# Analytical solution of Blasius problem.

Introduction

In fluid mechanics, the problem of flow past a flat plate was first introduced by Blasius (1908) by assuming a series solutions. Later, numerical methods were used in L.Howarth, (1938) to obtain the solution of the boundary layer equation. In (M.E.Eglit et al, 1996) the first derivative with respect to y of the velocity component in the x direction at the point y = 0 for the Blasius problem is computed numerically for the estimation of the shear-stress on the plate surface. Later in (P.Vadasz, 1997) one solved the problem above by assuming a finite power series where the objective is to determine the power series coefficients. The purpose of this study is to obtain the solutions for the Blasius problem for two dimensional boundary layer using the Adomian decomposition technique and to compute the admissible values of the shear-stress on the wall, imposing the constraint on the first derivative with respect to y of the velocity component in the x direction at the point y = 0.

Mathematical model:

The physical model considered here consists of a flat plate parallel to the x- axis with its leading edge at x = 0 and infinitely long down stream with constant component [u.sub.o] of the velocity. For the mathematical analysis we assume the properties of the fluid such as viscosity and conductivity, to a first approximation, are constant. Under these assumptions the basic equations required for the analysis of the physical phenomenon are the equations of continuity and motion. According to the Boussinesq approximation these equations get the following expressions (M.E.Eglit et al, 1996)

[[[partial derivative].sub.u]/[[partial derivative].sub.x]] + [[[partial derivative].sub.v]/[[partial derivative].sub.y]] = 0 (1)

u[[[partial derivative].sub.u]/[[partial derivative].sub.x]] + u[[[partial derivative].sub.u]/[[partial derivative].sub.x]] = v[[[[partial derivative].sub.2]u]/[[partial derivative][y.sub.2]]] (2)

with the boundary conditions imposed on the flow in (M.E.Eglit et al, 1996)

[psi] = [[partial derivative].sub.[psi]]/[[partial derivative].sub.y] = 0, y = 0,[lim.sub.y[right arrow][infinity]][[[partial derivative].sub.[psi]]/[[partial derivative].sub.y]] (3)

Where [psi] is a stream function related to the velocity components as:

u = [[partial derivative].sub.[psi]]/[[partial derivative].sub.y], v = [-[partial derivative].sub.[psi]]/[[partial derivative].sub.x] (4)

3. Analytical solution and convergence results:

In this section we provide the analytical solutions, i.e. the fluid velocity components as sums of convergent series using the Adomian decomposition technique and compute the admissible values of the shear-stress on the plate surface Consider the stream function [psi]

[psi](x,y) = [square root of v[u.sub.0]x]f([eta]), [eta] = y[square root of [[u.sub.0]/[v.sub.x]]] (5)

Where f is a function three times continuously differentiable on the interval [0, [[eta].sub.0]] and [[eta].sub.0] a constant positive real. Then the equations (1) and (2) with the boundary conditions (3) are transformed as

f'" + [1/2]ff" = 0, (0) = f'(0) = 0,f'(+[infinity]) = 1 (6)

where (.)' stands for [d(.)/d[eta]]

Definition:

The problem (6) is called the Blasius problem for boundary-layer flows of pure fluids (non-porous domains) over a flat plate.

Let us transform the problem (6) into the nonlinear integral equation. For this purpose, setting g'([eta]) = f([eta]) we can write the equation in (6) as

9""+[1/2]g'g'" = 0 (7)

Multiplying by [e.sup.[1/2]g] and integrating the result from 0 to [eta] we reduce (7) to

g'" = [Ke.sup.-[1/2]g], where K = g'"(0) [e.sup.[1/2]g(0)] (8)

Integrating three times the relation (8) from 0 to [eta], [tau], [sigma] and taking into account the boundary conditions in (6) we reduce (8) to the nonlinear integral equation

g([eta]) - K [[integral].sup.[eta].sub.0] [[integral].sup.[tau].sub.0] [[integral].sup.[sigma].sub.0][e.sup.- 1/2g(S)]dsd[sigma]d[tau] = [a.sub.0] [a.sub.0] = g(0) = const (9)

which is a functional equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

Here N(g) is a nonlinear operator from a Hilbert space H into H. In (G.Adomian, 1990) G. Adomian has developed a decomposition technique for solving nonlinear functional equation such as (10). We assume that (10) has a unique solution. The Adomian technique allows us to find the solution of (10) as an infinite series g = [[summation].sub.n[greater than or equal to]0[g.sub.n]] using the following scheme:

[g.sub.0] = [a.sub.0]

[g.sub.1] = [A.sub.0]

[g.sub.n+1] = [A.sub.n]([g.sub.0],[g.sub.1,...,][g.sub.n])

N(g) = [[SIGMA].sub.n[greater than or equal to]0][A.sub.n], where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

n = 0,1,2,....

