An Iteration Method Generating Analytical Solutions for Blasius Problem

Abstract

We derive a new iteration method for finding solution of the generalized Blasius problem. This method results in the analytical series solutions which are consistent with the existing series solutions for some special cases.

1. Introduction

We consider the generalized Blasius’ equation where or , with boundary conditions This problem describes the boundary layer flow over a moving plate with constant velocity . For a special case of and , the series solution of the Blasius problem becomes where . The Blasius series, however, converges for . In the literature [1–3], it was shown that the limitation can be overcome by Padé approximants or an Euler-accelerated series.

Lots of analytical methods such as Adomian decomposition methods [4–6], variational iteration methods [7–11], and homotopy analysis methods [12–14] have been proposed.

2. Derivation of an Iteration Formula

We develop a new iteration method to find the analytical series solution of the Blasius problem (1.1) subject to the boundary condition where the curvature of the solution is assumed to be known. It should be noted that in order to make the problem easy to be solved, we consider the one point boundary conditions in (2.1) instead of the two-point boundary conditions in (1.2).

First, for the Blasius equation (1.1) becomes From the boundary conditions in (2.1), it follows that This can be represented by which implies In the result, we have

If we denote by the th iterate solution and substitute it into the right hand side of (2.6), we have an iteration formula where From (2.3) and (2.7), the function can be represented by for with . Referring to the boundary conditions in (2.1), we may take the initial solution as The proposed method can be summarized by the following algorithm.

Algorithm A. We have the following steps.Step 1. Set initial guesses Step 2. For a large integer , perform the iteration (2.7)–(2.9) using symbolic computations
It should be noted that by performing this algorithm, we can also obtain the approximates to the velocity .

3. Analytical Solutions

Performing the above algorithm by using the symbolic calculation software Mathematica, we have the successive approximate solutions below

For the case of , we have One can see that the result is consistent with the known series solution [11, 12].

In particular, when and , it follows that In this case, . For comparison, we refer to another analytical solution obtained by the Adomian decomposition method as follows: This solution is based on with , and Adomian polynomial generated by the formula [6] where is an inverse operator of . Comparing the formulas in (3.3) and (3.4), one can see that the presented analytical solution has more terms than in each th iteration. In other words, for any integer . In practice, Figure 1 depicts that the presented solutions and their derivatives , approximate exact ones better than and . Therein, we chose the initial solution as given in (2.10) and took a numerical solution for the exact solution which is denoted by . Moreover, Table 1 includes numerical results of the errors and the CUP times spent in computations for the presented solution compared with those of . The error indicates the maximum error for the 50 nodes selected in the interval , where is a radius of convergence of the series solution given in the literature [2, 15]. In fact, for and for . The error means over the same interval.

(a) Presented solutions

(b) Adomian’s solutions

(a) Presented solutions
(b) Adomian’s solutions

Figure 1 

Graphs of the presented solutions and their derivatives in (a) and those of Adomian’s solutions and their derivatives in (b).

By the numerical performance, we can surmise the convergence of the algorithm proposed in this work, and the rate of convergence is better than that of the Adomian decomposition method though it spends more CPU time as shown above. In addition, for example, for the case of and , we may guess that the presented method has the same radius of convergence, as the well known Blasius’ series [2] as follows: where and Theoretical convergence analysis with extended application of the presented method to more general problems is left for further works.

Acknowledgments

The author would like to show his gratitude to the Department of Mathematics at the University of British Columbia, where the author worked as a visiting scholar for a year. Particularly, the author is heartily thankful to Professor Anthony Peirce for his favor and kind assistance. In addition, the author shows his sincere gratitude to reviewers for their helpful comments and valuable suggestions on the first draft of this paper. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science, and Technology (20110006106).

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Copyright

Copyright © 2011 Beong In Yun. 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.