International Journal of Engineering Mathematics

Volume 2016, Article ID 9014354, 6 pages

http://dx.doi.org/10.1155/2016/9014354

## Numerical Solution of the Blasius Viscous Flow Problem by Quartic B-Spline Method

Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Guilan, P.O. Box 1914, Rasht 41938, Iran

Received 1 March 2016; Accepted 28 March 2016

Academic Editor: Josè A. Tenereiro Machado

Copyright © 2016 Hossein Aminikhah and Somayyeh Kazemi. 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

A numerical method is proposed to study the laminar boundary layer about a flat plate in a uniform stream of fluid. The presented method is based on the quartic B-spline approximations with minimizing the error -norm. Theoretical considerations are discussed. The computed results are compared with some numerical results to show the efficiency of the proposed approach.

#### 1. Introduction

One of the well-known equations arising in fluid mechanics and boundary layer approach is Blasius differential equation. The classical Blasius [1] equation is a third-order nonlinear two-point boundary value problem, which describes two-dimensional incompressible laminar flow over a semi-infinite flat plate at high Reynolds number,with boundary conditionswhere the prime denotes the derivatives with respect to . In addition to the unknown function , the solution of (1) and (2) is characterized by the value of . Blasius [1] in 1908 found the exact solution of boundary layer equation over a flat plate. A highly accurate numerical solution of Blasius equation has been provided by Howarth [2], who obtained the initial slope . Liu and Chang [3] have developed a new numerical technique; they have transformed the governing equation into a nonlinear second-order boundary value problem by a new transformation technique, and then they have solved it by the Lie group shooting method. He [4, 5] gave a solution in a family of power series with parameter by means of the perturbation method for solving this equation. Bender et al. [6] proposed a simple approach using -expansion to obtain accurate totally analytical solution of viscous fluid flow over a flat plate. Aminikhah [7] used LTNHPM to obtain an analytical approximation to the solution of nonlinear Blasius viscous flow equation. Recently, the fixed point method (FPM) [8], which is based on the fixed point concept in functional analysis, is adopted to acquire the explicit approximate analytical solution to the nonlinear differential equation. Finally, efforts [9, 10] have been made to obtain the solution at the surface boundary and changed the problem from a boundary value differential equation into an initial value one. The solution in the entire domain, however, still requires computation.

In this paper, the quartic B-spline approximations are employed to construct the numerical solution for solving the Blasius equation. The unknowns are obtained with using optimization. The proposed method is applied to the problem and the computed results are compared with those of Howarth’s method to demonstrate its efficiency.

#### 2. Description of the Method

Let there be a uniform partition of an interval as follows: , where , . The quartic B-splines are defined upon an increasing set of knots over the problem domain plus 8 additional knots outside the problem domain; the 8 additional knots are positioned as The quartic B-splines , , at the knots are defined as [12] And the set of quartic B-splines forms a basis over the region .

Let be the quartic B-spline function at the nodal points. Then approximate solution of (1) can be written as [13]where are the quartic B-spline functions and are the unknown coefficients. Each B-spline covers the five elements so that an element is covered by five B-splines. The values of and its derivatives are tabulated in Table 1.