The Blasius equation describes the nondimensional velocity distribution in the laminar boundary layer over a flat plate. It describes the similarity solution of the fluid flow influenced by viscous effects [33]. The Navier Stokes equation and the boundary conditions on the problem are

(3.1)

(3.2)

where ** u** and

Figure 3.1 Laminar flow over a flat plate

Using a stream function the continuity equation, the first
equation in Equation 3.1, is satisfied. Introducing the dimensionless
coordinate *η**, *and a dimensionless stream function ** f
**[33], the second equation in Equation (3.1) is reduced to

(3.3)

The boundary conditions then translate to

(3.4)

This equation, termed as the Blasius equation, is a third
order nonlinear ordinary differential equation with boundary conditions at two
points. Published solutions indicate that the infinity boundary condition is
easily met at η = 6. The analytical solution was obtained by Blasius by a
series expansion at ** η **= 0 and asymptotic expansion for large

The Blasius problem is part of the set of problems defined
by the Falkner-Skan equations, also appears to be a bench mark for comparing
the different techniques for the solution to the NLBVP, particularly analytical
ones. Reference 35 transforms the BVP into a pair of initial value problems
using transformation groups. The interval is broken into 17 domains with a
different power series in each domain. They are able to obtain good agreement
with Howarth’s solution. Reference 36 applies the Adomain decomposition to
obtain the transformation of the Blasius equation to Banach space and then
solves for the missing initial condition necessary to convert it to an initial
value problem. A standard numerical technique is then used to obtain the
solution. The procedure is strongly problem dependent. Reference 37 applies
to problems that have asymptotic boundary conditions, which is true of the
Blasius equation. It uses a special numerical technique to obtain the
solution. The original differential equations are normalized and rewritten
over a set of intervals. A perturbation system is developed to solve the
normalized equation and the whole process is iterated. Reference 38 uses a
direct perturbation technique by imbedding a small parameter into the Blasius
equation. This parameter value is then iterated until desired accuracy is
reached. Reference 39 uses perturbation and calculus of variations to
establish the analytic solution of the Blasius equation, which resembles a
power series, but has imbedded in it a factor that is similar to a basis
definition. Reference 40 uses the ** δ**- perturbation scheme
which involves iteration on

The problem in Equation (3.3, 3.4) is solved using a *Bezier
function*. Find the Bezier function [x, y] or the vertices of the parametric
curve {P_{1}, P_{2},..,P_{m+1} } or [a_{j} ,b_{j}],
j = 1, .., m, that

*Minimizes*

(3.5)

** Subject to:** a

b_{2}=0, []
(3.6)

a_{m+1} = 6.

where ** i** is point on the curve. The number of
points (

The problem in Equation (3.5, 3.6)
is solved using ** fmincon** from the MATLAB optimization toolbox.
The version used is MATLAB v.7.04.

The solution to the Blasius equation is presented in Figure 3.2. There are five vertices used to define the curve. A fourth order Bezier function is used to generate the curve. The minimum value of the objective function in Equation (3.5) is 7.4e-05.

Figure 3.2 Final Solution

Figure 3.3 is the comparison of the Bezier solution and the analytical solution (obtained numerically through the BVP solver). The axes are interchanged with respect to Figure 3.2 as η represents the y-variable. The figure illustrates that the Bezier function is can match the solution produced by the BVP solver available in the MATLAB package.

Figure 3.3 Solution to the Blasius equation

The explicit Bezier solution is

(3.7)

The analytical solution from Blasius was not used here because of convenience. This means that Figure 3.3 remains the major source of comparison. Since the residuals are computed exactly, its low value for the final iteration should provide reasonable confidence that the Bezier function provides an excellent solution.