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ISRN Mathematical Analysis
Volume 2012 (2012), Article ID 957473, 10 pages
http://dx.doi.org/10.5402/2012/957473
Research Article

Analytical Approximation to the Solution of Nonlinear Blasius’ Viscous Flow Equation by LTNHPM

Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Guilan, P.O. Box 1914, Rasht, Iran

Received 16 November 2011; Accepted 8 December 2011

Academic Editor: S. Zhang

Copyright © 2012 Hossein Aminikhah. 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

Laplace transform and new homotopy perturbation methods are adopted to study Blasius’ viscous flow equation analytically. The solutions approximated by the proposed method are shown to be precise as compared to the corresponding results obtained by Howarth’s numerical method. A high accuracy of the new method is evident.

1. Introduction

One of the well-known equations arising in fluid mechanics and boundary layer approach is Blasius’ differential equation. Blasius [1] in 1908 found the exact solution of boundary layer equation over a flat plate. Afterwards it has been solved by Howarth [2] by means of some numerical methods. The solution of Blasius equation was studied recently by AbuSitta for the mixing layers of fluid past a flat plate and the existence of a solution well established in [3]. Asaithambi [4] presented an effective finite difference method which has improved the previous numerical methods by reducing the amount of computational work. Recently, He [5], Abbasbandy [6], Esmaeilpour and Ganji [7] obtained an approximate solution for Blasius equation using HPM, ADM, and VIM. In this work, we obtain an analytical approximation to the solution of the classical Blasius flat-plate problem using combination of Laplace transform and new homotopy perturbation method (LTNHPM). The results obtained via LTNHPM are compared with the numerical solutions [2] which confirms the validity of the proposed method.

2. Governing Equations

Boundary layer flow over a flat plate is governed by the continuity and the Navier-Stokes equations. For a two-dimensional, steady-state, incompressible flow with zero pressure gradient over a flat plate, governing equations are simplified to𝜕𝑢+𝜕𝑥𝜕𝜈𝑢𝜕𝑦=0,𝜕𝑢𝜕𝑥+𝜈𝜕𝜈𝜕𝜕𝑦=𝜈2𝑢𝜕𝑦2.(2.1) The boundary conditions are𝑦=0,𝑢=𝜈=0,𝑦=,𝑢=𝑈.(2.2) Assuming that the leading edge of the plate is 𝑥=0 and the plate is infinity long, this system can be simplified further to an ordinary differential equation. To do this, we have an equation that reads𝛿𝜈𝑥𝑈.(2.3) To make this quantity dimensionless, it can be divided by 𝑦 to obtain𝜂=𝑦𝑈𝜈𝑥,𝜑=𝜈𝑥𝑈𝑓(𝜂),(2.4) where 𝑓(𝜂) is the dimensionless stream function. The velocity component 𝑢 which is equal to 𝜕𝜑/𝜕𝑦 can be expressed as follows:𝑢=𝜕𝜑=𝜕𝑦𝜕𝜑𝜕𝜂𝜕𝜂=𝜕𝑦𝜈𝑥𝑈𝑓(𝜂)𝑈𝜈𝑥.(2.5) So 𝑢=𝑈𝑓(𝜂). Also the transverse velocity component can be expressed as𝜈=𝜕𝜑=1𝜕𝑥2𝜈𝑈𝑥𝜂𝑓(𝜂)𝑓(𝜂).(2.6) Now, inserting (2.5) and (2.6) into the second boundary layer flow equation𝑈22𝑥𝜂𝑓(𝜂)𝑓𝑈(𝜂)+22𝑥𝜂𝑓𝑈(𝜂)𝑓(𝜂)=𝜈2𝑓𝑥𝑣(𝜂),(2.7) therefore,𝑓1(𝜂)+2𝑓(𝜂)𝑓(𝜂)=0,(2.8) with boundary equations𝜂=0,𝑓=𝑑𝑓𝑑𝜂=0,𝜂,𝑑𝑓𝑑𝜂=1.(2.9) In 1908, Blasius [1] provided a solution in the following form:𝑓(𝜂)=𝑘=012𝐴𝑘𝜎𝑘+1𝜂(3𝑘+2)!3𝑘+2,(2.10) where 𝜎=𝑓(0), 𝐴0=𝐴1=1, 𝐴𝑘=𝑘1𝑟=03𝑘13𝑟𝐴𝑟𝐴𝑘𝑟1(𝑘2).

