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Analytical solution of heat transfer over an unsteady stretching permeable surface with prescribed wall temperature
Journal of the Taiwan Institute of Chemical Engineers 41 (2010) 169–177
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Analytical solution of heat transfer over an unsteady stretching permeable surface with prescribed wall temperature Z. Ziabakhsh, G. Domairry *, M. Mozaffari, M. Mahbobifar Department of Mechanical Engineering, Babol University of Technology, P.O. Box 484, Babol, Iran
A R T I C L E I N F O
A B S T R A C T
Article history: Received 9 May 2009 Received in revised form 21 July 2009 Accepted 30 August 2009
In this literature, the analytic solution of the nonlinear heat transfer over an unsteady stretching permeable surface with prescribed wall temperature is obtained utilizing the newly developed analytic method, namely the homotopy analysis method (HAM). The analytic results are compared with the numerical solution (NS) in which the comparison reveals good agreement between the numerical solution and HAM solution. In addition, the convergence of the obtained HAM solution is discussed explicitly. Finally, the validation of achieved analytic solution through all values of the Prandtl number (Pr) has been shown in the article. ß 2009 Taiwan Institute of Chemical Engineers. Published by Elsevier B.V. All rights reserved.
Keywords: Homotopy analysis method (HAM) Heat transfer Unsteady stretching permeable surface Wall temperature Numeric solution Auxiliary parameter
1. Introduction The heat transfer over an unsteady stretching permeable surface with prescribed wall temperature has been 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 (1970) 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 Grubka and Bobba (1985), Gupta and Gupta (1977), Ishak et al. (2007) 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; few of them are those considered by Andersson et al. (2000); Devi et al. (1991); Nazar et al. (2004), and very recently by Ali and Mehmood (2008). Ali and Mehmood (2008) and Nazar et al. (2004) introduced a similar transformation which was used by Williams and Rhyne (1980), in which
* Corresponding author. Tel.: +98 111 3234201; fax: +98 111 3234201. E-mail addresses: [email protected] (Z. Ziabakhsh), [email protected] (G. Domairry).
transforms the governing partial differential equations with three independent variables in two independent variables, which are more convenient for numerical computations. In addition, Nadeem and Awais (2008) considered unsteady shrinking sheet through porous medium with variable viscosity and pipe flow of a fourth grade fluid with Reynold and Vogel’s models of viscosities (Nadeem and Ali, 2009) by HAM analytically. Nadeem (2009) and Nadeem et al. (2009) applied homotopy analysis method to solving third and fourth grade fluid with variable viscosity as an analytical method. 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 HAM and numerical results given by Ishak et al. (2009). Homotopy analysis method (HAM) which was recently developed by Liao (1992) 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, Liao (2003): first and foremost, unlike all previous analytic techniques, the homotopy analysis method 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 homotopy analysis method always provides us with a family of solution expressions in the auxiliary parameter h, even if a nonlinear problem has a unique solution. Thirdly, unlike perturbation techniques, the homotopy
1876-1070/$ – see front matter ß 2009 Taiwan Institute of Chemical Engineers. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.jtice.2009.08.010
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Nomenclature Re A h2 h1 ; HAM H Pr L1 ; L2 Cf N1, N2 Nu u k T a b c
Reynolds number unsteadiness parameter auxiliary parameters homotopy analysis method auxiliary function Prandtl number linear operator of HAM skin friction coefficient nonlinear operator Nusselt number local macroscopic velocity thermal conductivity temperature constant parameter constant parameter constant parameter
Greek symbols a thermal diffusivity h dimensionless similarity variable m dynamic viscosity n kinematic viscosity r density of the fluid tw wall skin coefficient y velocity component in y-direction j suction parameter c stream function
analysis method is independent through any small or large quantities. So, the homotopy analysis method 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 Hayat and Khan (2005), Hayat and Sajid (2007), Hayat et al. (2006, 2007, 2008) and Liao (1995, 1999, 2003, 2004), 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 d-expansion method. Many authors such as Liao (2005, 2006) have applied HAM in solving permeable and impulsively stretched plate; Hayat et al. (2009) considered MHD flow of a micropolar fluid near a stagnation-point towards 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. (2008a). Abbasbandy (2007), Ali and Mehmood (2008), Ali et al. (2008), Domairry and Nadim (2008), Mehmood et al. (2008b), Ziabakhsh and Domairry (2009a,b) and Ziabakhsh et al. (2009a,b) have successfully applied HAM in solving different types of nonlinear problems, i.e. coupled, decoupled, homogeneous and non-homogeneous 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 h. 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 homotopy analysis method. 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 prior. So, theoretically, this assumption impairs the homotopy analysis method, although we can always attempt some base functions even if a given nonlinear problem is completely new for us. 2. Mathematical formulation Investigated the unsteady laminar boundary layer in a quiescent viscous and incompressible fluid, as shown in Fig. 1 at t = 0 the sheet is stretched with the velocity U w ðx; tÞ along the xaxis, keeping the origin in the fluid of ambient temperature T1. The stationary Cartesian coordinate system is originally placed at the leading edge of the plate in the whole positive x-axis of plate, and the y-axis is in a normal position to the surface of the sheet. According to this assumption, with the boundary layer evaluation and neglecting the viscous dissipation, the governing unsteady two-dimensional Navier–Stokes and energy equations can be written as:
@u @y þ ¼ 0; @x @y
(1)
@u @u @u @2 u þu þy ¼n 2; @t @x @y @y
(2)
@T @T @T @2 T þu þy ¼a 2; @t @x @y @y
(3)
With the boundary conditions: u ¼ U w ; y ¼ V w ; T ¼ T w at y ¼ 0: u ! 0; T ! T 1 as y ! 1:
(4)
where u and y are the velocity and velocity elements respectively, where T is the fluid temperature inside the boundary layer. Other components can be defined in this way: t is time a and n are the thermal diffusivity and the kinematic viscosity, V w ¼ ðvUw =xÞ1=2 f ð0Þ shows the mass transfer at the surface and can be define in two ways: V w > 0 for injection and V w < 0 for suction. The stretching velocity U w ðx; tÞ and surface temperature T w ðx; tÞ are in the form of (Ishak et al., 2009): U w ðx; tÞ ¼
ax ; 1 ct
T w ðx; tÞ ¼ T 1 þ
bx : 1 ct
(5)
where a, b and c are constants that a > 0, b 0, c 0 (with ct < 1), and both a and c have dimension time1. It is important to know that at t = 0 (initial motion), Eqs. (1)–(3) describe the steady flow over the stretching surface. T w ðx; tÞ and U w ðx; tÞ is written in this form to transform the governing practical equations (1)–(3) into a set of ordinary differential equations.