The proofs of convergence of the series [[SIGMA].sub.n[greater than or equal to]0][g.sub.n] and [[SIGMA].sub.n[greater than or equal to]0][A.sub.n] are given below. Without loss of generality we set [a.sub.0] = 0 and we have the following scheme:

[[d.sub.n]/d[g.sub.n]]N([g.sub.0]) = [(-[1/2]).sub.n] N([g.sub.0])

[g.sub.0] = 0

[g.sub.1] = [1/6]K[[eta].sup.3] [equivalent to] [b.sub.1]K[[eta].sub.3]

[g.sub.2] = -[1/72][(K[[eta].sup.3]).sup.2] [equivalent to] [b.sub.2][(K[[eta].sup.3]).sup.2]

[g.sub.3] = [1/576][(K[[eta].sup.3]).sup.3] [equivalent to] [b.sub.3][(K[[eta].sup.3]).sup.3]

[g.sub.4] = [1/3888][(K[[eta].sup.3]).sup.4] [equivalent to] [b.sub.4][(K[[eta].sup.3]).sup.4]

[g.sub.5] = [125/2985984][(K[[eta].sup.3]).sup.5] [equivalent to] [b.sub.5][(K[[eta].sup.3]).sup.5]

[g.sub.6] = [569/79626240][(K[[eta].sup.3]).sup.6] [equivalent to] [b.sub.6][(K[[eta].sup.3]).sup.6]

By induction, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

i.e [g.sub.n+1] [equivalent to] [b.sub.n+1][(K[[eta].sup.3]).sup.n+1], n [greater than or equal to] 1

h = [summation over (i [greater than or equal to] 0)] [[lamda].sup.i][g.sub.i]

where the [b.sub.n] are real numbers. Then we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

We arrive at the following results:

Lemma 3.1:

The velocity components of the fluid flow u, V of the equations of continuity and motion (1) and (2) satisfying the boundary conditions (3) are defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

[eta] = y[square root of [u.sub.0]/[v.sub.x]]

Remark 3.1:

The terms approximation of

g = [+[infinity].summation over (n=0)][g.sub.n]

is 18th degree polynomial in [eta] = [eta] = y[square root of [u.sub.0]/[v.sub.x]]

For the proofs of convergence of the series with general term (11), the following statement holds

Lemma 3.2:

The nonlinear operator IV Qj) can be developed in entire series with a convergence radius equal to infinity

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

Furthermore the following series

[+[infinity].summation over (n=0)][A.sub.n] and [+[infinity].summation over (n=0)][g.sub.n] (17)

are convergent.

Proof 3.1:

Taking into account the expansion of the function g [??] [e.sup.[-1/2]g] entire series and using the uniform convergence of entire series we obtain the formula. By the Cauchy-Hadamard formula for the radius, we prove that the radius is equal to infinity.

On the other side, taking into account the different expressions of N(g)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

we obtain that the series [[SIGMA].sub.i[greater than or equal to]0][A.sub.i] is convergent with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

then the series [[SIGMA].sub.i[greater than or equal to]0][g.sub.i] is convergent; that completes the proof.

4. Error estimate:

In this section we estimate the error by approximating the exact values of the shear-stress Y by the value Y of the shear-stress obtained in (M.E.Eglit et al 1996). The shear-stress Y at the plate surface is defined in (M.E.Eglit et al, 1996) by

[GAMMA] = [mu][([partial derivative]u/[partial derivative]y).sub.y=0] (19)

Taking into account (14) we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

As K is defined implicitly from (20) we suggest to compute the admissible values of [GAMMA], taking into account K [approximately equal to] 0.332 in (M.E.Eglit et al 1996). For this purpose we impose the constraint on the absolute error [absolute value of [GAMMA] - [??]]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)

It is observed from the table, that the absolute error decreases with the increase in the values of x The approximation may be efficient if the approximation precision is too small,i.e. there exists n [member of] [??]* such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We arrive at the following result

Lemma 4.1:

The admissible values of the shear-stress T on the plate surface obtained in (20) belong to the open interval

]0.332 [square root of [[mu][rho][u.sup.3.sub.0]/x]] - [10.sup.-n]; 0.332 [square root of [[mu][rho][u.sup.3.sub.0]/x]] + [10.sup.-n][ (22)

for each given value of x > 0 and for the given approximation precision depending on n [member of] [N.sup.*].

Conclusion:

In this paper, we have investigated the analytical solutions for the Blasius problem which are the sums of convergent series, using the Adomian decomposition technique. Then we estimated the error by approximating the exact values of the shear-stress on the plate surface obtained in this paper by the approximate values of the shear-stress obtained in (M.E.Eglit et al 1996). Doing so, we constructed the interval of admissible values of the shear-stress on the plate surface.
```Nomenclature

1. u               velocity in the x-direction
2. [u.sub.o]       velocity of the free stream
3. v               velocity in the y-direction
4. x               horizontal coordinate
5. y               vertical coordinate
6. [mu]            viscosity coefficient
7. [rho]           density
8. v = [mu]/[rho]  kinematic viscosity of the fluid
```

References

Busse, F., 1987.[u.sub.o]n the optimum theory of turbulence, Energy Stability and convection", Pitman Research Notes in Mathematics, editedby G. Galdi and B. Straughan, Wiley, New York.