Blasius evaluated 𝜎 by demonstrating another approximation of 𝑓(𝜂) at large 𝜂. Then, by means of matching two different approximations at a proper point, he obtained the numerical result 𝜎=0.332. In 1938, by means of a numerical technique, Howarth [2] gained a more accurate value 𝜎=0.332057 utilized to solve Blasius equation (2.8).

3. Analysis of the Method

To illustrate the basic ideas of this method, let us consider the following nonlinear differential equation 𝐴(𝑢)𝑓(𝑟)=0,𝑟Ω,(3.1) with the following initial conditions𝑢(0)=𝛼0,𝑢(0)=𝛼1,,𝑢(𝑛1)(0)=𝛼𝑛1,(3.2) where 𝐴 is a general differential operator and 𝑓(𝑟) is a known analytical function. The operator 𝐴 can be divided into two parts, 𝐿 and 𝑁, where 𝐿 is a linear and 𝑁 is a nonlinear operator. Therefore, (3.1) can be rewritten as𝐿(𝑢)+𝑁(𝑢)𝑓(𝑟)=0.(3.3) By the NHPM [8], we construct a homotopy 𝑈(𝑟,𝑝)Ω×[0,1], which satisfies𝐻𝐿(𝑈,𝑝)=(1𝑝)(𝑈)𝑢0[𝐴][]+𝑝(𝑈)𝑓(𝑟)=0,𝑝0,1,𝑟Ω,(3.4) or equivalently,𝐻(𝑈,𝑝)=𝐿(𝑈)𝑢0+𝑝𝑢0[]+𝑝𝑁(𝑈)𝑓(𝑟)=0,(3.5) where 𝑝[0,1] is an embedding parameter and 𝑢0 is an initial approximation of solution of (3.1). Clearly, we have, from (3.4) and (3.5), 𝐻(𝑈,0)=𝐿(𝑈)𝑢0𝐻=0,(𝑈(𝑥),1)=𝐴(𝑈)𝑓(𝑟)=0.(3.6) By applying Laplace transform on both sides of (3.5), we have𝐿(𝑈)𝑢0+𝑝𝑢0[𝑁]+𝑝(𝑈)𝑓(𝑟)=0.(3.7) Using the differential property of Laplace transform, we have𝑠𝑛{𝑈}𝑠𝑛1𝑈(0)𝑠𝑛2𝑈(0)𝑈(𝑛1)𝑢(0)=0𝑝𝑢0[]+𝑝𝑁(𝑈)𝑓(𝑟),(3.8) or1{𝑈}=𝑠𝑛𝑠𝑛1𝑈(0)+𝑠𝑛2𝑈(0)++𝑈(𝑛1)𝑢(0)+0𝑝𝑢0[]+𝑝𝑁(𝑈)𝑓(𝑟).(3.9) By applying inverse Laplace transform on both sides of (3.9), we have𝑈=11𝑠𝑛𝑠𝑛1𝑈(0)+𝑠𝑛2𝑈(0)++𝑈(𝑛1)𝑢(0)+0𝑝𝑢0[]+𝑝𝑁(𝑈)𝑓(𝑟).(3.10) According to the HPM, we can first use the embedding parameter 𝑝 as a small parameter and assume that the solutions of (3.10) can be represented as a power series in 𝑝 as𝑈(𝑥)=𝑛=0𝑝𝑛𝑈𝑛.(3.11) Now, let us write the (3.10) in the following form: 𝑛=0𝑝𝑛𝑈𝑛=11𝑠𝑛𝑠𝑛1𝑈(0)+𝑠𝑛2𝑈(0)++𝑈(𝑛1)𝑢(0)+0𝑝𝑢0𝑁+𝑝𝑛=0𝑝𝑛𝑈𝑛.𝑓(𝑟)(3.12) Comparing coefficients of terms with identical powers of 𝑝, leads to𝑝0𝑈0=11𝑠𝑛𝑠𝑛1𝑈(0)+𝑠𝑛2𝑈(0)++𝑈(𝑛1)𝑢(0)+0,𝑝1𝑈1=11𝑠𝑛𝑁𝑈0𝑢0,𝑝𝑓(𝑟)2𝑈2=11𝑠𝑛𝑁𝑈0,𝑈1,𝑝3𝑈3=11𝑠𝑛𝑁𝑈0,𝑈1,𝑈2,𝑝𝑗𝑈𝑗=11𝑠𝑛𝑁𝑈0,𝑈1,𝑈2,,𝑈𝑗1,(3.13) Suppose that the initial approximation has the form 𝑈(0)=𝑢0=𝛼0,  𝑈(0)=𝛼1,,𝑈(𝑛1)(0)=𝛼𝑛1; therefore, the exact solution may be obtained as follows:𝑢=lim𝑝1𝑈=𝑈0+𝑈1+𝑈2+.(3.14)