Fig. 1. Physical model and coordinate system.
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Now we introduce the following dimensionless function f and u, and similarity variables h (Ishak et al., 2009):
h¼
1=2 Uw y; yx
c ¼ ðyxU w Þ1=2 f ðhÞ;
uðhÞ ¼
T T1 : Tw T1
(6)
where c(x, y, t) is a stream function defined as u = @c/@y and y = @c/@x, which satisfies the mass conservation equation (1). Substituting (6) into (2) and (3) we have: 1 000 00 0 0 00 f ðhÞ þ f ðhÞ f ðhÞ f ðhÞ2 A f ðhÞ þ h f ðhÞ ¼ 0 (7) 2 1 00 1 u ðhÞ þ f ðhÞu0 ðhÞ f 0 ðhÞuðhÞ A uðhÞ þ hu0 ðhÞ ¼ 0 Pr 2
(8)
where A = c/a is a component that measures the unsteadiness and Pr = y/a is the Prandtl number. And the boundary conditions:
We defined f0 = j 1/j (with j > 0) and for 0 < j < 1 and j > 1 correspond to injection and suction, respectively. And ci (i = 1–5) are constants. Let P 2 [0,1] denotes the embedding parameter and h indicates non-zero auxiliary parameters. We then construct the following equations: 3.1. Zeroth–order deformation equations ð1 PÞL1 ½ f ðh; pÞ f 0 ðhÞ ¼ p hN1 ½ f ðh; pÞ; uðh; pÞ f ð0; pÞ ¼ 0;
0
xqw ; kðT w T 1 Þ
Cf ¼
tw : rUw2 =2
uð0; pÞ ¼ 1;
(21)
uð1; pÞ ¼ 0:
N1 ½ f ðh; pÞ; uðh; pÞ ¼
1 1=2 00 C Re ¼ f ð0Þ; 2 f x
Nux 1=2
Rex
0
¼ u ð0Þ;
(12)
3. Application of HAM As mentioned by Liao, a solution may be expressed with different base functions, among which some trigger convergency to the exact solution of the problem faster than others. Such base functions are obviously better suited for the final solution to be expressed in terms of noting these facts, we have decided to express f(h) and u(h) by a set of base functions of the form: f ðhÞ ¼
1 X 1 X 1 X akm;n hk expðnhÞ;
1 X 1 X 1 X uðhÞ ¼ bkm;n hk expðnhÞ;
(14)
(22)
1 @ uðh; pÞ @uðh; pÞ þ f ðh; pÞ Pr @h @h2
@ f ðh; pÞ uðh; pÞ @h
1 @uðh; pÞ A uðh; pÞ þ h ; 2 @h
(23)
For p = 0 and p = 1 we have f ðh; 0Þ ¼ f 0 ðhÞ
f ðh; 1Þ ¼ f ðhÞ
(24)
uðh; 0Þ ¼ u0 ðhÞ uðh; 1Þ ¼ uðhÞ
(25)
When p increases from 0 to 1 then f(h; p) and u(h; p) vary from f0(h) and u0(h) to f(h) and u(h). By Taylor’s theorem and using Eqs. (24) and (25), f(h; p) and u(h; p) can be expanded in a power series of p as follows: f ðh; pÞ ¼ f 0 ðhÞ þ
1 X
m
f m ðhÞ pm ;
f m ðhÞ ¼
m1
uðh; pÞ ¼ u0 ðhÞ þ
1 X
1 @ ð f ðh; pÞÞ m! @ pm
(26)
m
um ðhÞ pm ;
um ðhÞ ¼
m1
(13)
m¼0 n¼0 k¼0
@3 f ðh; pÞ @2 f ðh; pÞ þ f ðh; pÞ 3 @h @h2 @ f ðh; pÞ @ f ðh; pÞ @h @h ! @ f ðh; pÞ 1 @2 f ðh; pÞ A þ h ; 2 @h @h2 2
N2 ½ f ðh; pÞ; uðh; pÞ ¼
And r is the fluid density, t w is the wall shear stress, and qw is the surface heat flux and given: @u @T tw ¼ m ; qw ¼ k ; (11) @y y¼0 @y y¼0 where m and k are the dynamic viscosity and thermal conductivity, by using the dimensionless quantities (6), we obtain (Ishak et al., 2009)
(19) (20)
(9)
(10)
f ð1; pÞ ¼ 1:
(18)
ð1 PÞL2 ½uðh; pÞ u 0 ðhÞ ¼ p hN2 ½ f ðh; pÞ; uðh; pÞ
where f(0) = f0, with f0 < 0 and f0 > 0 corresponding to injection and suction, respectively. Local Nusselt number Nux and Cf the skin friction coefficient are defined as Nux ¼
0
f ð0; pÞ ¼ 0;
0
f ð0Þ ¼ f 0 ; f ð0Þ ¼ 1; uð0Þ ¼ 1; 0 f ðhÞ ! 0; uðhÞ ! 0; as h ! 1:
171
1 @ ðuðh; pÞÞ m! @ pm
(27)
In which h is chosen in such a way that these two series are convergent at p = 1, therefore we have through Eqs. (26) and (27) that
m¼0 n¼0 k¼0
The rule of solution expression provides us with a starting point. It is under the rule of solution expression that initial approximations, auxiliary linear operators, and the auxiliary functions are determined. So, according to the rule of solution expression, we choose the initial guess and auxiliary linear operator in the following form: f 0 ðhÞ ¼ f 0 þ 1 expðhÞ; L1 ð f Þ ¼ f
000
00
þ f ;
f ðhÞ ¼ f 0 ðhÞ þ
1 X
f m ðhÞ;
uðhÞ ¼ u0 ðhÞ þ
m1
1 X
um ðhÞ;
(28)
m1
3.2. mth–order deformation equations h1 HðhÞRmf ðhÞ; L1 ½ f m ðhÞ xm f m1 ðhÞ ¼
(29)
u0 ðhÞ ¼ expðhÞ;
(15)
00
(16)
f m ð0Þ ¼ 0;
f m ð1Þ ¼ 0;
(30)
(17)
h2 HðhÞRum ðhÞ L2 ½um ðhÞ xm um1 ðhÞ ¼
(31)
L2 ðuÞ ¼ u þ u
L1 ðc1 expðhÞ þ c2 h þ c3 Þ ¼ 0;
0
L2 ðc4 expðhÞ þ c5 Þ ¼ 0;
0
f m ð0Þ ¼ 0;
0
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172
Fig. 2. The h1 validity for j=1.0, A = 1/2 and Pr = 1.0.
