ASIA-PACIFIC JOURNAL OF CHEMICAL ENGINEERING Asia-Pac. J. Chem. Eng. 2011; 6: 669–678 Published online 28 June 2010 in Wiley Online Library (wileyonlinelibrary.com) DOI:10.1002/apj.472 Research article New applications of approximate methods in fluid mechanics Z. Ziabakhsh,1 * G. Domairry,2 * Ammar Domiri2 and S. M. Moghimi3 1 IOR/EOR Research Institute, National Iranian Oil Company (NIOC), Tehran, Iran Department of Mechanical Engineering, Babol University of Technology, PO Box 484, Babol, Iran 3 Department of Mechanical Engineering, Islamic Azad University, Qaemshahr branch, Qaemshahr, Iran 2 Received 22 February 2010; Revised 2 May 2010; Accepted 19 May 2010 ABSTRACT: In this paper, we have modeled boundary layer flows induced by continuous stretched surfaces by implementing one of the newest analytical methods of solving nonlinear differential equations called homotopy analysis method (HAM), which gives us a vast freedom to choose the answer type. We have used an iterating analytical method to cope with the nonlinearity. A new adapting boundary condition is proposed in this work that is based on an initial guess and then it is developed to the solution expression. The analytic results are compared with the numerical solution (NS) and the comparison reveals that a good agreement exists between the NS and HAM solution. Also the convergence of the obtained HAM solution is discussed explicitly. The obtained approximate solutions are valid for all values of the dimensionless parameter β, as it is shown later in the paper. 2010 Curtin University of Technology and John Wiley & Sons, Ltd. KEYWORDS: homotopy analysis method; boundary layer flows; numerical solution; continuous stretched surfaces; heat transfer; nonlinear problem INTRODUCTION The heat transfer over an unsteady stretching permeable surface with prescribed wall temperature has considered by many investigators throughout its applications in engineering practice, particularly in chemical industries. Some common examples are the cases of boundary layer control, transpiration cooling and gaseous diffusion. Crane[1] has presented an exact analytical solution for the steady two-dimensional flow due to a stretching surface in a quiescent fluid; additionally many authors such as Ishak et al .,[2] Gupta et al .[3] and Grubka et al .[4] have considered various aspects of this problem and achieved similarity solutions. It is necessary to mention that through above referred studies which discussed stretching surfaces, the flows were assumed to be steady. Unsteady flows due to stretching surfaces have received less heed; a few of them are those considered by Devi et al .,[5] Andersson et al .,[6] Nazar et al .,[7] and very recently Ali *Correspondence to: Z. Ziabakhsh, IOR/EOR Research Institute, National Iranian Oil Company (NIOC), Tehran, Iran. E-mail: ziagandji@yahoo.com G. Domairry, Department of Mechanical Engineering, Babol University of Technology, PO Box 484, Babol, Iran. E-mail: amirganga111@yahoo.com 2010 Curtin University of Technology and John Wiley & Sons, Ltd. Curtin University is a trademark of Curtin University of Technology and Mehmood[8 – 9] introduced a similar transformation which was used by Williams and Rhyne,[10] in which it transforms the governing partial differential equations with three independent variables in two independent variables, which are more convenient for numerical computations. In addition, S. Nadeem et al .[11] considered unsteady shrinking sheet through porous medium with variable viscosity. In this paper, we explore the analytic solution of the nonlinear heat transfer over an unsteady stretching permeable surface with prescribed wall temperature and will make a comparison between the homotopy analysis method (HAM) and numerical results given by Magyari et al .