internal heat generation and temperature dependent thermal conductivity, and it will be shown that the temperature distribution in the fin with temperature dependent internal heat generation and temperature dependent thermal conductivity can be accurately predicted with the HAM solution. The rest of this article has been arranged as follows: In Section 2, the basic ideas of the HAM are expressed. In Section 3, the problem formulation is presented. In Section 4, the method is applied to solve the fin with temperature dependent internal heat generation and temperature dependent thermal conductivity. In Section 5, a comparative study is made between Corresponding author. address: kamyar hosseini@yahoo.com (K. Hosseini) Copyright c World Academic Press, World Academic Union IJNS /656

2 202 International Journal of Nonlinear Science, Vol.14(2012), No.2, pp the results obtained using the HAM and a numerical procedure based on the finite difference method with Richardson extrapolation, and finally conclusion is given in Section 6. Nomenclature T local fin temperature, K ε internal heat generation parameter, K 1 θ dimensionless temperature, (T T )/(T b T ) X axial distance measured from the tip of the fin, dimensionless N fin parameter or convection-conduction parameter, dimensionless G generation number, dimensionless ε G internal heat generation parameter, dimensionless k thermal conductivity, W/mK thermal conductivity parameter, dimensionless ε C 2 Basic ideas of homotopy analysis method To illustrate the basic ideas of this method, consider the following general nonlinear differential equation N (ϑ (V)) = 0, where N is a nonlinear differential operator, V is an independent variable and ϑ (V) is an unknown function. For simplicity, we ignore all boundary or initial conditions, which can be treated in a similar way. By means of generalizing the traditional homotopy method, Liao constructed the following zeroth-order deformation equation (1 q) L(ϕ (V; q) ϑ 0 (V)) = qhh(v)n (ϕ (V; q)), (1) where q [0, 1] is the embedding parameter, L is an auxiliary linear operator, ϑ 0 (V) is an initial guess, h is a nonzero auxiliary parameter, H(V) is an auxiliary function and ϕ (V; q) is an unknown function of the independent variables V and q. Clearly, from the zeroth-order deformation Eq. (1), we have ϕ (V; 0) = ϑ 0 (V), ϕ (V; 1) = ϑ (V), in fact, by changing the value of q from zero to unity, ϕ (V; q) changes from initial guess ϑ 0 (V) to the solution ϑ (V). Expanding ϕ (V; q) in the Taylor series with respect to the embedding parameter q, gives where ϕ (V; q) = ϑ 0 (V) + + m=1 ϑ m (V) = 1 m ϕ (V; q) m! q m ϑ m (V)q m, (2) q = 0 If the auxiliary linear operator, the initial guess, the auxiliary parameter and the auxiliary function are properly chosen, the series (2) converges at q = 1, thus we have ϑ (V) = ϑ 0 (V) + + m=1 ϑ m (V). Differentiating the zeroth-order deformation Eq. (1), m times with respect to q and then setting q = 0 and finally dividing each side by m!, the mth-order deformation equation can be obtained as follows L(ϑ m (V) χ m ϑ m 1 (V)) = hh(v)r m,. where R m = 1 (m 1)! m 1 N (ϕ (V; q)) q m 1 q = 0, IJNS for contribution: editor@nonlinearscience.org.uk

