A study on mhd boundary layer flow over a nonlinear stretching sheet using implicit finite difference method

IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308
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Volume: 02 Issue: 12 | Dec-2013, Available @ http://www.ijret.org 287
A STUDY ON MHD BOUNDARY LAYER FLOW OVER A NONLINEAR
STRETCHING SHEET USING IMPLICIT FINITE DIFFERENCE
METHOD
Satish V Desale1
, V H Pradhan2
1
R.C.Patel Institute of Technology, Shirpur, India satishdesale@gmail.com
2
S.V.National Institute of Technology, Surat, India pradhan65@yahoo.com
Abstract
The problem of the MHD boundary layer flow of an incompressible viscous fluid over a non-linear stretching sheet is discussed.
Using similarity transformation governing equations are transformed in to nonlinear ordinary differential equation. The Implicit finite
difference Keller box method is applied to find the numerical solution of nonlinear differential equation. Graphical results of fluid
velocity have been presented and discussed for the related parameters.
Keywords: MHD, boundary layer, nonlinear differential equations, and Keller box method
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1. INTRODUCTION
In 1973, McCormack & Crane [8] introduced the stretching
sheet problem. The stretching problems for steady flow have
been used in various engineering and industrial processes, like
non-Newtonian fluids, MHD flows, porous plate, porous
medium and heat transfer analysis.
Magnetohydrodynamics (MHD) is the study of the interaction
of conducting fluids with electromagnetic phenomena. The
flow of an electrically conducting fluid in the presence of a
magnetic field is important in various areas of technology and
engineering such as MHD power generation, MHD flow
meters, MHD pumps, etc.
Chiam [7] investigated the MHD flow of a viscous fluid
bounded by a stretching surface with power law velocity. He
obtained the numerical solution of the boundary value
problem by using the Runge–Kutta shooting algorithm with
Newton iteration. Hayat et al. [6] applied the modified
Adomian decomposition method with the Padé approximation
and obtained the series solution of the governing nonlinear
problem.
Rashidi in [4] used the differential transform method with the
Padé approximant and developed analytical solutions for this
problem. In [1, 2] applied the HAM in order to obtain an
analytical solution of the governing nonlinear differential
equations.
In this paper, MHD boundary layer equation is solved with the
help of implicit finite difference Keller box method and
various results are discussed graphically.
2. GOVERNING EQUATIONS
Consider the Magnetohydrodynamic flow of an
incompressible viscous fluid over a stretching sheet at y = 0.
The fluid is electrically conducting under the influence of an
applied magnetic field B(x) normal to the stretching sheet. The
induced magnetic field is neglected. The resulting boundary
layer equations are as follows [6]
(1)0
u v
x y
 
 
 
2 2
2
(2)
( )u v u B x
u v u
x y y
 

  
  
  
where u and v are the velocity components in the x - and y –
directions respectively,  is the kinematic viscosity, ρ is the
fluid density and σ is the electrical conductivity of the fluid. In
equation (2), the external electric field and the polarization
effects are negligible and following Chiam [7] we assume that
the magnetic field B takes the form
  ( 1)/2
0
n
x BB x 

The boundary conditions corresponding to the non-linear
stretching of the sheet are [6]
( ,0) , ( ,0) 0n
u x cx v x 

__________________________________________________________________________________________
( , ) 0,u x y y   where c and n are constants.
using the following substitutions
1
2
( 1)
2
n
c n
t x y
v



'( )n
u cx f t
1
2
( 1) 1
( ) '( )
2 1
n
c n n
v x f t t f t
v n

  
    
Equation (1)-(2) are transformed to
 
2''' '' ' '
(3)( ) ( ) ( ) ( ) ( ) 0f t f t f t f t Mf t   
with the following boundary conditions
' '
(4)(0) 0, (0) 1, ( ) 0f f f   
Where
2
022
,
1 (1 )
Bn
M
n c n



 
  .
3. KELLER BOX METHOD
Equation (3) subject to the boundary conditions (4) is solved
numerically using implicit finite difference method that is
known as Keller-box in combination with the Newton’s
linearization techniques as described by Cebeci and Bradshaw
[3]. This method is unconditionally stable and has second-
order accuracy.
In this method the transformed differential equation (3) are
writes in terms of first order system, for that introduce new
dependent variable ,u v such that
(5)'f u
(6)'u v
Where prime denotes differentiation with t.
2
(7)
' 0v fv u Mu   
with the new boundary conditions
(8)(0) 0, (0) 1, ( ) 0f u u   
Now, write the finite difference approximations of the
ordinary differential equations (5)-(6) for the midpoint
1
2
,n
jx t 
 
 
  of the segment using centered difference
derivatives, this is called centering about
1
2
,n
jx t 
 
 
  .
1 1
(9)
2
n n n n
j j j j
j
f f u u
h
  

1 1
(10)
2
n n n n
j j j j
j
u u v v
h
  

Now, ordinary differential equation (7) is approximated by the
centering about the mid-point
1
12
2
,
n
jx t


 
 
  of the rectangle.
In first step we center about
1
2 ,
n
x t
 
 
  without specifying t.
     