[12] HAM which was recently developed by Liao[13] is one of the most successful and efficient methods in solving nonlinear equations. In comparison with previous analytic techniques, the following merits can be mentioned for HAM:[14] first and foremost, unlike all previous analytic techniques, the HAM provides us great freedom to express solutions of a given nonlinear problem by means of different base functions. Secondly, unlike all previous analytic techniques, the HAM always provides us with a family of solution expressions in the auxiliary parameter h̄, even if a nonlinear 670 Z. ZIABAKHSH et al. problem has a unique solution. Thirdly, unlike perturbation techniques, the HAM is independent through any small or large quantities. So, the HAM can be applied no matter if governing equations and boundary/initial conditions of a given nonlinear problem contain small or large quantities or not. Finally, in the works of previous authors such as Hayat et al .[15] and Liao,[16] it has also shown that the HAM method logically contains some previous techniques such as Adomian’s decomposition method, Lyapunov’s artificial small parameter method and the δ-expansion method. Many authors such as Liao[17] have applied HAM in solving permeable and impulsively stretched plate; Hayat et al .[18] considered Magnetohydrodynamics (MHD) flow of a micropolar fluid near a stagnation-point toward a nonlinear stretching surface by HAM. Unsteady boundary layer flow adjacent to permeable stretching surface in a porous medium by HAM was considered by Mehmood et al .[19] Domairry and Nadim,[20] Nadeem et al .,[21] Abbasbandy[22] and Ziabakhsh et al .[23 – 24] and others[25 – 33] have successfully applied HAM in solving different types of nonlinear problems i.e. coupled, decoupled, homogeneous and nonhomogeneous equations arising in different physical problems such as heat transfer, fluid flow, oscillatory systems, etc. Above all, there are no rigorous theories to direct us to choose the initial approximations, auxiliary linear operators, auxiliary functions and auxiliary parameter h̄. From practical viewpoints, there are some fundamental rules such as the rule of solution expression, the rule of coefficient ergodicity and the rule of solution existence, which play important roles within the HAM. Unfortunately, the rule of solution expression implies such an assumption that we should have, more or less, some knowledge about a given nonlinear problem a priori. So, theoretically, this assumption impairs the HAM, although we can always attempt some base functions even if a given nonlinear problem is completely new for us. Therefore in the present work, we reexamine the heat transfer characteristics of boundary layer flows induced by continuous stretched surfaces, and attempt to obtain its solution using the HAM. DESCRIPTION OF THE PROBLEM As the stretching sheet interacts with the surrounding fluid both thermally and mechanically, different sets of prescribed boundary mechanical and thermal conditions can be considered. Consider the steady velocity and thermal boundary layers induced by a continuous (in general nonisothermal) stretching surface moving through a quiescent incompressible fluid of constant temperature T∞ (Magyari and Keller[12] ). As shown in Fig. 1 with the usual boundary layer approximations, the governing equations of momentum, energy balance 2010 Curtin University of Technology and John Wiley & Sons, Ltd. Asia-Pacific Journal of Chemical Engineering Figure 1. Schematic diagram of the problem