3 K. Hosseini, B. Daneshian, N. Amanifard, R. Ansari : Homotopy Analysis Method For a Fin With Temperature Dependent 203 and The partial sum χ m = ϑ 0 (V) + { 0 m 1, 1 m > 1. M ϑ m (V), m=1 yields the M th-order approximation of the considered problem. It is worth noting that the HAM provides us with great freedom in choosing the initial guess ϑ 0 (V) and the auxiliary linear operator L. More importantly, the freedom of choice of the auxiliary parameter h gives way to ensure the convergence of the solution series. In order to apply the HAM for solving the fin with temperature dependent internal heat generation and temperature dependent thermal conductivity, it is necessary to express the following Theorems. Theorem 2.1 Let D m (ϕ κ ) denote 1 m (ϕ κ ) m! q m Proof. Please see [30]. D m (ϕ κ ) = m Corollary 2.1 From Theorem 2.1, we have D m (ϕ κ 1 ϕ ) = q = 0 r 1 r 2 ϑ m r1 ϑ r1 r 2 r 1 =0 r 2 =0 r 3 =0 m r κ 2 r κ 1 =0 where ϕ = ϑ m=1 ϑ mq m, then ϑ r2 r 3 ϑ rκ 2 r κ 1 ϑ rκ 1. r 1 r 2 ϑ m r1 ϑ r1 r 2 r 1 =0 r 2 =0 r 3 =0 ϑ r2 r 3... r κ 3 r κ 2 =0 r κ 3 r κ 2 =0 ϑ rκ 3 r κ 2 ϑ rκ 3 r κ 2 and D m (ϕ κ 1 ϕ ) = m r κ 2 r κ 1 =0 ϑ rκ 2 r κ 1 ϑ r κ 1, r 1 r 2 ϑ m r1 ϑ r1 r 2 r 1=0 r 2=0 r 3=0 ϑ r2 r 3... r κ 3 r κ 2=0 ϑ rκ 3 r κ 2 r κ 2 r κ 1 =0 ϑ rκ 2 r κ 1 ϑ r κ 1. 3 Problem formulation 3.1 Fin with temperature dependent internal heat generation and constant thermal conductivity Consider a longitudinal fin of profile area A, length L, perimeter P and a constant thermal conductivity k 0. The coordinate x is measured from the tip of the fin. The fin is attached to a primary surface at fixed temperature T b and loses heat by convection to the surrounding medium. The fin contains an internal heat source of strength q which is assumed to depend linearly on the local fin temperature. The sink temperature for convection is taken to be T. The convective heat transfer coefficient h over the exposed surface of the fin is considered to be a constant. The heat loss from the tip of the fin is assumed to be negligible compared to the top and bottom surfaces of the fin. For the fin to be effective, the transverse Biot number should be small, which leads to neglecting the temperature variation in the transverse direction. Thus, the heat conduction is considered to take place solely in the longitudinal direction. For the problem described, the governing IJNS homepage:

4 204 International Journal of Nonlinear Science, Vol.14(2012), No.2, pp differential equation and the boundary conditions can be written as [39] d 2 T dx 2 hp k 0 A (T T ) + q = 0, (3) k 0 x = 0 : dt = 0, dx (4) x = L : T = T b, (5) where q = q (1+ε (T T )) such that q is the internal heat generation at temperature T. The boundary condition (4) represents a tip with no heat loss (adiabatic) while the boundary condition (5) describes the constant temperature condition at the base of the fin. Now, by introducing the following dimensionless quantities θ = (T T ) (T b T ), X = x L, N 2 hp L2 = k 0 A, G = q A hp (T b T ), ε G = ε (T b T ), Eqs. (3)-(5) are converted into d 2 θ dx 2 N 2 θ + N 2 G (1 + ε G θ) = 0, X = 0 : dθ dx = 0, X = 1 : θ = Fin with temperature dependent internal heat generation and temperature dependent thermal conductivity Assuming the thermal conductivity of the fin to vary linearly with temperature, we have [39] k = k 0 (1 + β (T T )), (6) where the constant β is a measure of the thermal conductivity variation with temperature. Eq. (6) can be written in dimensionless form as the following k k 0 = 1 + ε C θ, in which ε C = β (T T ). The governing equation in this case is d dx ((1 + ε Cθ) dθ dx ) N 2 θ + N 2 G(1 + ε G θ) = 0, (7) with the following boundary conditions X = 0 : dθ = 0, dx (8) X = 1 : θ = 1. (9) 4 Implementation of the HAM to the governing equations To solve Eq. (7) with (8) and (9) by means of the HAM, we choose a set of base functions as follows {X 2n n = 0, 1, 2,... }, also, we choose θ 0 (X) = 1 as the initial guess and as the linear operator, with the property L(ϕ (X; q)) = d2 ϕ (X; q) dx 2, L(c 1 + c 2 X)= 0, IJNS for contribution: editor@nonlinearscience.org.uk