4. LTNHPM Applied to the Nonlinear Blasius Ordinary Differential Equation

Consider the nonlinear Blasius ordinary differential equation (2.8). For solving this equation by applying the new homotopy perturbation method, we construct the following homotopy:𝐻(𝐹(𝜂),𝑝)=𝐹(𝜂)𝑓0𝑓(𝜂)+𝑝01(𝜂)+2𝐹(𝜂)𝐹(𝜂)=0,(4.1) where 𝑝[0,1] is an embedding parameter and 𝑓0(𝜂) is an initial approximation of solution of (2.8).

Clearly, we have from (4.1) 𝐻(𝐹(𝜂),0)=𝐹(𝜂)𝑓0(𝜂)=0,𝐻(𝐹(𝜂),1)=𝐹1(𝜂)+2𝐹(𝜂)𝐹(𝜂)=0.(4.2) By applying Laplace transform on both sides of (4.1), we have𝐹(𝜂)𝑓0𝑓(𝜂)+𝑝01(𝜂)+2𝐹(𝜂)𝐹(𝜂)=0.(4.3) Using the differential property of Laplace transform, we have𝑠3{𝐹(𝜂)}𝑠2𝐹(0)𝑠𝐹(0)𝐹𝑓(0)=0𝑓(𝜂)𝑝01(𝜂)+2𝐹(𝜂)𝐹(𝜂),(4.4) or1{𝐹(𝜂)}=𝑠3𝑠2𝐹(0)+𝑠𝐹(0)+𝐹𝑓(0)+0𝑓(𝜂)𝑝01(𝜂)+2𝐹(𝜂)𝐹(𝜂).(4.5) By applying inverse Laplace transform on both sides of (4.5), we have 𝐹(𝜂)=11𝑠3𝑠2𝐹(0)+𝑠𝐹(0)+𝐹𝑓(0)+0𝑓(𝜂)𝑝01(𝜂)+2𝐹(𝜂)𝐹(𝜂).(4.6) According to the HPM, we use the embedding parameter 𝑝 as a small parameter and assume that the solutions of (4.6) can be represented as a power series in 𝑝 as𝐹(𝜂)=𝑛=0𝑝𝑛𝐹𝑛(𝜂).(4.7)