under consideration. and mass transfer are as follows: ∂u ∂v + =0 ∂x ∂y ∂u ∂u u +v =υ ∂x ∂y u (1) ∂ 2u ∂y 2 ∂T ∂T ∂ 2T +υ =α 2 ∂x ∂y ∂y (2) (3) Accordingly to stretching velocity and surface temperature distribution (uw , Tw ), the boundary condition is u(x , 0) = uw (x ), v (x , 0) = 0, u(x , ∞) = 0 (4) T (x , 0) = Tw (x ), T (x , ∞) = T∞ = constant (5) Equations 1–3 lead to an independent flow boundary value problem and a forced thermal convection problem. In terms of the stream function ψ = ψ(x , y) defined by u = ∂ψ/∂y, v = −∂ψ/∂x . Equations 1–3 is reduced to: ∂ψ ∂ 2 ψ ∂ 3ψ ∂ψ ∂ 2 ψ − = v ∂y ∂x ∂y ∂x ∂y 2 ∂y 3 (6) ∂ψ ∂T ∂ψ ∂T ∂ 2T − =α 2 ∂y ∂x ∂x ∂y ∂y (7) By using the transformation as described by Schlichting and Gersten[34] ψ(x , y) = A(x )f (x ), η = B (x )y, T (x , y) = T∞ + C (x )θ (η) (8) Asia-Pac. J. Chem. Eng. 2011; 6: 669–678 DOI: 10.1002/apj Asia-Pacific Journal of Chemical Engineering FLUID MECHANICS where A(x ) = const and B (x ) > 0, and by using similarity solution A(x ) = A0 x m+1/2 , B (x ) = B0 x m−1/2 , m = −1, C (x ) = C0 .x n (9) where A0 , B0 , n and m are constant. With the power low similarity (Eqn 9) and the boundary conditions (Eqns 4 and 5), the quantities of physical become: ψ(x , y) = A0 .x m+1/2 f (η), η = B0 x in which ci (i = 1 − 5) are constants. Let P ∈ [0, 1] denotes the embedding parameter and h̄ indicates nonzero auxiliary parameters. We then construct the following equations: m−1/2 The zeroth-order deformation equations (1 − P )L1 [f (η; p) − f0 (η)] = p h̄ 1 N1 [f (η; p), θ (η; p)] y, (21) m = −1, B0 > 0, T (x , y) = T∞ + C0 x θ (η) n f (0; p) = 0; (10) f (0; p) = 1; f (∞; p) = 0 (22) (1 − P )L2 [θ (η; p) − θ0 (η)] = p h̄ 2 N2 [θ (η; p), f (η; p)] m u(x , y) = A0 B0 x f (η) m −1 m−1/2 m + 1 v (x , y) = −A0 x f (η) + ηf (η) 2 2 (11) (23) θ (0; p) = 1; N1 [f (η; p), θ (η; p)] = m uw (x ) = A0 B0 x f (0), v (x , ∞) m + 1 m−1/2 x f∞ = −A0 2 Tw (x ) = T∞ + C0 x n θ (0) (12) (13) The balance Eqns 6–7 are reduced to the ordinary differential equations for f (η) and θ (η). Then we have bf (η) + f (η)f (η) − βf (η)2 = 0 b θ (η) + f (η)θ (η) − γ f (η)θ (η) = 0 Pr (14) (15) 2vB0 , β = 2m , γ = 2n and where b = m +1 m +1 (m + 1)A0 Pr = v /α is the Prandtl number. The boundary conditions are f (0) = 0, f (0) = 1, f (∞) = 0 (16) θ (0) = 1, θ (∞) = 0 (17) θ (∞; p) = 0 −β df (η; p) dη L1 (f ) = f +f , =0 L2 (θ ) = θ + θ L2 (c4 exp(−η) + c5 ) = 0 When p increases from 0 to 1, then f (η; p) and θ (η; p) vary from f0 (η) and θ0 (η) to f (η) and θ (η). By Taylor’s theorem and using Eqns 27 and 28, f (η; p) and θ (η; p) can be expanded in a power series of p as follows: ∞ f (η; p) = f0 (η) + fm (η)p m , fm (η) = m−1 1 ∂ m (f (η; p)) m! ∂p m (29) ∞ θm (η)p m , θm (η) = 1 ∂ m (θ (η; p)) m! ∂p m (30) where h̄ is chosen in such a way that these two series are convergent at p = 1, therefore we have through Eqns 29 and 30 that ∞ m−1 2010 Curtin University of Technology and John Wiley & Sons, Ltd. (26) (28) f (η) = f0 (η) + (20) 1 d 2 θ (η; p) dθ (η; p) + f (η; p) Pr dη2 dη θ (η; 0) = θ0 (η) θ (η; 1) = θ (η) (19) L1 (c3 exp(−η) + c2 η + c3 ) = 0, (25) (27) m−1 θ0 (η) = exp(−η) (18) dη2 f (η; 0) = f0 (η) f (η; 1) = f (η) APPLICATIONS OF HAM d 2 f (η; p) For p = 0 and p = 1, we have θ (η; p) = θ0 (η) + f0 (η) = 1 − exp(−η), + f (η; p) df (η; p) =0 dη For special case we take b = 1 in Eqns14 and 15 and solve them by HAM. For HAM solutions, we choose the initial guesses and auxiliary linear operators in the following form: dη3 2 N2 [f (η; p), θ (η; p)] = − γ θ (η; p) d 3 f (η; p) (24) fm (η), θ (η) = θ0 (η) + ∞ θm (η) m−1 (31) Asia-Pac. J. Chem. Eng. 2011; 6: 669–678 DOI: 10.1002/apj 671 672 Z. ZIABAKHSH