5 K. Hosseini, B. Daneshian, N. Amanifard, R. Ansari : Homotopy Analysis Method For a Fin With Temperature Dependent 205 where c 1 and c 2 are the constants. The auxiliary function H(X) is considered to be one. From Eq. (7), we define N (ϕ (X; q)) = d dx ((1 + ε dϕ (X; q) Cϕ (X; q)) dx ) N 2 ϕ (X; q) + N 2 G(1 + ε G ϕ (X; q)). Now, one can construct the zeroth-order deformation equation (1 q) L(ϕ (X; q) θ 0 (X)) = qh( d dx ((1 + ε dϕ (X; q) Cϕ (X; q)) dx ) N 2 ϕ (X; q) + N 2 G(1 + ε G ϕ (X; q))), (10) with the following boundary conditions ϕ (1; q) = 1, (11) ϕ (0; q) = 0. (12) Differentiating the zeroth-order deformation Eq. (10) and the boundary conditions (11) and (12) m times (m 1) with respect to q and then setting q = 0 and finally dividing them by m!, one can obtain the so-called mth-order deformation equation L(θ m (X) χ m θ m 1 (X)) = hr m, (13) with the boundary conditions where R m and χ m are in the following form R m = d2 θ m 1 dx 2 + ε C θ m (1) = 0, θ m (0) = 0, (14) m 1 n=0 d 2 m 1 θ n θ m 1 n dx 2 + ε dθ m 1 n dθ n C dx dx n=0 N 2 θ m 1 + N 2 G (1 + ε G θ m 1 ) χ m N 2 G, and χ m = { 0 m 1, 1 m > 1. The general solution of Eq. (13) can be written as the following θ m (X) = θ m (X) + c 2 X + c 1, where c 1 and c 2 are the constants and θ m (X) is the special solution. From the boundary conditions (14), we have c 2 = (θ m) (0), c 1 = c 2 θ m (1). The problems above can be readily solved by symbolic computation packages such as Maple and Mathematica. Thereupon, successive solving of these problems yields θ 1 (X) = 1 2 ( hn 2 G hn 2 + hn 2 G ε G ) X hn 2 G hn hn 2 G ε G, θ 2 (X) = 1 24 h2 N 4 X 4 (G ε G + G 1) (G ε G 1) ( h2 N 2 + h 2 N 2 G ε G +h 2 N 2 G hn 2 + hn 2 G ε G + hn 2 G h 2 N 2 ε C + h 2 N 2 G ε G ε C + h 2 N 2 G ε C + h 2 N 4 G ε G 1 2 h2 N 4 G 2 ε 2 G 1 2 h2 N 4 G 2 ε G 1 2 h2 N h2 N 4 G)X h2 N 4 G ε G h2 N 4 G 2 ε 2 G h2 N 4 G 2 ε G h2 N h2 N 4 G h2 N h2 N 2 G ε G 1 2 h2 N 2 G hn hn 2 G ε G 1 2 hn 2 G h2 N 2 ε C 1 2 h2 N 2 G ε G ε C 1 2 h2 N 2 G ε C,. IJNS homepage:

6 206 International Journal of Nonlinear Science, Vol.14(2012), No.2, pp where a 5th-order approximation of the fin with temperature dependent internal heat generation and temperature dependent thermal conductivity is as follows θ(x) = θ 0 (X) + 5 θ m (X). (15) It is noted that setting ε C = 0 in (15), yields a 5th-order approximation of the fin with temperature dependent internal heat generation and constant thermal conductivity. m=1 5 Results and discussion In this section, to illustrate the efficiency of the method, comparisons are made. For problem 3-1, we first plot the so-called h-curves of θ (0) obtained by the 5th-order approximation of the HAM at N = 1, when G = ε G = 0.2, G = ε G = 0.4 and G = 0.6, ε G = 0.4; and N = 0.5, when G = ε G = 0.2, G = ε G = 0.4 and G = 0.6, ε G = 0.4; as shown in Figs. 1 and 2. By means of these so-called h-curves, an appropriate range for h can be easily recognized so that the convergence of the solution series is assured. Fig. 3 demonstrates the temperature distribution in a fin with temperature dependent internal heat generation and constant thermal conductivity at N = 1, when G = ε G = 0.2, G = ε G = 0.4 and G = 0.6, ε G = 0.4. From this Figure, the efficiency and robustness of the HAM in solving approximately the fin with temperature dependent internal heat generation and constant thermal conductivity is evident. Fig. 4 shows the temperature distribution in a fin with temperature dependent internal heat generation and constant thermal conductivity at N = 0.5, when G = ε G = 0.2, G = ε G = 0.4 and G = 0.6, ε G = 0.4. From this Figure, it is observed that the results obtained by the HAM and numerical procedure are in good agreement. This confirms that the temperature distribution in the fin with temperature dependent internal heat generation and constant thermal conductivity can be accurately predicted with the HAM solution. Figure 1: The h-curves of θ (0) obtained by the 5thorder approximation of the HAM, when: Dashdot (N = 1, G = 0.2 and ε G = 0.2); Dash (N = 1, G = 0.4 and ε G = 0.4); Longdash (N = 1, G = 0.6 and ε G = 0.4). Figure 2: The h-curves of θ (0) obtained by the 5thorder approximation of the HAM, when: Dashdot (N = 0.5, G = 0.2 and ε G = 0.2); Dash (N = 0.5, G = 0.4 and ε G = 0.4); Longdash (N = 0.5, G = 0.6 and ε G = 0.4). Similarly, for problem 3-2, we depict the so-called h-curves of θ (0) obtained by the 5th-order approximation of the HAM for different values of ε C, when N = 1 and G = ε G = 0.2; N = 1 and G = ε G = 0.4; and N = 1 and G = 0.6, ε G = 0.4; as illustrated in Figs. 5, 6 and 7. With the use of these h-curves, an appropriate range for h can be readily IJNS for contribution: editor@nonlinearscience.org.uk

