Abstract and Applied Analysis

Volume 2013 (2013), Article ID 324869, 10 pages

http://dx.doi.org/10.1155/2013/324869

Research Article

Solution of Boundary Layer Problems with Heat Transfer by Optimal Homotopy Asymptotic Method

H. Ullah, S. Islam, M. Idrees, and M. Arif

Department of Mathematics, Abdul Wali Khan University Mardan, Mardan 23200, Pakistan

Received 17 June 2013; Accepted 7 August 2013

Academic Editor: Carlo Bianca

Copyright © 2013 H. Ullah et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Application of Optimal Homotopy Asymptotic Method (OHAM), a new analytic approximate technique for treatment of Falkner-Skan equations with heat transfer, has been applied in this work. To see the efficiency of the method, we consider Falkner-Skan equations with heat transfer. It provides us with a convenient way to control the convergence of approximate solutions when it is compared with other methods of solution found in the literature as finite difference (N. S. Asaithambi, 1997) and shooting method (Cebeci and Keller, 1971). The obtained solutions show that OHAM is effective, simpler, easier, and explicit.

1. Introduction

Most of the problems in engineering sciences are nonlinear, particularly some of heat transfer problems. Limited analytic methods are presented for the solution of such problems in the literature. Therefore, the researcher’s profound attention is to hunt some new analytic methods for the solution of the problems. Most of the methods like Adomian Decomposition Method (ADM) [1], Variational Iteration Method (VIM) [2], Differential Transform Method (DTM) [3], Radial basis function [4] and Homotopy Perturbation Method (HPM) [5], are used for the solution of weakly nonlinear problems, and limited for strongly nonlinear problems. For the solution of the strongly nonlinear problems the perturbation methods were studied [6–8]. These methods comprise a small parameter which cannot be found easily. To overcome this issue, some new analytic methods such as Artificial Parameters Method [9], Homotopy Analysis Method (HAM) [10], and Homotopy Perturbation Method (HPM) [5] were introduced. These methods pooled the homotopy with the perturbation techniques. Recently, Marinca et al. introduced Optimal Homotopy Asymptotic Method (OHAM) [11–15] for the solution of nonlinear problems which made the perturbation methods independent of the assumption of small parameters.

The Falkner-Skan equation has been considered in the last forty years due to its importance in the boundary layer theory. The boundary layer theory plays a vital role in the diverse area of engineering and scientific applications. The solution of the Falkner-Skan equation has been studied numerically first by Hartree [16]. Smith and Cebeci [17, 18] solved this equation by shooting method. Maksyn [19] solved the Falkner-Skan equation by analytic approximation. Asithambi [20–22] found its solution by finite differences, Liao [23] applied homotopy analysis to solve Falkner-Skan equation, and recently Vera [24] found its solution by Fourier series. An important case is the Blasius equation. This problem was solved by Rosales and Valencia [25] using Fourier series. Boyd [26] found the solution of Falkner-Skan equation by numerical method. An enormous amount of research work has been invested in the study of nonlinear boundary value problems [27–38]. In this paper, we will deal with the Falkner-Skan equations with heat transfer, a nonlinear boundary value problem [24] in different forms.

The motivation of this paper is to enhance OHAM for the solution of Falkner-Skan equation with heat transfer. In [11–15], OHAM has been proved to be useful for obtaining an approximate solution of nonlinear boundary value problems. In this work, we have proved that OHAM is also useful and reliable for the solution of the Falkner-Skan equation with heat transfer, hence showing its validity and great potential for the solution of transient physical phenomenon in science and engineering.

In the succeeding section, the basic idea of OHAM [11–15] is formulated for the solution of boundary value problems arising in heat transfer. In Section 3, the effectiveness of the enhanced formulation of OHAM for Falkner-Skan equation with heat transfer has been studied. Two special cases [24] of Falkner-Skan equation with heat transfer problems have been analyzed.

2. Basic Mathematical Theory of OHAM

Let us consider the following differential equation:

along with boundary conditions of the form

where is the linear operator, is an unknown function, is a known function, is a nonlinear differential operator, and is a boundary operator.

According to OHAM, one can construct an optimal homotopy : which satisfies

where is an embedding parameter, is an unknown function, and is a nonzero auxiliary function. The auxiliary function is nonzero for and . Equation (3) is the structure of OHAM homotopy.

It is defined that

respectively. Thus, as varies from to, the solution varies from to , where is obtained from (1) and (2) for as follows:

Next, we choose auxiliary function in the form

where are constants and can be found latter.