et al. Asia-Pacific Journal of Chemical Engineering The mth-order deformation equations L1 [fm (η) − χm fm−1 (η)] = h̄ 1 Rmf (η) fm (0) = 0; fm (0) = 0; L2 [θm (η) − χm θm−1 (η)] θm (0) = 0; fm (∞) = = h̄ 2 Rmθ (η) (32) 0 (33) (34) θm (∞) = 0 Rmf (η) = f m−1 + m−1 n=0 fn fm−1−n −β (35) m−1 fn fm−1−n n=0 (36) Rmθ (η) = m−1 m−1 1 θm−1 + fm−1−n θn − γ fm−1−n θn Pr n=0 n=0 (37) We have found the solution by Maple analytic solution device. Figure 2. The h̄1 -validity for f (0) when Pr = 1, γ = 0 and β = 3. CONVERGENCE OF THE HAM SOLUTION As mentioned by Liao,[14] HAM provides us with great freedom in choosing the solution of a nonlinear problem by different base functions. This has a great effect on the convergence region because the convergence region and rate of a series are chiefly determined by the base functions used to express the solution. Therefore, we can approximate a nonlinear problem more efficiently by choosing a proper set of base functions and ensure its convergency. On the other hand, as pointed out by Liao, the convergence and rate of approximation for the HAM solution strongly depends on the value of auxiliary parameter h̄. Even if the initial approximation f0 (η) and θ0 (η) and the auxiliary linear operator L, are given, we still have great freedom to choose the value of the auxiliary parameter h̄. So, the auxiliary parameter h̄ provides us with an additional way to conveniently adjust and control the convergence region and rate of solution series. By means of the so-called h̄-curves, it is easy to find out the so-called valid regions of h̄ to gain a convergent solution series. When the valid region of h̄ is a horizontal line segment then the solution (converges) is converged. In our case study, according to Figs 2 and 3, the convergence and rate of approximation for the HAM solution strongly depend on the values of auxiliary parameters h̄ 1 and h̄ 2 . According to Figs 2 and 3 for Pr = 1, γ = 0 and β = 3 the ranges for values of f (0) and θ (0) are −0.4 < h̄ 1 < −0.15 and −1.6 < h̄ 2 < −0.4, respectively. Figures 4 and 5 show how auxiliary parameters h̄ 1 and h̄ 2 varies with changing β. If β increases, then the range of convergency of solution is restricted and then decreased. As well as according to Figs 6 and 7 the 2010 Curtin University of Technology and John Wiley & Sons, Ltd. Figure 3. The h̄2 -validity for θ (0) when Pr = 1, γ = 0 and β = 3. range of convergency is restricted with increasing in Pr and β for θ (0). Finally, Figs 8 and 9 illustrate the effect of parameter γ on convergency range; at these figures by increasing of γ the range of convergency is decreased. NUMERICAL METHOD (NM) The best approximate for solving Eqns 14 and 15 that can be used is fourth order Runge–Kutta method. Asia-Pac. J. Chem. Eng. 2011; 6: 669–678 DOI: 10.1002/apj Asia-Pacific Journal of Chemical Engineering Figure 4. The h̄1 -validity of f (0) for various value of β when γ = 1, Pr = 1 and 11th order of approximation. FLUID MECHANICS Figure 6. The h̄2 -validity of θ (0) for various value of β when γ = 5, Pr = 1.2 and 11th order of approximation. Figure 5. The h̄2 -validity of θ (0) for various value of β when γ = 1, Pr = 1 and 11th order of approximation. It is often utilized to solve differential equation systems. Third-order differential equations can be usually changed into second-order equations and then first order. After that, it can be solved through Runge–Kutta method. RESULT AND DISCUSSION Figures 10–12 illustrate effect of β on the nondimensional boundary layer flow f (η), its derivative f (η) 2010 Curtin University of Technology and John Wiley & Sons, Ltd. Figure 7. The h̄2 -validity of θ (0) for