Chebyshev finite difference method for the solution of boundary-layer equations

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Abstract

A new Chebyshev finite difference method is proposed for solving the governing equations of the boundary-layer flow. The Falkner–Skan equation has been solved as a model problem. The more general problem of the equations governing magnetohydrodynamic three dimensional free convection on a vertical stretching surface is solved. The comparisons between the data resulting from the present method and those obtained by others are made. The results indicate that the suggested method yields more accurate results than those computed by others.

Introduction

Chebyshev polynomials are used widely in numerical computations. One of the advantages of using Chebyshev polynomials Tn(x) as expansion functions is the good representation of smooth functions by finite Chebyshev expansions, provided that the function u(x) is infinitely differentiable. Chebyshev polynomials have proven successfully in the numerical solution of various boundary value problems [10], [12] and in computational fluid dynamics [3], [15].

The finite difference methods have been used extensevly for the evaluation of the boundary-layer flow. Beckett [1], Wadia and Payne [16], Rosenhead [14] and White [17] have discussed different finite difference schemes for the solution of the Falkner–Skan equation.

Nasr et al. [13] use Chebyshev expansion procedure based on El-Gendi method [7] to solve the Falkner–Skan equation. El-Gindy et al. [8] use the Chebyshev spectral method with modified Rayleigh–Ritz method to solve the Falkner–Skan equation. El-Hawary [9] use a deficient spline function to solve the Falkner–Skan equation.

The present work deals with application of a radically new approach to computation of the boundary-layer equations in magnetohydrodynamic (MHD) flows. This approach requires the definition of a grid as the finite difference and elements techniques also do and it is applied to satisfy the differential equation and the boundary conditions at the grid points. It can be regarded as a non-uniform finite difference scheme. The derivatives of the function f(x) at a point xj is linear combination from the values of the function f(x) at the Gauss–Lobatto points xk=−cosN, where k=0,1,2,…,N, and j is an integer 0⩽jN. The suggested method is more accurate in comparison to the finite difference and finite elements methods because the approximation of the derivatives are defined over the whole domain. In [6], this approach is applied successfully for solving the boundary value problems. The novelties of this techniques which we will call Chebyshev finite difference method (or shortly ChFD), can broadly be summarized in the next section, in which we applied the suggested method to the Falkner–Skan equation as a model problem. In Section 3, we apply the present method to more general boundary-layer problem which considered by Chamkha [4].

Section snippets

The method of solution for a model problem

The Falkner–Skan equation which is derived by White [17] was chosen as the equation of interest because of the inherent non-linearity that it exhibits. Illustrating the basic ideas of the ChFD method we will consider the Falkner–Skan equation as a model problem.f(η)+αf(η)f(η)+β(1−f(η)2)=0with the boundary conditionsf(0)=f(0)=0,f(η)→1asη→∞.Here α is assumed constant, β is a measure of the pressure gradient. The special case of the Blasius similarity relation for incompressible viscus flow

More general boundary-layer problem

In this section we will apply the ChFD method to more general problem. That is the problem of steady, laminar, three-dimensional free convection flow over a vertical porous surface in the presence of magnetic field and heat generation or absorption. This problem was considered by Chamkha [4]. He used the implicit, iterative, tri-diagonal finite difference method which is similar to the Kellers box method and it is discussed by Blottner [2].

Conclusions

The problem of boundary-layer flow is investigated numerically by new accurate Chebyshev finite difference method. Such procedure has been established throughout the paper with details, and has been showed when applies it on boundary-layer equations highly accurate results, as shown in the given tables, better than those obtained by other techniques.

References (17)

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