To obtain an approximate solution, we expand by Taylor’s series about in the following form:

Now substituting (8) into (1) and (2) and equating the coefficient of like powers of , we obtain the zeroth-order problem given by (6), the first- and second-order problems given by (9)–(11), respectively, and the general governing equations for given by (11):

where is the coefficient of in the expansion series of about the embedding parameter as follows:

It should be underscored that the for are governed by the linear equations with linear boundary conditions that come from the original problem, which can be easily solved.

It has been observed that the convergence of the series (8) depends upon the auxiliary constants . If it is convergent at , one has

Substituting (13) into (1), it results in the following expression for residual:

If , then is the exact solution of the problem. Generally it does not happen, especially in nonlinear problems.

For the determinations of auxiliary constants, , there are different methods like Galerkin’s method, Ritz method, least squares method, and collocation method. One can apply the method of  least squares as follows:

where and are two values, depending on the nature of the given problem.

The auxiliary constants can be optimally found from

The th order approximate solution can be obtained bythese constants. The constants can also be determined by another method as follows:

The convergence of OHAM is directly proportional to the number of optimal constants which is determined by (16).

It is easy to observe [13] that the Homotopy Perturbation Method (HPM) proposed by He [4] is a special case of (3) when , and on the other hand, the Homotopy Analysis Method (HAM) proposed by Liao [11] is another special case of (3) when where is chosen from “-curves” [12].

3. Application of OHAM to Falkner-Skan Equations with Heat Transfer

To demonstrate the effectiveness of OHAM formulation, two models are studied.

Model 1 (see [23]). When an incompressible fluid passes in the vicinity of solid boundaries, the Navier-Stokes equations may be reduced drastically into the boundary layer equations:

where is the free stream velocity, and are velocity components in - and -directions, and is the kinematic viscosity. In case of two-dimensional flow, the incompressible boundary layer flow over a wedge, when the free stream velocity is of the form , is the following similarity transformation:

Using (19) into (18), we obtained the Falkner-Skan equation

along with boundary conditions

According to (1), we have

The boundary conditions are

Applying the method formulation mentioned in Section 2 leads to the following.

Zeroth-Order Problem. Consider

from which we obtain

First-Order Problem. Consider

Its solution is

Second-Order Problem. Consider

with BC

whose solution is

Adding (25), (27), and (29), we obtain

where

For the computation of the constants and applying the method of least square mentioned in (14)–(16), we get

Putting these values in (30), we obtained the approximate solution of the form

Now consider the energy equation of an incompressible fluid which passes through the vicinity of the solid boundaries:

Upon using the transformation in (19) into (37), we obtained

with boundary conditions

where “” is the thermal diffusivity and is the prandtl number.

Applying the method formulated in Section 2 leads to the following.

Zeroth-Order Problem. Consider

Its solution is

First-Order Problem. Consider

whose solution is

Second-Order Problem. Consider

We obtain the following solution:

Adding (41), (43), and (45), we obtain

where

Using (46) in (38) and applying the method of least square, we obtain

Substituting these values in (46) for and , we obtain

Model 2 (see [23]). In case of in (52), we obtain the Blasius equation which is the famous equation of fluid dynamics and represent the problem of an incompressible fluid that passes through on a semi-infinite flat plate.

One has

According to (1), we define the operators

where and represent the second and third derivatives of with respect to .

Applying the method formulated in Section 2 leads to the following.

Zeroth-Order Problem. Consider

Its solution is

First-Order Problem. Consider

whose solution is

Second-Order Problem. Consider

We obtain the following solution:

Third-Order Problem. Consider

Its solution is

From (53), (57), (59), and (61), we obtain

where

Using (62) in (50) and applying the technique as discussed in (14)–(16), we obtain

Substituting these values in (62), we have

4. Results and Discussions

The formulation presented in Section 2 provides highly accurate solutions for the problems demonstrated in Section 3. We have used Mathematica 7 for most of our computational work. In Table 1, we have presented the initial slope for different values of obtained by OHAM. Table 1 shows that the results obtained by OHAM are in excellent agreement with the results found in the literature. Table 1 shows a benchmark for the initial slope found by different authors. It is found that the method present in this work is very good and provides the same values that optimized numerical methods with which it is compared. Figure 1 shows the variation of the function against for different values of with OHAM, while Figure 2 shows the variation of the function with respect to for different values of . Table 2 shows the solution of the Blasius equation obtained by the present method. In order to verify the accuracy of the present method, we have compared the results obtained by OHAM to the results available in the literature and found an excellent agreement. Figures 2 and 3 show the variation of with respect to for the Blasius equation which is identical to results in the literature [41].

Table 1: Initial slope obtained by OHAM for different values of .