various value of Pr when γ = 1, β = 2 and 11th order of approximation. and temperature θ (η) for Pr = 1 and γ = 0 as well. Figure 13 shows the result of θ (η) for various β when Pr = 1 and γ = 1. Figures 14 and 15 show the result of θ (η) and f (η) for various β when Pr = 1.2 and γ = 5, respectively. Figures 13 and 14 illustrate the decrement of the velocity and temperature profiles as β increases. According to Figs 16 and 17 by increasing γ , the nondimensional temperature θ (η) decreases. By Asia-Pac. J. Chem. Eng. 2011; 6: 669–678 DOI: 10.1002/apj 673 674 Z. ZIABAKHSH et al. Figure 8. The h̄2 -validity of θ (0) for various value of γ when Pr = 0.2, β = 5 and 11th order of approximation. Asia-Pacific Journal of Chemical Engineering Figure 10. The result of f (η) for various β when Pr = 1 and γ = 0. Figure 11. The result of f (η) for various β when Pr = 1 and γ = 0. Figure 9. The h̄2 -validity of θ (0) for various value of γ when Pr = 1, β = 1.5 and 11th order of approximation. increasing Pr in Figs 18 and 19, the nondimensional temperature θ (η) decreases. As shown in Figs 20 and 21, it has been attempted to show the accuracy, capabilities and wide-range applications of the HAM in comparison with the numerical solution (NS) of heat transfer characteristics of boundary layer flows induced by continuous stretched surfaces. Figure 20 shows the boundary layer flow 2010 Curtin University of Technology and John Wiley & Sons, Ltd. f (η), and Fig. 21 indicates the comparison of the nondimensional temperature θ (η) and NS for known values of the parameters Pr = 1, γ = 1 and β = 0. According to the Tables 1–3 these approximate solutions are in excellent agreement with the corresponding NSs. CONCLUSIONS In the present work, we have applied the HAM to compute the heat transfer characteristics of boundary Asia-Pac. J. Chem. Eng. 2011; 6: 669–678 DOI: 10.1002/apj Asia-Pacific Journal of Chemical Engineering Figure 12. The result of θ(η) for various β when Pr = 1 and γ = 0. Figure 13. The result of θ(η) for various β when Pr = 1 and γ = 1. layer flows induced by continuous stretched surfaces. The convergency of the solution series and the power of HAM in controlling and adjusting the convergence region and rate of solution series was discussed. When compared with other analytic methods, it is clear that HAM provides highly accurate analytic solutions for nonlinear problems. The results show that: 1. HAM can give much better approximations for nonlinear differential equations than do the previous solutions including perturbation method in the 2010 Curtin University of Technology and John Wiley & Sons, Ltd. FLUID MECHANICS Figure 14. The result of θ(η) for various β when Pr = 1.2 and γ = 5. Figure 15. The result of f (η) for various β when Pr = 1.2 and γ = 5. heat transfer characteristics of boundary layer flows induced by continuously stretched surfaces problem. 2. The proper range of the auxiliary parameters h̄ 1 and h̄ 2 to ensure the convergency of the solution series were obtained through the so-called h̄-curves. In HAM, the auxiliary parameter h̄ provides us with a convenient way to adjust and control the convergence and its rate for the solutions series. Asia-Pac. J. Chem. Eng. 2011; 6: 669–678 DOI: 10.1002/apj 675 676 Z. ZIABAKHSH et al. Figure 16. The result of θ(η) for various γ when Pr = 0.2 and β = 5. Figure 17. The result of θ(η) for various γ when Pr = 1 and β = 1.5. 3. Solutions of HAM can be expressed with different functions and therefore they can be originated from the nature of the problems. 4. The boundary conditions can be applied not only on the initial guess but also on other terms as well. 5. In analytic approach, the phenomena with same physical manners have the same solution expressions. 