Analysis of mhd non darcian boundary layer flow and heat transfer over an exponentially vertically strtching surface with thermal radiation

IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308
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Volume: 03 Special Issue: 03 | May-2014 | NCRIET-2014, Available @ http://www.ijret.org 987
ANALYSIS OF MHD NON-DARCIAN BOUNDARY LAYER FLOW AND HEAT TRANSFER OVER AN EXPONENTIALLY VERTICALLY STRTCHING SURFACE WITH THERMAL RADIATION M. Subhas Abel1, Veena.M.Basangouda2, M.Narayana3, Prashant.G.metri4 1Dept of Mathematics Gulbarga University, Gulbarga 585106, Karnataka, India 2Dept of Mathematics Gulbarga University, Gulbarga 585106, Karnataka, India 3School of Advanced science, VIT University, Vellore 632014, Tamilnadu, India 4Dept of Mathematics Gulbarga University, Gulbarga 585106, Karnataka, India Abstract This paper deals with the numerical study of MHD Non-Darcianlayer flow on an exponentially stretching surface and free convection heat transfer with a presence of Thermal Radiation. The flow is considered over a stretching sheet in the presence of non dimensional parameters. Conversion of governing nonlinear boundary layer equations to coupled higher order non-linear ordinary differential equations using similarity transformations. The obtained governing equations were solved numerically by using keller box method. The various nondimentional parameters effects with velocity profile and thermal profile are discussed in detail with graphically. Keywords: Thermal Radiation, Statching Surface
-----------------------------------------------------------------------***---------------------------------------------------------------------- 1. INTRODUCTION In industrial manufacturing process the heat and mass transfer problems are well used. This phenomena applicable in wire and fibre coatings and transpiration cooling etc. In astrophysics and geophysics the MHD flow basically used. Basically the MHD flow has wide applications. Usually used in Engineering and industrial.T he fluid subjected to a magnetic field become a good agreement results. There is a wide application in Mechanical Engineering field.. After the pioneering work of Sakiadis [1, 2] many researchers gave attention to study flow and heat transfer of Newtonian and non-Newtonian fluids over a linear stretching sheet. By considering quadratic stretching sheet, Kumaran and Ramanaiah [3] analyzed the problem of heat transfer. Ali [4] investigated the thermal boundary layer flow on a power law stretching surface with suction or injection.
Elbashbeshy [5] analyzed the problem of heat transfer over an exponentially stretching sheet with suction. Magyari and Keller [6] discussed the heat and mass transfer in boundary layers on an exponentially stretching continuous surface. Sanjayanand and Khan [7, 8] extended the work of Elbashbeshy [5] to viscoelastic fluid flow, heat and mass transfer over an exponentially stretching sheet .Raptis et al.[9] constructed similarity solutions for boundary layer near a vertical surface in a porous medium with constant temperature and concentration. Bejan and Khair [10] used Darcy’s law to study the features of natural convection boundary layer flow driven by temperature and concentration gradients. Forchheimer[11] proposed quadratic term in Darcian velocity to describe the inertia effect in porous medium. Plumb and Huenefeld[12] studied the problem of non-Darcian free convection over a vertical isothermal flat plate. Rees and Pop[13] also studied yhe free convection flow along a vertical wavy surface with constant wall temperature. Rees and Pop[14] studied the case where the heated surface displays waves while the Darcys law is supplemented by the Forchheimerterms. They stated that the boundary flow remains self similar in the presence of surface waves but where inertia is absent, and when inertia is present but surface waves are absent. However, the combination of the two effects yields non similarity. Tsou et al.[15] studied flow and heat transfer in the boundary layer on a continuous moving surface while Gupta and Gupta[16] solved boundary layer flow with suction and injection. Andresson and Bech[17] have studied the MHD flow of the power law fluid over stretching sheet. Pavlov[18] gave an exact similarity solution to the MHD boundary layer equation for the steady and two dimensional flow caused solely by the stretching if an elastic surface in the presence of uniform magnetic field. M S Abel and Mahesha [19] heat transfer in MHD visco elastic fluid flow over a stretching sheet with variable thermal conductivity non uniform heat, source andradiation. In the paper we analysed thermal radiation effect in a exponentially vertically stretching surface on a MHD flow. And effect of various physically parameters are also discussed in detail.

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2. MATHEMATICAL FORMULATION
Under the usual boundary layer approximations, the flow and heat transfer in the presence of radiation effects are governed by the
following equations:
0
u v
x y
 
 
 
(1)
2 2
2 0
2 b ( ) u u u C B
u v u u g T T u
x y y k k

  
 
  
      
   (2)
2
2 2 2
2 0
1
( ) ( ) r
p p p p
T T T u Q q
u v B u T T
x y y C C y C c y
 
 
    
    
      
    
(3)
The associated boundary conditions to the problem are
U= ( ), 0, ( ), w w U x v  T T x at y=0, (4)
, u 0,T T as    y   (5)
0 ( ) ,
x
L
w U x U e (6)
2
0 ( ) ( ) ,
ax
L
w T x T T T e      (7)