um ð0Þ ¼ 0;
um ð1Þ ¼ 0;
000
Rmf ðhÞ ¼ f m1 þ A
Rum ðhÞ ¼
xm ¼
m 1 X
0
(32) 00
f m1n f n
n¼0
f m1 þ
m 1 X
0
n¼0
1 h f 00m1 ; 2
m1 ; m>1
Fig. 3. The h2 validity for j=1.0,A = 1/2 and Pr = 1.0.
The general solutions then become: f m ðhÞ xm f m1 ðhÞ ¼ fm ðhÞ þ C1m expðhÞ þ C2m h þ C3m ; um ðhÞ xm um1 ðhÞ ¼ um ðhÞ þ C4m expðhÞ þ C5m ;
0
f n f m1n
m 1 m 1 X X 1 00 u þ f u0 um1n f 0n Pr m1 n¼0 m1n n n¼0 1 0 A um1 þ hum1 ; 2
0 1
Fig. 4. The h1 validity for various A when j = 1/2 and Pr = 1.0.
(33)
(34)
(35)
(36)
where C1m to C5m are constants that can be obtained by applying the boundary conditions in Eqs. (30) and (32). As discussed by Liao the rule of coefficient ergodicity and the rule of solution existence play important roles in determining the auxiliary function and ensuring that the high-order deformation equations are closed and have solutions. In many cases, by means of the rule of solution expression and the rule of coefficient ergodicity, auxiliary functions can be uniquely determined. So we define the auxiliary function H(h) which for both velocity field and temperature is true and same. It is in
Fig. 5. The h2 validity curve for various A when j = 1/2, h1 ¼ 1 and Pr = 1.0.
Z. Ziabakhsh et al. / Journal of the Taiwan Institute of Chemical Engineers 41 (2010) 169–177
Fig. 6. The h1 validity for various j when A = 1/2 and Pr = 1.0.
173
Fig. 8. The h2 validity for various Pr when j = 2.0 and Pr = 1.0.
As was mentioned in Section 1, 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 parameters h. Even, if the initial approximations f0(h) and u0(h), the auxiliary linear operator L, and the auxiliary function H(h) are given, we still have great freedom to choose the value of the auxiliary parameters h1 and h2 . So, the auxiliary parameters provide us with an additional way to conveniently adjust and control the convergence region and rate of solution series. By means of the so-called h-curves, it is easy to find out the socalled valid regions of auxiliary parameters to gain a convergent solution series. When the important physical parameters such as: f00 (0) and u0 (0) considering auxiliary parameters, if they do
Fig. 7. The h2 validity curve for various j when A = 1/2, h1 ¼ 1 and Pr = 1.0.
Fig. 9. The analytic approximation for f(n), solid curve: 14-order approximate; symbols: 15-order approximate; dotted curve: initial approximation of by HAM for Pr = 1.0, A = 1/2, j = 2.0 and h1 ¼ 0:9.
the following form: HðhÞ ¼ expðhÞ
(37)
4. Convergence of the HAM solution
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Table 1 The results of HAM and numerical solution for f00 (0) and u0 (0) for various j when A = 1.0 and Pr = 1.0.
j
h1
h2
2.0 1.0 1.5 2.0 2.5
0.8 0.8 0.8 0.8 0.8
1 1 1 1 1
f00 (0)
u0 (0)
NM
HAM
NM
HAM
0.8095115 1.3205220 1.7706579 2.2223554 2.6853999
0.8095291 1.3203526 1.7706091 2.2222147 2.6853923
0.8095115 1.3205220 1.7706579 2.2223554 2.6853999
0.8095291 1.3203526 1.7706091 2.2222147 2.6853923
Table 2 The results of HAM and numerical solution for f00 (0) and u0 (0) for various A when j = 2 and Pr = 1.0. A
0 1.0 1.5 2.0 2.5
h1
0.8 0.8 0.8 0.8 0.8
h2
1 1 1 1 1
f00 (0)
u0 (0)
NM
HAM
NM
HAM
2.0000001 2.2223554 2.3310177 2.4360794 2.5371027
1.9989219 2.2222147 2.3309709 2.4360762 2.5372085
2.0000001 2.2223554 2.3310177 2.4360794 2.5371027
1.9989219 2.2222147 2.3309709 2.4360762 2.5372085
Table 3 The results of HAM and numerical solution for u0 (0) for various Pr when j = 2 and A = 1. Pr
h1
h2
0.5 0.7 1.0 1.5 2.0
1 1 1 1 1
0.5 0.8 1 1 1
u0 (0) NM
HAM
1.2920441 1.6749745 2.2223554 3.0924822 3.9312369
1.2940513 1.6750373 2.2222147 3.0926679 3.9313481
Fig. 11. The f0 (n) by homotopy analysis method for various j when A = 1/2, Pr = 1.0 and h1 ¼ 0:9.