7 K. Hosseini, B. Daneshian, N. Amanifard, R. Ansari : Homotopy Analysis Method For a Fin With Temperature Dependent 207 Figure 3: Temperature distribution in a fin with temperature dependent internal heat generation and constant thermal conductivity at N = 1, when G = ε G = 0.2, G = ε G = 0.4 and G = 0.6, ε G = 0.4. Figure 4: Temperature distribution in a fin with temperature dependent internal heat generation and constant thermal conductivity at N = 0.5, when G = ε G = 0.2, G = ε G = 0.4 and G = 0.6, ε G = 0.4. chosen such that the convergence of the solution series is assured. Fig. 8 shows the temperature distribution in a fin with temperature dependent internal heat generation and temperature dependent thermal conductivity for different values of ε C, when N = 1, G = 0.2 and ε G = 0.2. From this Figure, an excellent agreement is found between the results of the HAM and the numerical procedure. In fact, the freedom of choice of the auxiliary parameter h gives way to ensure the convergence of the solution series, which can be considered as a fundamental difference between the homotopy analysis method and other methods. It is also observed that the temperature θ(x) increases as the thermal conductivity parameter ε C increases. Figs. 9 and 10 indicate the effect of thermal conductivity parameter ε C on the temperature distribution in a fin with N = 1 and G = ε G = 0.4; and N = 1 and G = 0.6, ε G = 0.4; respectively. From these Figures, a good agreement is found between the results of the HAM and the numerical procedure based on the finite difference method with Richardson extrapolation, which confirms that the temperature distribution in the fin with temperature dependent internal heat generation and temperature dependent thermal conductivity can be accurately predicted with the HAM solution. Figs. 9 and 10 also show that the temperature θ(x) increases as the thermal conductivity parameter ε C increases. 6 Conclusion In this article, an approximate but very accurate solution of a fin with temperature dependent internal heat generation and temperature dependent thermal conductivity was successfully obtained using a powerful analytic method called the homotopy analysis method. It was shown that 1) The temperature distribution in the fin with temperature dependent internal heat generation and temperature dependent thermal conductivity can be accurately predicted with the HAM solution. 2) The freedom of choice of the auxiliary parameter h gives way to adjust and control the convergence of the solution series, which can be considered as a fundamental difference between the homotopy analysis method and other existing methods such as the Adomian decomposition method, homotopy perturbation method and variational iteration method. Moreover, the effect of thermal conductivity parameter ε C on the temperature distribution in the fin was investigated in this paper. It is believed that the homotopy analysis method is an efficient and capable technique in handling a wide variety of engineering problems. IJNS homepage:

8 208 International Journal of Nonlinear Science, Vol.14(2012), No.2, pp Figure 5: The h-curves of θ (0) obtained by the 5thorder approximation of the HAM, when: Dashdot (N = 1, G = 0.2, ε G = 0.2 and ε C = 0.2); Dash (N = 1, G = 0.2, ε G = 0.2 and ε C = 0.4); Longdash (N = 1, G = 0.2, ε G = 0.2 and ε C = 0.6). Figure 6: The h-curves of θ (0) obtained by the 5thorder approximation of the HAM, when: Dashdot (N = 1, G = 0.4, ε G = 0.4 and ε C = 0.2); Dash (N = 1, G = 0.4, ε G = 0.4 and ε C = 0.4); Longdash (N = 1, G = 0.4, ε G = 0.4 and ε C = 0.6). Figure 7: The h-curves of θ (0) obtained by the 5thorder approximation of the HAM, when: Dashdot (N = 1, G = 0.6, ε G = 0.4 and ε C = 0.2); Dash (N = 1, G = 0.6, ε G = 0.4 and ε C = 0.4); Longdash (N = 1, G = 0.6, ε G = 0.4 and ε C = 0.6). Figure 8: Temperature distribution in a fin with temperature dependent internal heat generation and temperature dependent thermal conductivity for different values of ε C (0.2, 0.4 and 0.6), when N = 1, G = 0.2 and ε G = 0.2. IJNS for contribution: editor@nonlinearscience.org.uk

9 K. Hosseini, B. Daneshian, N. Amanifard, R. Ansari : Homotopy Analysis Method For a Fin With Temperature Dependent 209 Figure 9: Temperature distribution in a fin with temperature dependent internal heat generation and temperature dependent thermal conductivity for different values of ε C (0.2, 0.4 and 0.6), when N = 1, G = 0.4 and ε G = 0.4. Figure 10: Temperature distribution in a fin with temperature dependent internal heat generation and temperature dependent thermal conductivity for different values of ε C (0.2, 0.4 and 0.6), when N = 1, G = 0.6 and ε G = 0.4. References [1] S.J. Liao, The proposed homotopy analysis technique for the solution of nonlinear problems, Ph.D. Thesis, Shanghai Jiao Tong University, [2] S.J. Liao, Beyond perturbation: Introduction to the homotopy analysis method, Boca Raton: Chapman and Hall/CRC Press; [3] S.J. Liao, An explicit, totally analytic approximation of Blasius viscous flow problems, Int. J. Nonlinear Mech., 34 (1999): [4] S.J. Liao, On the homotopy analysis method for nonlinear problems, Appl. Math. Comput., 147 (2004): [5] J.H. He, Homotopy perturbation technique, Comput. Methods Appl. Mech. Engng., 178 (1999): [6] J.H. He, Homotopy perturbation method for solving boundary value problems, Phys. Lett. A, 350 (2006): [7] J.H. He, Application of homotopy perturbation method to nonlinear wave equations, Chaos, Solitons and Fractals, 26 (2005): [8] J. Biazar, H. Ghazvini, Exact solution for non-linear Schrodinger equations by He s homotopy perturbation method, Phys. Lett. A, 366 (2007): [9] E. Hetmaniok, I. Nowak, D. Słota, R. Wituła, Application of the homotopy perturbation method for the solution of inverse heat conduction problem, Int. Commun. Heat Mass Transfer, 39 (2012): [10] J. Biazar, R. Ansari, K. Hosseini, P. Gholamin, Application of homotopy perturbation method to Poisson equation, J. Appl. Math., 5 (2008): 1-6. [11] J.H. He, A new approach to non-linear partial differential equations, Commun. Nonlinear Sci. Numer. Simul., 2 (1997): [12] J.H. He, Variational iteration method- a kind of non-linear analytical technique: some examples, Int. J. Nonlinear Mech., 34 (1999): [13] J. Biazar, R. Ansari, K. Hosseini, P. Gholamin, Obtaining D Alembert s wave formula from variational iteration and homotopy perturbation methods, Math. Sci. Quarterly J., 5 (2011): [14] J. Biazar, P. Gholamin, K. Hosseini, Variational iteration method for solving Fokker-Planck equation, J. Franklin Inst., 347 (2010): [15] J. Biazar, H. Aminikhah, Exact and numerical solutions for non-linear Burger s equation by VIM, Math. Comput. Modelling, 49 (2009): [16] R. Ansari, M. Hemmatnezhad, Nonlinear vibrations of embedded multiwalled carbon nanotubes using a variational approach, Math. Comput. Modelling, 53 (2011): [17] G. Adomian, Nonlinear stochastic differential equations, J. Math. Anal. Appl., 55 (1976): IJNS homepage:

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