Eglit M.E. et al., 1996. "Problemes de mecanique des milieux continus".Tomes 1,2, Lycee Moscovite.

Adomian, G., 1991."A review of the decomposition method and some recent results for nonlinear equations". Comput.Math.Applic, 21(5): 101-127.

Adomian, G., 1990."A review of the decomposition method and some recent results for nonlinear equations", Math.Comp.Modelling, 13(7): 17-43.

Blasius, H., 1908."Grenzschichten inFlussigkeiten mit kleiner Reibung". Z.Math.Phys., 56(1).

Cherruault, Y., 1989."Convergence of Adomian's method", Kybernetes, 18(2): 31-38.

Howarth, L., 1938.[u.sub.o]n the solution of the laminar boundary layer equations". Proc.Roy.Soc.London, A164,547.

Lu, L., C.R. Doering and F.H. Busse, 2004. "Bounds on convection driven by internal heating". J. Math.Phys., 45: 2967-2986.

Vadasz, P., 1997."Free convection in rotating porous media.Transport Phenomena in porous media". Pergamon Elsevier Science, pp: 285-312.

Zadrzynska, E. and W.M. Zajaczkowski, 2009. "Global Regular Solutions with Large Swirl to the Navier Stokes Equations in a cylinder". J.Math.Fluid Mechanics,11: 126-169.

* Francois de Paule Codo received the Mining Engineer, M.Sc.in Mining Sciences and Ph.D.degrees in Mining Sciences from the Heavy Industries Technical University of Miskolc, Hungary He is currently Assistant Professor of Applied Fluid Mechanics and Hydraulics in Department of Civil Engineering at the University of Abomey-Calavi,Benin. His principal research interests are Applied Fluid Mechanics and Hydraulics at the Applied Mechanics and Energy Laboratory. e-mail:fdepaule2003@yahoo.fr

* Villevo Adanhounme received the M.Sc. and Ph.D. degrees in Mathematics from the Russian People University of Moscow, Federation of Russia. He is currently Assistant Professor of Variational Calculus and Advanced Probability at the International Chair of Mathematical Physics and Applications-University of Abomey-Calavi, Benin. His principal research interests are applied mechanics, partial differential equations and optimal control in the International Chair of Mathematical Physics and Applications. e-mail:adanhounm@yahoo.fr

* Alain ADOMOU received the M.Sc. and Ph.D. degrees in Theoretical Physics from the Russian People University of Moscow, Federation of Russia. He is currently Assistant Professor of Mechanical Theory of continua at the Technological Institute of Lokossa, University of Abomey-Calavi, Benin. His principal research interests are applied mechanics and theory of gravitation. e-mail: denisadomou@yahoo.fr

(1) Institut Universitaire de Technologie, Universite d'Abomey-Calavi, B.P. 133 Lokossa, Republic of Benin.

(2) Laboratoire d'Energetique et de Mecanique Appliquee (LEMA-EPAC), Universite d'Abomey-Calavi, 01 B.P.2009 Cotonou, Republic of Benin.

(3) Institut (1) International Chair of Mathematical Physics and Applications, (ICMPA-UNESCO Chair), Universite d'Abomey-Calavi, 072B.P. 50 Cotonou, Republic of Benin.

Corresponding Author: A. Adomou, Institut Universitaire de Technologie, Universite d'Abomey-Calavi, B.P. 133 Lokossa, Republic of Benin.
```Table

X     [absolute value of [[GAMMA] - [??])]

1     [absolute value of (K - 0.332)] [square root of [mu] [rho]
[u.sup.3.sub.0])]
4     1/2 [absolute value of (K - 0.332)] [square root of [mu] [rho]
[u.sup.3.sub.0])]
9     1/3 [absolute value of (K - 0.332)] [square root of [mu] [rho]
[u.sup.3.sub.0])]
16    1/4 [absolute value of (K - 0.332)] [square root of [mu] [rho]
[u.sup.3.sub.0])]
20    1/2[square root of 5] [absolute value of (K - 0.332)]
[square root of [mu] [rho] [u.sup.3.sub.0])]
25    1/5 [absolute value of (K - 0.332)] [square root of [mu] [rho]
[u.sup.3.sub.0])]
36    1/6 [absolute value of (K - 0.332)] [square root of [mu] [rho]
[u.sup.3.sub.0])]
49    1/7 [absolute value of (K - 0.332)] [square root of [mu] [rho]
[u.sup.3.sub.0])]
60    1/2[square root of 15] [absolute value of (K - 0.332)]
[square root of [mu] [rho] [u.sup.3.sub.0])]
64    1/8 [absolute value of (K - 0.332)] [square root of [mu] [rho]
[u.sup.3.sub.0])]
81    1/9 [absolute value of (K - 0.332)] [square root of [mu] [rho]
[u.sup.3.sub.0])]
```
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