Figs. 4–8 show how auxiliary parameters varied with changing A, j and Pr. The range of convergence for various A would be restricted if A increases. According to Fig. 6 by increasing Pr, the range of convergency is increased. Figs. 9 and 10 indicate the analytic approximation for f(h) and u(h) in which solid curve is 14order approximate, symbols show the 15-order approximate and dotted curve is initial approximation of by HAM for Pr = 1, A = 0.5, j = 2, h1 ¼ 0:9 and h2 ¼ 1. 5. Numerical method
not change, corresponds region is known as the convergence region. In our case study, according to Figs. 2 and 3, the acceptable range of auxiliary parameters for j = 2, A = 1.0 and Pr = 1 are 1:4 < h1 < 0:3 and 1:4 < h2 < 0:4.
The best approximate for solving Eqs. (7) and (8) with the boundary conditions define as Eq. (9) is finite-difference scheme known as the Keller-box method. It is often utilized to solve
Fig. 10. The analytic approximation for u(h), solid curve: 14-order approximate; symbols: 15-order approximate; dotted curve: initial approximation of by HAM for Pr = 1.0, A = 1/2, j = 2.0, h1 ¼ 0:9 and h2 ¼ 1:0.
Fig. 12. The u(h) by homotopy analysis method for various j when A = 1/2, Pr = 1.0 and h1 ¼ 0:9.
Z. Ziabakhsh et al. / Journal of the Taiwan Institute of Chemical Engineers 41 (2010) 169–177
Fig. 13. The f(n) by homotopy analysis method for various j when A = 1/2 and Pr = 1.0.
Fig. 14. The f0 (h) by homotopy analysis method for various A when Pr = 1.0 and j = 1/ 2.
differential equation systems. As indicated in Tables 1–3, it has been attempted to show the accuracy, capabilities and wide-range applications of the homotopy analysis method in comparison with the numerical solution. 6. Result and discussion Figs. 11–13 show u(h), f0 (h) and f(h) that are obtained by using homotopy analysis method for various values of j when A = 0.5, Pr = 1.0 and h1 ¼ 0:9, h2 ¼ 1:0. Additionally
175
Fig. 15. The u(h) by homotopy analysis method for various A when Pr = 1.0, j = 1/2 and h1 ¼ 0:9.
Fig. 16. The f(h) by homotopy analysis method for various A when Pr = 1.0, j = 1/2 and h1 ¼ 0:9.
Figs. 14–16 illustrate u(h), f0 (h) and f(h) for various values of A when Pr = 1.0, j = 0.5, h2 ¼ 1:0 and h1 ¼ 0:9, and finally Fig. 17 shows the effect of the Prandtl number (Pr) on the thermal boundary layer’s thickness respectively. The figures show that by increasing A, j and Prandtl number (Pr), the velocity boundary layer thickness and thermal boundary layer’s thickness would decrease. It can be considered through out Tables 1 and 2 that f0 (h) and u(h) are equal, which advocate the accuracy of our calculations in compassion with Ishak et al. (2009) paper.
176
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homotopy analysis method in comparison with the numerical solution of heat transfer over an unsteady stretching permeable surface with prescribed wall temperature. References
Fig. 17. The u(h) by homotopy analysis method for various Pr when A = 1.0, j = 1/2, h1 ¼ 1 and h2 ¼ 0:9.
Fig. 18. The f0 (h) and u(h) by homotopy analysis method when Pr = 1.0, A = 1.0, j = 1.0, h1 ¼ 1, h2 ¼ 0:9 and point symbol is numerical method.
7. Conclusion In this study, heat transfer over an unsteady stretching permeable surface with prescribed wall temperature was analyzed utilizing HAM. The recurrence formula was obtained and the accuracy of convergency for the solution series was discussed. The power of HAM in controlling and adjusting the convergence region and rate of solution series were discussed. The proper range of auxiliary parameter h for ensuring convergency of the solution series was obtained through the so-called h-curves. By comparing with other analytic methods, it is clear that HAM provides highly accurate analytic solutions for nonlinear problems. As shown in Fig. 18 and Tables 1–3, it has been attempted to show the accuracy, capabilities and wide-range applications of the
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J., ‘‘A New Branch of Solutions of Boundary-Layer Flows over an Impermeable Stretched Plate,’’ International Journal of Heat Mass Transfer, 48, 2529 (2005). Liao, S. J., ‘‘An Analytic Solution of Unsteady Boundary-Layer Flows Caused by an Impulsively Stretching Plate,’’ Communications in Non-linear Science and Numerical Simulation, 11, 326 (2006). Mehmood, A., A. Ali, and T. Shah, ‘‘Unsteady Boundary-Layer Viscous Flow Due to an Impulsively Started Porous Plate,’’ Canadian Journal of Physics, 86, 1079 (2008a). Mehmood, A., A. Ali, H. S. Takhar, and T. Shah, ‘‘Corrigendum: Unsteady ThreeDimensional MHD Boundary Layer Flow Due to the Impulsive Motion of the Stretching Sheet,’’ Acta Mechanica, 199, 241 (2008b). Nadeem, S., ‘‘Thin Film Flow of a Third Grade Fluid With Variable Viscosity,’’ Zeitschrift fu¨r Naturforschung, (2009). Nadeem, S. and M. Ali, ‘‘Analytical Solutions for Pipe Flow of a Fourth Grade Fluid with