     
2
12 1
'
1 1'
n n
n n
n nv fv u Mu
n nv fv u Mu


 
  
      
  
Now centering about
1
2
j 
(for simplicity, remove n)
   
1 1 1
1 1 1
1
1 2
1/21/2 1/2
(11)
2 2
2 2 2
j j j j j j
j
j j j j j j
n
j j
jj j
j
v v f f v v
h
u u u u u u
M
v v
fv u Mu
h


  
  


 
      
            
       
      
     
 
    
  
Now linearize the nonlinear system of equations (9)-(11) using
the Newton’s quasi-linearization method [5]
For that use,

__________________________________________________________________________________________
1
1
1
n n n
j j j
n n n
j j j
n n n
j j j
f f f
u u u
v v v






 
 
 
Equation (8) to(10) can be rewritten as
   1 1 1 (12)
2
j
j j j j j
h
f f u u r       
   1 1 2 (13)
2
j
j j j j j
h
u u v v r       
 
1 2 1 3 4 1
5 6 1 3 (14)
( ) ( ) ( ) ( )
( ) ( )
j j j j j j j j
j j j j j
a v a v a f a f
a u a u r
   
 
 

   
 
The linearized difference system of equations (12)-(14) has a
block tridiagonal structure. In a vector matrix form, it can be
written as
   
     
     
     
   
1 11 1
2 22 2 2
3 33 3 3
1 1 1
[ ] [ ]
[ ] [ ]
[ ] [ ]
:: : :
: :
[ ] [ ]
J J
J J J
J J
rA C
rB A C
rB A C
B A C
B A r




  

    
    
    
    
    
    
    
         
This block tridiagonal structure can be solved using LU
method explained by Na [5].
4. RESULT AND DISCUSSION
Graphically, effects of magnetic parameter and non-
dimensional parameter  on velocity profile are shown in
following figures. Fig.1 and 2 shows, as the magnetic
parameter M and non-dimensional parameter  increases, the
fluid velocity decreases. Also it can be observed that as
magnetic parameter M increases velocity profile decreases
very rapidly as compared to increase of non-dimensional
parameter  . Fig 3 and 4 shows the effect of magnetic
parameter on f.
Fig -1:.Effect of Magnetic Parameter on fluid velocity obtained by Keller Box Method for
1 

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Fig -2: Effect of on fluid velocity obtained by Keller Box Method when 2M 
Fig -3: Effect of Magnetic Parameter on f (t) obtained by Keller Box Method for 1 
Fig -4: Effect of Magnetic Parameter on f (t) obtained by Keller Box Method for
1  

__________________________________________________________________________________________
CONCLUSIONS
In this study, MHD viscous flow over a stretching sheet is
considered. Keller Box Method is applied to solve the
governing nonlinear differential equation. The results indicate
that the velocity will be reduced by increasing the two
parameters of M and β; however, effect of “M” in reduction of
the velocity components is more than “β”. Therefore, it can be
concluded that Keller Box is one of the best method to study
on MHD viscous flow numerically and get the appropriate
results.
REFERENCES
[1] A.Mehmood, S. Munawar, A. Ali, Comments to:
''Homotopy analysis method for solving the MHD flow
over a non-linear stretching sheet (Commun. Nonlinear
Sci. Numer. Simulat. 14 (2009) 2653-2663)'',
Communications in Nonlinear Science and Numerical
Simulation, 15 (2010) 4233-4240.
[2] A.R. Ghotbi, Homotopy analysis method for solving
the MHD flow over a non-linear stretching sheet,
Communications in Nonlinear Science and Numerical
Simulation,14 (2009) 2653-2663.
[3] Cebeci T., Bradshaw P. Physical and computational
Aspects of Convective Heat Transfer, New York:
Springer (1988), 391-411.
[4] M.M. Rashidi, The modified differential transform
method for solving MHD boundary-layer equations,
Computer Physics Communication, 180 (2009), 2210-
2217.
[5] Na T.Y., Computational Methods in Engineering
Boundary Value Problem. New York: Academic Press
(1979), 111-118.
[6] T. Hayat, Q. Hussain, T. Javed, The modified
decomposition method and Padé approximants for the
MHD flow over a non-linear stretching sheet,
Nonlinear Analysis: Real World Applications, 10
(2009), 966–973.
[7] T.C. Chiam, Hydromagnetic flow over a surface
stretching with a power-law velocity, Int. J. Eng. Sci.
33 (1995), 429-435.
[8] McCormack P.D., Crane L., "Physics of Fluid
Dynamics", New York, Academic Press, (1973).

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