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Solving Blasius Problem by Adomian Decomposition Method
« on: February 18, 2012, 02:11:07 am »
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Author : V. Adanhounme, F.P. Codo
International Journal of Scientific & Engineering Research Volume 3, Issue 1, January-2012
ISSN 2229-5518
Download Full Paper : PDF

Abstract - Using the Adomian decomposition method we solved the Blasius problem for boundary-layer flows of pure fluids (non-porous domains) over a flat plate. We obtained the velocity components as sums of convergent series. Furthermore we constructed the interval of admissible values of the shear-stress on the plate surface.

Index Terms - Convergent series, Decomposition technique, Fluid flow, Shear-stress.

Nomenclature
1.       velocity in the x-direction
2.         velocity of the free stream
3.       velocity in the y-direction
4.       horizontal coordinate
5.       vertical coordinate
6.       viscosity coefficient
7.        density
8.           kinematic viscosity of the fluid

1  INTRODUCTION
The problem of flow past a flat plate is one of interesting problems in fluid mechanics which was first  solved by Blasius [5]  by assuming a series solutions . Later, numerical methods were used in [7]  to obtain the solution of the boundary layer equation. In [2]  the first derivative with respect to     of the velocity component in the    direction at the point    for the Blasius problem is computed numerically for the estimation of the shear-stress on the plate surface. Later in [9]  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 of the velocity component in the    direction at the point .

2   MATHEMATICAL MODEL
The physical model considered here consists of a flat plate parallel to the   - axis with its leading edge at   and infinitely long down  stream  with constant component    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 [2]

                (1)
             (2)
with the boundary conditions imposed on the flow in [2]
  ,  ,         (3)
Where   is a stream function related to the velocity components as:
 ,                         (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     defined by
  ,        (5)
Where   is a function three times continuously differentiable on the interval   and   a constant positive real. Then  (1) and (2) with (3) are transformed as
  ,  (6)
where  stands for 

Definition 3.1
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 (6) into the nonlinear integral equation. For this purpose, setting  we can write the equation in (6) as
             (7)
Multiplying by  and integrating the result from   to  we reduce (7) to
  where        (8)
Integrating three times (8) from   to   and taking into account the boundary conditions in (6) we reduce  (8) to the nonlinear integral equation
  (9)
 
which is a functional equation
  where            (10)
       
             
Here  is a nonlinear operator from a Hilbert space  into  . In [4] 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   using the following scheme:
 
 , where
 
The proofs of convergence of the series   and  are given below.Without loss of generality we set  and we have the following scheme:
           
By induction, we have
 
 i.e.     , 
 
where the  are real numbers.Then we obtain

 
  We arrive at the following result
  Lemma 4.1
   The admissible values of the shear-stress  on the plate surface obtained in (20) belong to the open interval
          (22)
   for each given value of  and for the given approximation precision depending on     .

5  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 [2].Doing so, we constructed the interval of admissible values of the shear-stress on the plate surface.

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