2010 Curtin University of Technology and John Wiley & Sons, Ltd. Asia-Pacific Journal of Chemical Engineering Figure 18. The result of θ(η) for various Pr when β = 2 and γ = 1. Figure 19. The result of θ(η) for various Pr when β = 3 and γ = 3. 6. It is necessary to determine the validity space of auxiliary parameters to the convergence of the results and the assumption must be alike for all coupled equations. 7. The solutions, though restricted to the first-order expansion, are quite elegant and fully acceptable in accuracy. Governing equations are easily solved by the approximate method. Asia-Pac. J. Chem. Eng. 2011; 6: 669–678 DOI: 10.1002/apj Asia-Pacific Journal of Chemical Engineering FLUID MECHANICS Table 1. The results of HAM and NS for f (0) and θ (0) when γ = 1 and Pr = 1. f (0) β 0.0 0.5 1.0 1.5 2.0 θ (0) h̄ 1 h̄ 2 NM HAM NM HAM −0.5 −0.5 −0.5 −0.5 −0.4 −0.6 −0.6 −0.6 −0.6 −0.5 −0.627563 −0.829954 −1.000000 −1.148601 −1.281816 −0.627553 −0.829941 −1.000000 −1.148594 −1.281746 −1.062324 −1.028183 −1.000000 −0.975917 −0.954811 −1.062326 −1.028180 −1.000000 −0.975902 −0.954762 Figure 20. The comparison between HAM solution and NS for f (η) and f (η) that solid lines are HAM solutions and symbols are NS when Pr = 1, h̄1 = −0.5, β = 0 and γ = 1. Table 2. The results of HAM and NS for θ (0) when γ = 1 and β = 0. θ (0) Figure 21. The comparison between HAM solution and NS for θ(η) that solid line is HAM solution and symbol is NS when Pr = 1, h̄1 = −0.5, h̄2 = −0.6, β = 0 and γ = 1. Table 3. The results of HAM and NS for θ (0) when Pr = 1 and β = 0. θ (0) γ Pr h̄ 1 h̄ 2 NM HAM 0.5 1.5 2.0 3.0 5.0 −0.4 −0.4 −0.4 −0.4 −0.4 −0.5 −0.5 −0.6 −0.8 −1.6 −0.682933 −1.351373 −1.593483 −1.997453 −2.634798 −0.682150 −1.351726 −1.593503 −1.994660 −2.633785 Consequently, these equations are solved by utilizing MAPLE 10, mathematical software, whose results are given in following figures and tables. Acknowledgements The authors are grateful to the reviewers for their helpful and useful comments and suggestions. 2010 Curtin University of Technology and John Wiley & Sons, Ltd. 0.0 1.0 2.0 3.0 4.0 5.0 h̄ 1 h̄ 2 NM HAM −0.5 −0.5 −0.5 −0.5 −0.5 −0.5 −0.6 −0.6 −0.6 −0.6 −0.6 −0.6 −0.627563 −1.062324 −1.403498 −1.689876 −1.940194 −2.164833 −0.627559 −1.062324 −1.403471 −1.689899 −1.940223 −2.164841 NOMENCLATURE A(x ) A0 B (x ) B0 b C0 f Scaling function Constant parameter Scaling function Constant parameter Constant parameter Constant parameter Dimensionless stream function Asia-Pac. J. Chem. Eng. 2011; 6: 669–678 DOI: 10.1002/apj 677 678 Z. ZIABAKHSH et al. h̄ 1 , h̄ 2 k L1 , L2 m n N1 , N2 Pr T u v x y Asia-Pacific Journal of Chemical Engineering Auxiliary parameters Thermal conductivity Linear operator of HAM Stretching exponent Temperature exponent Nonlinear operator Prandtl parameter Temperature Dimensional longitudinal velocity Dimensional transversal velocity Dimensional wall coordinate Dimensional transversal coordinate Greek symbols α β γ η θ υ ψ Thermal diffusivity Constant parameter Constant parameter Dimensionless similarity variable Dimensionless temperature Kinematic viscosity Stream function Subscripts w ∞ Wall conditions Conditions at infinity REFERENCES [1] L.J. Crane. Z. Angew. Math. Phys., 1970; 21, 645–647. [2] A. Ishak, R. Nazar, I. Pop. Nonlinear Anal. Real World Appl., 2909; 10(5), 2909–2913. [3] P.S. Gupta, A.S. Gupta. Can. J. Chem. Eng., 1977; 55, 744–749. [4] L.J. Grubka, K.M. Bobba. ASME J. Heat Transf., 1985; 107, 248–250. [5] C.D.S. Devi, H.S. Takhar, G. Nath. J. 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