Numerical solutions of the classical Blasius flat-plate problem

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Abstract

This paper presents a numerical study of the non-linear differential equationaf+ff=0,where a prime denotes differentiation with respect to the similarity variable η and a is a parameter. For a = 1 and a = 2 this equation is a form of the Blasius relation for the flat-plate flow in fluid mechanics. Several numerical solution are obtained using a Runge-Kutta algorithm for high-order initial value problems for 1  a  2.

Introduction

In recent years, the study of the steady flow of viscous incompressible fluid has gained considerable interest because of its extensive engineering applications. Since the pioneering work of Howarth [1] various aspects of the problem have been investigated by many authors. In [1], hand computations using the Runge-Kutta numerical method were performed for the flat-plate flow. Lock [2], [3] studied the laminar boundary layer between parallel streams. Later, Potter [4] investigated laminar boundary layer solutions for mass transfer across the plane interface between two-current parallel fluid streams. Blasius solution for flow past a flat-plate was investigated by Abussita [5] and the existence of a solution was established. Asaithambi [6] presented a finite-difference method for the solution of the Falkner-Skan equation and very recently, Wang [7] obtained an approximate solution for classical Blasius equation using Adomian decomposition method. In this work, we obtain numerical and complete solutions of the classical Blasius flat-plate problem by using a Runge-Kutta algorithm for high-order initial value problems (see Ref. [8]).

Section snippets

Problem definition and solution procedure

The equations of motion for the classical Blasius flat-plate flow problem can be summarized by the following boundary value problem [1]2f+ff=0,f=f=0,atη=0;f1asη,where a prime denotes differentiation with respect to η. This is a form of the Blasius relation for the flat-plate. Sometimes, the factor of 2 in Eq. (1) has been omitted in favour of a canonical form of this equation. Furthermore, we consider in this paper the equationaf+ff=0,and obtain numerical solutions of the problem

Illustrative results

Following Section 2, the problem (1), (2) was solved with Δη = 0.1 and the numerical results are shown in Fig. 1 and Table 1. From this Table, it is obvious that our numerical results are in good agreement with those given by Howarth [1].

In the same manner, numerical results were obtained for several values of a in the interval 1  a < 2. Some of the computed results for the variations with η of the functions f,f′ and f′′ are listed in Table 2. From this Table, one sees that the effect of the

Conclusion

In this brief paper an initial value problem (IVP) is employed to give numerical solutions of the classical Blasius flat-plate flow in fluid mechanics (see Table 1 and the case a = 1 in Table 2). To illustrate the accuracy and efficiency of the proposed procedure, various different examples in the interval 1 < a < 2 have also been analyzed and the numerical results are listed in Table 2. We found that, f and f′ decreases with increasing a.

On the other hand, we have compared in Table 1 the numerical

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