Reynold and Vogel’s Models of Viscosities,’’ Communications in Nonlinear Science and Numerical Simulation, 14, 2073 (2009). Nadeem, S. and M. Awais, ‘‘Thin Film Flow of an Unsteady Shrinking Sheet through Porous Medium with Variable Viscosity,’’ Physics Letters A, 372, 4965 (2008). Nadeem, S., T. Hayat, S. Abbasbandy, and M. Ali, ‘‘Effects of Partial Slip on a Fourthgrade Fluid with Variable Viscosity: An Analytic Solution,’’ Nonlinear Analysis: Real World Applications, doi:10.1016/j.nonrwa.2009.01.030. Nazar, R., N. Amin, and I. Pop, ‘‘Unsteady Boundary Layer Flow Due to a Stretching Surface in a Rotating Fluid,’’ Mechanics Research Communications, 31, 121 (2004). Williams, J. C. and T. B. Rhyne, ‘‘Boundary Layer Development on a Wedge Impulsively Set into Motion,’’ SIAM Journal of Applied Mathematics, 38, 215 (1980). Ziabakhsh, Z. and G. 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Journal of the Taiwan Institute of Chemical Engineers journal homepage: www.elsevier.com/locate/jtice
Analytical solution of heat transfer over an unsteady stretching permeable surface with prescribed wall temperature Z. Ziabakhsh, G. Domairry *, M. Mozaffari, M. Mahbobifar Department of Mechanical Engineering, Babol University of Technology, P.O. Box 484, Babol, Iran
A R T I C L E I N F O
A B S T R A C T
Article history: Received 9 May 2009 Received in revised form 21 July 2009 Accepted 30 August 2009
In this literature, the analytic solution of the nonlinear heat transfer over an unsteady stretching permeable surface with prescribed wall temperature is obtained utilizing the newly developed analytic method, namely the homotopy analysis method (HAM). The analytic results are compared with the numerical solution (NS) in which the comparison reveals good agreement between the numerical solution and HAM solution. In addition, the convergence of the obtained HAM solution is discussed explicitly. Finally, the validation of achieved analytic solution through all values of the Prandtl number (Pr) has been shown in the article. ß 2009 Taiwan Institute of Chemical Engineers. Published by Elsevier B.V. All rights reserved.
Keywords: Homotopy analysis method (HAM) Heat transfer Unsteady stretching permeable surface Wall temperature Numeric solution Auxiliary parameter
1. Introduction The heat transfer over an unsteady stretching permeable surface with prescribed wall temperature has been 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 (1970) 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 Grubka and Bobba (1985), Gupta and Gupta (1977), Ishak et al. (2007) 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; few of them are those considered by Andersson et al. (2000); Devi et al. (1991); Nazar et al. (2004), and very recently by Ali and Mehmood (2008). Ali and Mehmood (2008) and Nazar et al. (2004) introduced a similar transformation which was used by Williams and Rhyne (1980), in which
* Corresponding author. Tel.: +98 111 3234201; fax: +98 111 3234201. E-mail addresses: [email protected] (Z. Ziabakhsh), [email protected] (G. Domairry).
transforms the governing partial differential equations with three independent variables in two independent variables, which are more convenient for numerical computations. In addition, Nadeem and Awais (2008) considered unsteady shrinking sheet through porous medium with variable viscosity and pipe flow of a fourth grade fluid with Reynold and Vogel’s models of viscosities (Nadeem and Ali, 2009) by HAM analytically. Nadeem (2009) and Nadeem et al. (2009) applied homotopy analysis method to solving third and fourth grade fluid with variable viscosity as an analytical method. 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 HAM and numerical results given by Ishak et al. (2009). Homotopy analysis method (HAM) which was recently developed by Liao (1992) 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, Liao (2003): first and foremost, unlike all previous analytic techniques, the homotopy analysis method 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 homotopy analysis method always provides us with a family of solution expressions in the auxiliary parameter h, even if a nonlinear problem has a unique solution. Thirdly, unlike perturbation techniques, the homotopy
1876-1070/$ – see front matter ß 2009 Taiwan Institute of Chemical Engineers. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.jtice.2009.08.010
Z. Ziabakhsh et al. / Journal of the Taiwan Institute of Chemical Engineers 41 (2010) 169–177
170
Nomenclature Re A h2 h1 ; HAM H Pr L1 ; L2 Cf N1, N2 Nu u k T a b c
Reynolds number unsteadiness parameter auxiliary parameters homotopy analysis method auxiliary function Prandtl number linear operator of HAM skin friction coefficient nonlinear operator Nusselt number local macroscopic velocity thermal conductivity temperature constant parameter constant parameter constant parameter