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Where 0 T and a are parameters of temperature distribution in the stretching surface. T is the temperature,K is the thermal
conductivity, p C is the Specific heat and r q is the radiative heat flux.
4 4
,
3 r
T
q
K y


 
 

(8)
Where K
is the mean absorption coefficient and   is the Stefan-Boltzmann Constant.
4 T is expressed as a linear function of
temperature,hence
T 4 4T3T 3T 4     (9)
Introducing the following non- dimensional parameter
2 2 Re
, ( , ) 2Re ( ),
2
x x
L L y
e x ve f
L
     
(10)
2
0 ( , ) ( ) ( ),
ax
T x y T T T e L       (11)
Where is the stream function which is defined in the usual form as
u
y


 &
v
x

 
 (12)
Substituting (10)&(11) in (12).We obtain u and v as follows
'
0 ( , ) ( ),
x
u x y  u eL f  2 ' Re
( , ) [ ( ) ( )].
2
x
L v
v x y e f f
L
    
(13)
Eqns (1) to(5) istronsformed into the ordinary differential equation with the aid of equations( 10)-(13).Thus, the governing equations
using the diemensionalessfuction f() and  ( ) become

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2
2
''' '' ' 2 2 '
2 1 (2 ) 2 2 ( ) 0
Re
ax
x x Ha
f ff N f Gre e  e f N         
(14)
(15)
The boundary conditions (4) and (5) reduce to
'' ' Re 2 (0).f(0)=0,f (0) 1, (0) 1, f x C  X f    (16)
' f ()  0, ()  0, (17)
Where,
x
X
L

2 2 1
( 0 )2
B L
Ha


 is Hartman number,
2
0 0 / ( ) p Ec U c T T   is Eckert
number,
2 /( Re) p  QL c is the dimensionless heat generation/absorption
parameter,
3
1 0 2 ( )
L
Gr g T T
v
    is the Grashof number, 0 Re U L/ v is Reynolds
number,
2
1 Gr=Gr /Re is the thermal buoyancy parameter, and Pr=


is the
Prandtlnumber,Where
2
1 , porous parameter
Re
L
N is the
k

2
2
b C L
N is the inertia coefficient
k
 ,
2   L,Z  Ha /Re,a W .K=
3 4 T
K K
 

 Radiation number.In the above system of local similarity equations ,the effect of the magnetic field is included as a ratio
of the Hartman number to the Reynolds number.
The physical quantities of interest in the problem are the local skin friction acting on the surface in contact with the ambient fluid of
constant density which is defined as
2 2
(2 ) 2
1 '' ' ' 2 ' '' 4
Pr (1 ) (2 ) 2 0
3 Re
X a
X X K Ha
 f af  e Ec f f e e 

        

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1
0 2 2 ''
0
Re
( ) ( )( ) (0)
2
x
wx y
u vU
v e f
y L

  

 
 (18)
And the non-dimensional skin friction coefficient, f C , which can be written as,
2
2
( )
wx
f
w
C
U



or
'' Re 2 (0). f x C  X f (19)
The local surface heat flux through the wall with k as thermal conductivity of the fluid is given by
1 ( 1)
0 2 2 '
0
( )Re
( ) ( ) (0).
2
a
wx y
T k T T
q k e
y L




 
  
 (20)
The local Nusselt number, x, Nu which is defined as
( )
,
( )
wx
x
w
xq x
Nu
k T T

 (21)
1
/ Re ( / 2)2 '(0), x x Nu   X  (22)
Where Rex is the local Reynolds number based on the surface velocity and is given by
( )
Re w
x
xU x
v

. (23)
3. NUMERICAL METHOD
The above Non linear equations that is 14 and 15 are subjected with similarity transformations and the obtained governing equations
solved by finite difference scheme kellor box method by gauss elimination method.
2
2
''' '' ' 2 2 '
2 1 (2 ) 2 2 ( ) 0
Re
ax
x x Ha
f ff N f Gre e  e f N         
(24)
2 2
(2 ) 2
1 '' ' ' 2 ' '' 4
Pr (1 ) (2 ) 2 0 (25)
3 Re
    

        
X a
X X K Ha
f af e Ec f f e e

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(0) 0, '(0) 1, (0) 1 0
'( ) 0, ( ) 0
 
 
   
     
f f as
f as
(26)
In this method the third and second order non linear
differential equations,
3.1 Finite Difference Scheme
This scheme involves 5 steps
Step 1: Decomposing of given differential equations into a set
of first order ordinary differential equations.
Step2: a) Approximate the first order derivatives with standard
forward difference
i i 1 dy y y
dx x
 


b) Approximate the dependent variables with two point
averages
1
2
i i y y
y  

using these approximation the
ordinary differential equations is transformed to finite
difference equations. solution, say i i i y y  y