Hartree [16]
Asaithambi [22]
Asaithambi [21]
Salama [39]
Zhang [40]
Vera [24]
OHAM

2
1.687
1.687222
1.687218
1.687218
1.687218
1.687218
1.67054

1
1.233
1.23589
1.232588
1.232588
1.232587
1.232587
1.23567

0.5
0.927
0.927682
0.927680
0.927680
0.927680
0.927680
0.92697

−0.1
0.319
0.319270
0.319270
0.319270
0.319270
0.319270
0.31927

Table 2: Solution of the Blasius equation by OHAM.

0.0
0.000000000
0.000000000

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

1.0000000000

Figure 1: for different values of .

Figure 2: for different values of .

Figure 3: Plot of with respect to .

5. Conclusion

In this work, we have seen the effectiveness of OHAM [11–15] to Falkner-Skan, Energy and Blasius equations. By applying the basic idea of OHAM to Falken-Skan, Energy and Blasius equations, we found that it is simpler in applicability and, more convenient to control convergence and involved less computational overhead. Therefore, OHAM shows its validity and great potential for the solution Falken-Skan, Energy and Blasius equations, with heat transfer problems arising in science and engineering.

References

S. H. Chowdhury, “A comparison between the modified homotopy perturbation method and adomian decomposition method for solving nonlinear heat transfer equations,” Journal of Applied Sciences, vol. 11, no. 7, pp. 1416–1420, 2011.

D. D. Ganji, G. A. Afrouzi, and R. A. Talarposhti, “Application of variational iteration method and homotopy-perturbation method for nonlinear heat diffusion and heat transfer equations,” Physics Letters A, vol. 368, no. 6, pp. 450–457, 2007.

H. Yaghoobi and M. Torabi, “The application of differential transformation method to nonlinear equations arising in heat transfer,” International Communications in Heat and Mass Transfer, vol. 38, no. 6, pp. 815–820, 2011.

D. D. Ganji, “The application of He's homotopy perturbation method to nonlinear equations arising in heat transfer,” Physics Letters A, vol. 355, no. 4-5, pp. 337–341, 2006.

J. H. He, “Homotopy perturbation technique,” Computer Methods in Applied Mechanics and Engineering, vol. 178, pp. 257–262, 1999.

R. Bellman, Perturbation Techniques in Mathematics, Physics, and Engineering, Holt, Rinehart and Winston, New York, NY, USA, 1964.

J. D. Cole, Perturbation Methods in Applied Mathematics, Blaisdell Publishing Co. Ginn and Co, Waltham, Mass, USA, 1968.

R. E. O'Malley Jr., Introduction to Singular Perturbations, Academic Press, New York, NY, USA, 1974.

G. L. Liu, “New research direction in singular perturbation theory: artificial parameter approach and inverse perturbation technique,” in Proceedings of the 7th Conference of 7th Modern Mathematics and Mechanics, 1997.

S. J. Liao, The proposed homotopy analysis technique for the solution of nonlinear problems [Ph.D. thesis], Shanghai Jiao Tong University, 1992.

N. Herişanu and V. Marinca, “Explicit analytical approximation to large-amplitude non-linear oscillations of a uniform cantilever beam carrying an intermediate lumped mass and rotary inertia,” Meccanica, vol. 45, no. 6, pp. 847–855, 2010.

V. Marinca and N. Herişanu, “Application of Optimal Homotopy Asymptotic Method for solving nonlinear equations arising in heat transfer,” International Communications in Heat and Mass Transfer, vol. 35, no. 6, pp. 710–715, 2008.

V. Marinca, N. Herişanu, C. Bota, and B. Marinca, “An optimal homotopy asymptotic method applied to the steady flow of a fourth-grade fluid past a porous plate,” Applied Mathematics Letters, vol. 22, no. 2, pp. 245–251, 2009.

V. Marinca, N. Herisanu, and I. Nemes, “A new analytic approach to nonlinear vibration of an electrical machine,” Proceedings of the Romanian Academy, vol. 9, pp. 229–236, 2008.

V. Marinca and N. Herişanu, “Determination of periodic solutions for the motion of a particle on a rotating parabola by means of the optimal homotopy asymptotic method,” Journal of Sound and Vibration, vol. 329, no. 9, pp. 1450–1459, 2010.

D. R. Hartree, “On an equation occurring in Falkner-Skan’s approximate treatment of the equations boundary layer,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 33, pp. 233–239, 1921.

A. M. O. Smith, “Improved solution of the Falkner-Skan equation boundary layer equation,” Sherman M. Fairchild Fund Paper FF-10, 1954.

T. Cebeci and H. B. Keller, “Shooting and parallel shooting methods for solving the Falkner-Skan boundary-layer equation,” Journal of Computational Physics, vol. 7, no. 2, pp. 289–300, 1971.