Greek symbols a thermal diffusivity h dimensionless similarity variable m dynamic viscosity n kinematic viscosity r density of the fluid tw wall skin coefficient y velocity component in y-direction j suction parameter c stream function
analysis method is independent through any small or large quantities. So, the homotopy analysis method 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 Hayat and Khan (2005), Hayat and Sajid (2007), Hayat et al. (2006, 2007, 2008) and Liao (1995, 1999, 2003, 2004), 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 d-expansion method. Many authors such as Liao (2005, 2006) have applied HAM in solving permeable and impulsively stretched plate; Hayat et al. (2009) considered MHD flow of a micropolar fluid near a stagnation-point towards 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. (2008a). Abbasbandy (2007), Ali and Mehmood (2008), Ali et al. (2008), Domairry and Nadim (2008), Mehmood et al. (2008b), Ziabakhsh and Domairry (2009a,b) and Ziabakhsh et al. (2009a,b) have successfully applied HAM in solving different types of nonlinear problems, i.e. coupled, decoupled, homogeneous and non-homogeneous 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 h. 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 homotopy analysis method. 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 prior. So, theoretically, this assumption impairs the homotopy analysis method, although we can always attempt some base functions even if a given nonlinear problem is completely new for us. 2. Mathematical formulation Investigated the unsteady laminar boundary layer in a quiescent viscous and incompressible fluid, as shown in Fig. 1 at t = 0 the sheet is stretched with the velocity U w ðx; tÞ along the xaxis, keeping the origin in the fluid of ambient temperature T1. The stationary Cartesian coordinate system is originally placed at the leading edge of the plate in the whole positive x-axis of plate, and the y-axis is in a normal position to the surface of the sheet. According to this assumption, with the boundary layer evaluation and neglecting the viscous dissipation, the governing unsteady two-dimensional Navier–Stokes and energy equations can be written as:
@u @y þ ¼ 0; @x @y
(1)
@u @u @u @2 u þu þy ¼n 2; @t @x @y @y
(2)
@T @T @T @2 T þu þy ¼a 2; @t @x @y @y
(3)
With the boundary conditions: u ¼ U w ; y ¼ V w ; T ¼ T w at y ¼ 0: u ! 0; T ! T 1 as y ! 1:
(4)
where u and y are the velocity and velocity elements respectively, where T is the fluid temperature inside the boundary layer. Other components can be defined in this way: t is time a and n are the thermal diffusivity and the kinematic viscosity, V w ¼ ðvUw =xÞ1=2 f ð0Þ shows the mass transfer at the surface and can be define in two ways: V w > 0 for injection and V w < 0 for suction. The stretching velocity U w ðx; tÞ and surface temperature T w ðx; tÞ are in the form of (Ishak et al., 2009): U w ðx; tÞ ¼
ax ; 1 ct
T w ðx; tÞ ¼ T 1 þ
bx : 1 ct
(5)
where a, b and c are constants that a > 0, b 0, c 0 (with ct < 1), and both a and c have dimension time1. It is important to know that at t = 0 (initial motion), Eqs. (1)–(3) describe the steady flow over the stretching surface. T w ðx; tÞ and U w ðx; tÞ is written in this form to transform the governing practical equations (1)–(3) into a set of ordinary differential equations.
Fig. 1. Physical model and coordinate system.
Z. Ziabakhsh et al. / Journal of the Taiwan Institute of Chemical Engineers 41 (2010) 169–177
Now we introduce the following dimensionless function f and u, and similarity variables h (Ishak et al., 2009):
h¼
1=2 Uw y; yx
c ¼ ðyxU w Þ1=2 f ðhÞ;
uðhÞ ¼
T T1 : Tw T1
(6)
where c(x, y, t) is a stream function defined as u = @c/@y and y = @c/@x, which satisfies the mass conservation equation (1). Substituting (6) into (2) and (3) we have: 1 000 00 0 0 00 f ðhÞ þ f ðhÞ f ðhÞ f ðhÞ2 A f ðhÞ þ h f ðhÞ ¼ 0 (7) 2 1 00 1 u ðhÞ þ f ðhÞu0 ðhÞ f 0 ðhÞuðhÞ A uðhÞ þ hu0 ðhÞ ¼ 0 Pr 2
(8)
where A = c/a is a component that measures the unsteadiness and Pr = y/a is the Prandtl number. And the boundary conditions:
We defined f0 = j 1/j (with j > 0) and for 0 < j < 1 and j > 1 correspond to injection and suction, respectively. And ci (i = 1–5) are constants. Let P 2 [0,1] denotes the embedding parameter and h indicates non-zero auxiliary parameters. We then construct the following equations: 3.1. Zeroth–order deformation equations ð1 PÞL1 ½ f ðh; pÞ f 0 ðhÞ ¼ p hN1 ½ f ðh; pÞ; uðh; pÞ f ð0; pÞ ¼ 0;
0
xqw ; kðT w T 1 Þ
Cf ¼
tw : rUw2 =2
uð0; pÞ ¼ 1;
(21)
uð1; pÞ ¼ 0:
N1 ½ f ðh; pÞ; uðh; pÞ ¼
1 1=2 00 C Re ¼ f ð0Þ; 2 f x
Nux 1=2
Rex
0
¼ u ð0Þ;
(12)
3. Application of HAM As mentioned by Liao, a solution may be expressed with different base functions, among which some trigger convergency to the exact solution of the problem faster than others. Such base functions are obviously better suited for the final solution to be expressed in terms of noting these facts, we have decided to express f(h) and u(h) by a set of base functions of the form: f ðhÞ ¼