Substituting (4.7) into (4.6) and equating the terms with the identical powers of 𝑝 lead to 𝑝0𝐹0(𝜂)=1𝐹(0)𝑠+𝐹(0)𝑠2+𝐹(0)𝑠3𝑓+0,𝑝(𝜂)1𝐹1(𝜂)=11𝑠3𝑓01(𝜂)+2𝐹0(𝜂)𝐹0,𝑝(𝜂)2𝐹2(𝜂)=112𝑠3𝐹0(𝜂)𝐹1(𝜂)+𝐹1(𝜂)𝐹0,𝑝(𝜂)3𝐹3(𝜂)=112𝑠3𝐹0(𝜂)𝐹2(𝜂)+𝐹1(𝜂)𝐹1(𝜂)+𝐹2(𝜂)𝐹0,𝑝(𝜂)𝑗𝐹𝑗(𝜂)=112𝑠3𝑗1𝑘=0𝐹𝑘(𝜂)𝐹𝑗𝑘1,(𝜂)(4.8) To complete the solution, we choose 𝑓0(𝜂)=𝐹(0)=𝑓(0)=0, 𝐹(0)=𝑓(0)=0, and 𝐹(0)=𝑓(0)=𝜎. Solving (4.8) for 𝐹𝑗(𝜂), 𝑗=0,1,, leads to the results 𝐹01(𝜂)=2𝜎𝑥2,𝐹11(𝜂)=𝜎2402𝑥5,𝐹2(𝜂)=11𝜎1612803𝑥8,𝐹35(𝜂)=𝜎42577924𝑥11,𝐹4(𝜂)=9299𝜎4649508864005𝑥14,𝐹5(𝜂)=127239𝜎37939992330240006𝑥17,𝐹6(𝜂)=19241647𝜎34601273005178880007𝑥20,(4.9) Therefore, we gain the solution of (2.8) as 𝑓(𝜂)=lim𝑝1𝑛=0𝑝𝑛𝐹𝑛(𝜂)=𝐹0(𝜂)+𝐹1(𝜂)+𝐹2=1(𝜂)+2𝜎𝑥21𝜎2402𝑥5+11𝜎1612803𝑥85𝜎42577924𝑥11+.(4.10) According to Howarth calculation [2], inserting 𝜎=0.332057, therefore, the analytical approximation to the solution of Blasius equation can be expressed as𝑓(𝜂)=0.1660285000𝜂20.0004594243800𝜂5+0.000002497181392𝜂81.427697248×108𝜂11+8.074067341×1011𝜂144.495676921×1013𝜂17+.(4.11) Suppose 𝑓(𝜂)=12𝑛=0𝐹𝑛(𝜂), 𝑓(𝜂)=7𝑛=0𝐹𝑛(𝜂), 𝑔(𝜂)=(𝑑/𝑑𝜂)𝑓(𝜂), and 𝑔(𝜂)=(𝑑/𝑑𝜂)𝑓(𝜂), some numerical results of these solutions are presented in Tables 1 and 2, Figures 1 and 2.

tab1
Table 1: Comparison between the Howarth and LTNHPM methods for 𝑓(𝜂).
tab2
Table 2: Comparison between the Howarth and LTNHPM methods for 𝑓(𝜂).
957473.fig.001
Figure 1: The comparison of answers obtained by LTNHPM and Howarth’s results for 𝑓(𝜂).
957473.fig.002
Figure 2: The comparison of answers obtained by LTNHPM and Howarth’s results for 𝑓(𝜂).

Table 1 is made to compare between present results and results given by Howarth [2]. In Figures 1 and 2, one can also see the comparison between LTNHPM results and Howarth's results.

5. Conclusion

In this paper, the combined Laplace transform and homotopy perturbation methods are employed to give numerical solutions of the classical Blasius flat-plate flow in fluid mechanics. To illustrate the accuracy and efficiency of the proposed procedure, various different examples in the interval 0𝜂5 have also been analyzed and the numerical results are listed in Tables 1 and 2. Also, we have compared, in Figures 1 and 2, the numerical values of 𝑓 and 𝑓 with those of Howarth [2]. The results are found to be in good agreement. The results show that the LTNHPM is an effective mathematical tool which can play a very important role in nonlinear sciences.

References

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