D. Meksyn, New Methods in Laminar Boundary Layer Theory, Pergamon Press, 1961.

A. Asaithambi, “Numerical solution of the Falkner-Skan equation using piecewise linear functions,” Applied Mathematics and Computation, vol. 159, no. 1, pp. 267–273, 2004.

A. Asaithambi, “A finite-difference method for the Falkner-Skan equation,” Applied Mathematics and Computation, vol. 92, no. 2-3, pp. 135–141, 1998.

N. S. Asaithambi, “A numerical method for the solution of the Falkner-Skan equation,” Applied Mathematics and Computation, vol. 81, no. 2-3, pp. 259–264, 1997.

S.-J. Liao, “A uniformly valid analytic solution of two-dimensional viscous flow over a semi-infinite flat plate,” Journal of Fluid Mechanics, vol. 385, pp. 101–128, 1999.

M. Rosales-Vera and A. Valencia, “Solutions of Falkner-Skan equation with heat transfer by Fourier series,” International Communications in Heat and Mass Transfer, vol. 37, no. 7, pp. 761–765, 2010.

M. Rosales and A. Valencia, “A note on solution of blasius equation by Fourier series,” Advances in Applied Mechanics, vol. 6, pp. 33–38, 2009.

J. P. Boyd, “The Blasius function in the complex plane,” Experimental Mathematics, vol. 8, no. 4, pp. 381–394, 1999.

S. Liao, “On the homotopy analysis method for nonlinear problems,” Applied Mathematics and Computation, vol. 147, no. 2, pp. 499–513, 2004.

M. Idrees, S. Islam, and A. M. Tirnizi, “Application of optimal homotopy asymptotic method of the Korteqag-de-Varies equation,” Computers and Mathematics with Applications, vol. 63, pp. 695–707, 2012.

S. Haq, M. Idrees, and S. Islam, “Application of optimal homotopy asymptotic method to eight order initial and boundary value problem,” Applied Mathematics and Computation, vol. 4, pp. 73–80, 2010.

S. Iqbal, M. Idrees, A. M. Siddiqui, and A. R. Ansari, “Some solutions of the linear and nonlinear Klein-Gordon equations using the optimal homotopy asymptotic method,” Applied Mathematics and Computation, vol. 216, no. 10, pp. 2898–2909, 2010.

S. Haq, M. Idrees, and S. Islam, “Application of optimal homotopy asymptotic method to eight order boundary value problem,” Journal of Computational and Applied Mathematics, vol. 2, pp. 73–80, 2010.

M. Idrees, S. Haq, and S. Islam, “Application of optimal hommotopy asymptotic method to fourth order boundary value problem,” World Applied Sciences Journal, vol. 9, pp. 131–137, 2010.

M. Idrees, S. Islam, S. Haq, and S. Islam, “Application of the optimal homotopy asymptotic method to squeezing flow,” Computers & Mathematics with Applications, vol. 59, no. 12, pp. 3858–3866, 2010.

M. Idrees, S. Haq, and S. Islam, “Application of optimal homotopy asymptotic method to sixth order boundary value problems,” World Applied Sciences Journal, vol. 9, pp. 138–143, 2010.

H. Ullah, S. Islam, M. Idrees, and M. Arif, “Application of optimal homotopy asymptotic method to heat transfer problems,” Research Journal of Recent Sciences. In press

J. Ali, S. Islam, S. Islam, and G. Zaman, “The solution of multipoint boundary value problems by the optimal homotopy asymptotic method,” Computers & Mathematics with Applications, vol. 59, no. 6, pp. 2000–2006, 2010.

J. Ali, S. Islam, H. Khan, and S. I. Ali Shah, “The optimal homotopy asymptotic method for the solution of higher-order boundary value problems in finite domains,” Abstract and Applied Analysis, vol. 2012, Article ID 401217, 14 pages, 2012.

R. Nawaz, H. Ullah, S. Islam, and M. Idrees, “Application of optimal homotopy asymptotic method to Burger’s equations,” Journal of Applied Mathematics, vol. 2013, Article ID 387478, 8 pages, 2013.

A. A. Salama, “Higher-order method for solving free boundary-value problems,” Numerical Heat Transfer B, vol. 45, no. 4, pp. 385–394, 2004.

J. Zhang and B. Chen, “An iterative method for solving the Falkner-Skan equation,” Applied Mathematics and Computation, vol. 210, no. 1, pp. 215–222, 2009.

S. Abbasbandy, “A numerical solution of Blasius equation by Adomian's decomposition method and comparison with homotopy perturbation method,” Chaos, Solitons and Fractals, vol. 31, no. 1, pp. 257–260, 2007.