﻿Some of Semi Analytical Methods for Blasius Problem

Applied Mathematics
Vol. 3  No. 7 (2012) , Article ID: 19871 , 5 pages DOI:10.4236/am.2012.37106

Some of Semi Analytical Methods for Blasius Problem

Seval Catal

Department of Civil Engineering, Faculty of Engineering, Dokuz Eylul University, Izmir, Turkey

Email: seval.catal@deu.edu.tr

Received July 20, 2011; revised June 1, 2012; accepted June 8, 2012

Keywords: Blasius Problem; Non-Linear Differential Equation; Adomian Decomposition Method; Differential Transform Method; Taylor Series Method

ABSTRACT

In this paper, the Adomian methods, differential transform methods, and Taylor series methods are applied to non-linear differential equations which is called Blasius problem in fluid mechanics. The solutions of the Blasius problem for two cases are obtained by using these methods and their results are shown in table.

1. Introduction

On the fluid mechanics of non-parallel flows are called Blasius flows which is an important problem is interested in recently by authors [1-10]. In addition, this non-linear third order ordinary differential equation on a half-infinite interval is solved by using perturbation method , transformation of independent variable and finite difference method , homotopy analysis method [13-16], Adomian’s method [16-18], differential transform method [19,20].

In generally, it is considered the two-dimensional flow over a semi-infinite flat plain, is governed by (1.1)

with the boundary conditions (1.2)

where the prime denotes the derivatives with respect to a non-dimensional variable h, and is a non-dimensional function related to the stream function , where U is the velocity at infinity; n is the kinematic viscosity coefficient; x and y are the two independent coordinates.

In this work, when and , in Equation (1.1), we obtain the following equation which is called Blasius (1908) equation : (1.3)

with the boundary conditions as (1.4)

The physical character of boundary layer apparently needs close and far away solution to match as was done by Blasius. So, to obtain the solution of this problem, we consider the boundary conditions as:

For inner case  ; boundary conditions are; (1.5)

For inner-outer case  ; boundary conditions are; (1.6)

For outer case  ; boundary conditions are; (1.7)

The paper is organized as follows: Blasius problem is solved by using ADM in &2; by using DTM in &3 and also, by using Taylor Series Method in &4, then it is given some concluding remarks in &5.

For solving the following equation of the form (2.1)

where is non-linear mapping, X is Banach space, f is known function, by using Adomian’s decomposition method is taken that the solution u can be following convergent series form: (2.2)

with un Î X for all n. An is a polynomial depending on u0, u1, ∙∙∙, un, An Î X for all n, are obtained from the equality; (2.3)

In putting the Equations (2.2) and (2.3) into the Equation (2.1), it gives (2.4)

where (2.5)

to determine the so-called Adomian’s polynomials An from un, where h is a scalar parameter (2.6)

and (2.7)

then (2.8)

Thus, (2.9)

where shows the kth Fréchet derivative of N at .

To demonstrate Adomian’s solution of the Blasius problem, the differential operator and the inverse operator are treated. Operating with N−1 on Equation (1.3), then it gives: (2.10) where (2.11)

For inner case: f0 determined from the boundary conditions (1.5) and than the other components determined from Equation (2.11) as follows [16-18]: (2.12)

For outer case: Similarly, f0 determined from Equations (1.7) and than f1 determined from Equation (2.11), so on: (2.13)

3. Solution of Blasius Problem by Differential Transform Method (DTM)

In this study, similarly , it is applied DTM for the Equation (1.3): (3.1)

where shows the differential transform of .

For inner case: (1.5) boundary conditions (BCs) are transformed (3.2)

where c is an arbitrary constant. Then, we get the following equation: (3.3)

For outer case: (1.7) boundary conditions (BCs) are transformed (3.4)

where a and b are any arbitrary constants. Then, we obtain the following equation: (3.5)

For inner-outer case: From (1.6) boundary conditions, 4 < h0 < 5 taking the interval of h0 is found a, b, and c constants, is obtained similar result as in .

4. Solution of Blasius Problem by Taylor Series (TS)

Taylor’s series method is used for solving Blasius problem. This method assumes that the solution and derivative of can be taken power series as Table 1. Solutions of Blasius problem. (4.1)

Then Equation (4.1) is substituting into Equation (1.3);

For inner case: With the boundary conditions (1.5), it gives (4.2)

where (4.3)

with the definition .

c = 0.332 c = 0.332057 c = 0.333338 (this paper).

For outer case: Under the boundary conditions (1.7), it gives (4.4)

5. Conclusion

In this study, some of the semi analytical methods were applied by author in the non-linear Blasius problem which names are Adomian decomposition method, differential transform method, and Taylor series method. It was obtained their results for two cases by using these methods. Their results were presented in the Table 1. The results are shown that all of these methods are powerful and efficient technique for finding semi analytical solutions for Blasius problem in the fluid mechanics.

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