1 X 1 X 1 X akm;n hk expðnhÞ;
1 X 1 X 1 X uðhÞ ¼ bkm;n hk expðnhÞ;
(14)
(22)
1 @ uðh; pÞ @uðh; pÞ þ f ðh; pÞ Pr @h @h2
@ f ðh; pÞ uðh; pÞ @h
1 @uðh; pÞ A uðh; pÞ þ h ; 2 @h
(23)
For p = 0 and p = 1 we have f ðh; 0Þ ¼ f 0 ðhÞ
f ðh; 1Þ ¼ f ðhÞ
(24)
uðh; 0Þ ¼ u0 ðhÞ uðh; 1Þ ¼ uðhÞ
(25)
When p increases from 0 to 1 then f(h; p) and u(h; p) vary from f0(h) and u0(h) to f(h) and u(h). By Taylor’s theorem and using Eqs. (24) and (25), f(h; p) and u(h; p) can be expanded in a power series of p as follows: f ðh; pÞ ¼ f 0 ðhÞ þ
1 X
m
f m ðhÞ pm ;
f m ðhÞ ¼
m1
uðh; pÞ ¼ u0 ðhÞ þ
1 X
1 @ ð f ðh; pÞÞ m! @ pm
(26)
m
um ðhÞ pm ;
um ðhÞ ¼
m1
(13)
m¼0 n¼0 k¼0
@3 f ðh; pÞ @2 f ðh; pÞ þ f ðh; pÞ 3 @h @h2 @ f ðh; pÞ @ f ðh; pÞ @h @h ! @ f ðh; pÞ 1 @2 f ðh; pÞ A þ h ; 2 @h @h2 2
N2 ½ f ðh; pÞ; uðh; pÞ ¼
And r is the fluid density, t w is the wall shear stress, and qw is the surface heat flux and given: @u @T tw ¼ m ; qw ¼ k ; (11) @y y¼0 @y y¼0 where m and k are the dynamic viscosity and thermal conductivity, by using the dimensionless quantities (6), we obtain (Ishak et al., 2009)
(19) (20)
(9)
(10)
f ð1; pÞ ¼ 1:
(18)
ð1 PÞL2 ½uðh; pÞ u 0 ðhÞ ¼ p hN2 ½ f ðh; pÞ; uðh; pÞ
where f(0) = f0, with f0 < 0 and f0 > 0 corresponding to injection and suction, respectively. Local Nusselt number Nux and Cf the skin friction coefficient are defined as Nux ¼
0
f ð0; pÞ ¼ 0;
0
f ð0Þ ¼ f 0 ; f ð0Þ ¼ 1; uð0Þ ¼ 1; 0 f ðhÞ ! 0; uðhÞ ! 0; as h ! 1:
171
1 @ ðuðh; pÞÞ m! @ pm
(27)
In which h is chosen in such a way that these two series are convergent at p = 1, therefore we have through Eqs. (26) and (27) that
m¼0 n¼0 k¼0
The rule of solution expression provides us with a starting point. It is under the rule of solution expression that initial approximations, auxiliary linear operators, and the auxiliary functions are determined. So, according to the rule of solution expression, we choose the initial guess and auxiliary linear operator in the following form: f 0 ðhÞ ¼ f 0 þ 1 expðhÞ; L1 ð f Þ ¼ f
000
00
þ f ;
f ðhÞ ¼ f 0 ðhÞ þ
1 X
f m ðhÞ;
uðhÞ ¼ u0 ðhÞ þ
m1
1 X
um ðhÞ;
(28)
m1
3.2. mth–order deformation equations h1 HðhÞRmf ðhÞ; L1 ½ f m ðhÞ xm f m1 ðhÞ ¼
(29)
u0 ðhÞ ¼ expðhÞ;
(15)
00
(16)
f m ð0Þ ¼ 0;
f m ð1Þ ¼ 0;
(30)
(17)
h2 HðhÞRum ðhÞ L2 ½um ðhÞ xm um1 ðhÞ ¼
(31)
L2 ðuÞ ¼ u þ u
L1 ðc1 expðhÞ þ c2 h þ c3 Þ ¼ 0;
0
L2 ðc4 expðhÞ þ c5 Þ ¼ 0;
0
f m ð0Þ ¼ 0;
0
Z. Ziabakhsh et al. / Journal of the Taiwan Institute of Chemical Engineers 41 (2010) 169–177
172
Fig. 2. The h1 validity for j=1.0, A = 1/2 and Pr = 1.0.
um ð0Þ ¼ 0;
um ð1Þ ¼ 0;
000
Rmf ðhÞ ¼ f m1 þ A
Rum ðhÞ ¼
xm ¼
m 1 X
0
(32) 00
f m1n f n
n¼0
f m1 þ
m 1 X
0
n¼0
1 h f 00m1 ; 2
m1 ; m>1
Fig. 3. The h2 validity for j=1.0,A = 1/2 and Pr = 1.0.
The general solutions then become: f m ðhÞ xm f m1 ðhÞ ¼ fm ðhÞ þ C1m expðhÞ þ C2m h þ C3m ; um ðhÞ xm um1 ðhÞ ¼ um ðhÞ þ C4m expðhÞ þ C5m ;
0
f n f m1n
m 1 m 1 X X 1 00 u þ f u0 um1n f 0n Pr m1 n¼0 m1n n n¼0 1 0 A um1 þ hum1 ; 2
0 1
Fig. 4. The h1 validity for various A when j = 1/2 and Pr = 1.0.
(33)
(34)
(35)
(36)
where C1m to C5m are constants that can be obtained by applying the boundary conditions in Eqs. (30) and (32). As discussed by Liao the rule of coefficient ergodicity and the rule of solution existence play important roles in determining the auxiliary function and ensuring that the high-order deformation equations are closed and have solutions. In many cases, by means of the rule of solution expression and the rule of coefficient ergodicity, auxiliary functions can be uniquely determined. So we define the auxiliary function H(h) which for both velocity field and temperature is true and same. It is in
Fig. 5. The h2 validity curve for various A when j = 1/2, h1 ¼ 1 and Pr = 1.0.
Z. Ziabakhsh et al. / Journal of the Taiwan Institute of Chemical Engineers 41 (2010) 169–177
Fig. 6. The h1 validity for various j when A = 1/2 and Pr = 1.0.
173
Fig. 8. The h2 validity for various Pr when j = 2.0 and Pr = 1.0.
As was mentioned in Section 1, 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 parameters h. Even, if the initial approximations f0(h) and u0(h), the auxiliary linear operator L, and the auxiliary function H(h) are given, we still have great freedom to choose the value of the auxiliary parameters h1 and h2 . So, the auxiliary parameters provide us with an additional way to conveniently adjust and control the convergence region and rate of solution series. By means of the so-called h-curves, it is easy to find out the socalled valid regions of auxiliary parameters to gain a convergent solution series. When the important physical parameters such as: f00 (0) and u0 (0) considering auxiliary parameters, if they do
Fig. 7. The h2 validity curve for various j when A = 1/2, h1 ¼ 1 and Pr = 1.0.
Fig. 9. The analytic approximation for f(n), solid curve: 14-order approximate; symbols: 15-order approximate; dotted curve: initial approximation of by HAM for Pr = 1.0, A = 1/2, j = 2.0 and h1 ¼ 0:9.
the following form: HðhÞ ¼ expðhÞ
(37)
4. Convergence of the HAM solution
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Table 1 The results of HAM and numerical solution for f00 (0) and u0 (0) for various j when A = 1.0 and Pr = 1.0.
j
h1
h2
2.0 1.0 1.5 2.0 2.5
0.8 0.8 0.8 0.8 0.8
1 1 1 1 1
f00 (0)
u0 (0)
NM
HAM
NM
HAM
0.8095115 1.3205220 1.7706579 2.2223554 2.6853999
0.8095291 1.3203526 1.7706091 2.2222147 2.6853923
0.8095115 1.3205220 1.7706579 2.2223554 2.6853999
0.8095291 1.3203526 1.7706091 2.2222147 2.6853923
Table 2 The results of HAM and numerical solution for f00 (0) and u0 (0) for various A when j = 2 and Pr = 1.0. A
0 1.0 1.5 2.0 2.5
h1
0.8 0.8 0.8 0.8 0.8
h2
1 1 1 1 1
f00 (0)
u0 (0)
NM
HAM
NM
HAM
2.0000001 2.2223554 2.3310177 2.4360794 2.5371027
1.9989219 2.2222147 2.3309709 2.4360762 2.5372085
2.0000001 2.2223554 2.3310177 2.4360794 2.5371027
1.9989219 2.2222147 2.3309709 2.4360762 2.5372085
Table 3 The results of HAM and numerical solution for u0 (0) for various Pr when j = 2 and A = 1. Pr
h1
h2
0.5 0.7 1.0 1.5 2.0
1 1 1 1 1
0.5 0.8 1 1 1
u0 (0) NM
HAM
1.2920441 1.6749745 2.2223554 3.0924822 3.9312369
1.2940513 1.6750373 2.2222147 3.0926679 3.9313481
Fig. 11. The f0 (n) by homotopy analysis method for various j when A = 1/2, Pr = 1.0 and h1 ¼ 0:9.
Figs. 4–8 show how auxiliary parameters varied with changing A, j and Pr. The range of convergence for various A would be restricted if A increases. According to Fig. 6 by increasing Pr, the range of convergency is increased. Figs. 9 and 10 indicate the analytic approximation for f(h) and u(h) in which solid curve is 14order approximate, symbols show the 15-order approximate and dotted curve is initial approximation of by HAM for Pr = 1, A = 0.5, j = 2, h1 ¼ 0:9 and h2 ¼ 1. 5. Numerical method
not change, corresponds region is known as the convergence region. In our case study, according to Figs. 2 and 3, the acceptable range of auxiliary parameters for j = 2, A = 1.0 and Pr = 1 are 1:4 < h1 < 0:3 and 1:4 < h2 < 0:4.
The best approximate for solving Eqs. (7) and (8) with the boundary conditions define as Eq. (9) is finite-difference scheme known as the Keller-box method. It is often utilized to solve
Fig. 10. The analytic approximation for u(h), solid curve: 14-order approximate; symbols: 15-order approximate; dotted curve: initial approximation of by HAM for Pr = 1.0, A = 1/2, j = 2.0, h1 ¼ 0:9 and h2 ¼ 1:0.
Fig. 12. The u(h) by homotopy analysis method for various j when A = 1/2, Pr = 1.0 and h1 ¼ 0:9.
Z. Ziabakhsh et al. / Journal of the Taiwan Institute of Chemical Engineers 41 (2010) 169–177
Fig. 13. The f(n) by homotopy analysis method for various j when A = 1/2 and Pr = 1.0.
Fig. 14. The f0 (h) by homotopy analysis method for various A when Pr = 1.0 and j = 1/ 2.
differential equation systems. As indicated in Tables 1–3, it has been attempted to show the accuracy, capabilities and wide-range applications of the homotopy analysis method in comparison with the numerical solution. 6. Result and discussion Figs. 11–13 show u(h), f0 (h) and f(h) that are obtained by using homotopy analysis method for various values of j when A = 0.5, Pr = 1.0 and h1 ¼ 0:9, h2 ¼ 1:0. Additionally
175
Fig. 15. The u(h) by homotopy analysis method for various A when Pr = 1.0, j = 1/2 and h1 ¼ 0:9.
Fig. 16. The f(h) by homotopy analysis method for various A when Pr = 1.0, j = 1/2 and h1 ¼ 0:9.
Figs. 14–16 illustrate u(h), f0 (h) and f(h) for various values of A when Pr = 1.0, j = 0.5, h2 ¼ 1:0 and h1 ¼ 0:9, and finally Fig. 17 shows the effect of the Prandtl number (Pr) on the thermal boundary layer’s thickness respectively. The figures show that by increasing A, j and Prandtl number (Pr), the velocity boundary layer thickness and thermal boundary layer’s thickness would decrease. It can be considered through out Tables 1 and 2 that f0 (h) and u(h) are equal, which advocate the accuracy of our calculations in compassion with Ishak et al. (2009) paper.
176
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homotopy analysis method in comparison with the numerical solution of heat transfer over an unsteady stretching permeable surface with prescribed wall temperature. References
Fig. 17. The u(h) by homotopy analysis method for various Pr when A = 1.0, j = 1/2, h1 ¼ 1 and h2 ¼ 0:9.
Fig. 18. The f0 (h) and u(h) by homotopy analysis method when Pr = 1.0, A = 1.0, j = 1.0, h1 ¼ 1, h2 ¼ 0:9 and point symbol is numerical method.
7. Conclusion In this study, heat transfer over an unsteady stretching permeable surface with prescribed wall temperature was analyzed utilizing HAM. The recurrence formula was obtained and the accuracy of convergency for the solution series was discussed. The power of HAM in controlling and adjusting the convergence region and rate of solution series were discussed. The proper range of auxiliary parameter h for ensuring convergency of the solution series was obtained through the so-called h-curves. By comparing with other analytic methods, it is clear that HAM provides highly accurate analytic solutions for nonlinear problems. As shown in Fig. 18 and Tables 1–3, it has been attempted to show the accuracy, capabilities and wide-range applications of the
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