  And
substituting this in the finite difference equation and drop
terms non-linear in i  y to arrive at linear F.D.E’s.
Step3: Linearise F.D.E using Newton’s method this involves to
start with a guess seidel, or Jacobi method]and obtain i  y
and add the correction to initial solution.
Step4: Solve the linearised F.D.E’s using the standard method
Gauss elimination.
Step 5.Repeat step 3 & Step 4 until we obtain the required
result.
4. RESULTS AND DISCUSSIONS:
Present results, are displayed in Table 1 and are noticed to be
well in agreement with the present work
Fig. 2 Represents the effect of magnetic field parameter
2
Re
Ha
,
on velocity profile f ' .Here magnetic field produces a drag in
the form of Lorentz force.Due to this effect,the magnitude of
velocity decreases and the thermal boundary layer thickness
increases.
Fig 3 Represents the various values of parameter a with
velocity profile . From this figure, it is observed that the value
of a increases with increase in the velocity flow.and maximum
velocity occurs at a=7.
Fig. 4Represents the dimensionless parameter X with
horizontal velocity profile. From this figure, it is noticed that
the value of X increases with decreases in the velocity
profile.here the flow is adjacent to a stretching sheet.
Fig. 5 It is observed from this figure that temperature decreases
with increase in the values of a. Further, it is noticed that the
thermal boundary layer thickness increases with increase in the
value of a.. for positive value of a,heat transfer decreases.
which indicates that, the flow of heat transfer is directed from
the wall to the ambient fluid whereas the rate of heat transfer in
the boundary layer increases near the wall.
Fig. 6 depicts the temperature profile in the fluid for various
values of
2
Re
Ha
,for a = -2 and Gr = 0, 0.5. It is noticed that an
increase in the strength of magnetic field i.e Lorentz force
leads to an increase in the temperature far away from the wall,
within the thermal boundary layer but the effect of magnetic
field near the wall is to decrease the temperature in the absence
of Grash of Number. When the magnetic field increases, the
thermal boundary layer thickness increases.
fig7,and it is noticed that increase in Grash of number ,increase
in temperature up to certain value of n and suddenly decreases
and decays asymptotically to zero. Further it is observed that
this increase in temperature is due to the temperature
difference between stretched wall and the surrounding fluid.
When Grash of number leads to increases, the thermal
boundary layer thickness decreases
Fig. 8 Represents the temperature profile  () for various
values of X along  for different values of a = -1, -2 and also
Grash off number Gr = 1.0. It is noticed that the effect of
increasing X on () is more effective for a = -2 than
compared to the results obtained in the case when a = -1. It is
interesting to note the behaviour of X on  () , is that the
temperature overshoots near the wall for small value of X,for a
= -2, whereas the overshoot diminishes when a is enhanced to -
1 for all other values of X. It is also observed that the boundary
layer thickness decreases with an increase in X.
Fig. 9 Represents the variation of temperature profiles  ()
for various values of magnetic field parameter (Ha2/Re = 0, 6,
8) for two values of X. when X increases temperature
decreases all other fixed values of other involved parameters

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except when the value of parameter a = 5. It is also to be
noticed that thermal boundary layer thickness increases as X
decreases and the effect of magnetic field is to increase the
temperature for both valuesof X.This is due to the Lorentz
force the temperature increases.
Fig10 Represents the effect of Prandtl number Pr on
dimensionless heat transfer parameter . It is noticed from this
figure that as Prandtl number Pr increases,temperature profile
decreases. When Prandtl number Pr is small, heat diffuses
quickly compared to the velocity (momentum), especially for
liquid metals,(low Prandtl number) the thickness of the thermal
boundary layer is much bigger than the momentum boundary
layer. Fluids with lower Prandtl number have higher thermal
conductivities where.Hence the rate of cooling in conducting
flows increases due to the Prandtl number.
Fig 11 Represents the effect of porous parameter N1 over
velocity profile.Porous parameter increases ,velocity
decreases.Due to this,the velocity decreases in the boundary
layer.
Fig12 Represents the effect of inertia coefficient N2 in the
velocity profile.From this we conclude that due to the N2,the
thickness of momentum of boundary layer decreases.
Fig 13: Represents the effect of heat source/sink parameter 
.It is noticed that, when   0
, the temperature increases.
when, th
  0
temperature falls.
Fig14: depicts dimensionless temperature field for various
values of K,with fixed values of other involved parameters. It
is observed from the figure that ,K increases, the temperature
profiles and the thermal boundary layer thickness also increase.
Fig15: Effect of porous parameter N1 on a temperature profiles
and it is noticed that, temperature increases with the increase of
porous parameter, which offers resistance to the flow resulting
in the increase of temperature in the boundary layer.
Fig16: Effect of drag coefficient of porous medium N2.From
the figure it is noticed that the effect of drag coefficient is to
increase the temperature profile in the boundary layer. Which
implies boundary layer thickness also increases.
Table: Values of heat transfer coefficient ,  '(0) for various values of K and Ec with Pr=1.0 and all parameters taken as 0.0
K Ec=0.0 Ec=0.5 Ec=1.0
1.0 -1.641723 -0.6609 0.3198
2.0 -0.57579 -0.29001 -0.00423
3.0 -0.4714 -0.26390 -0.05638

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0 2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
1.0
Fig 2 Effect of magnetic field on velocity profiles with 
Ha2/Re=Z,a=W,L
Gr=2.0,pr=1.0,Ec=0.1,a=-1.5,X=1.5,=0.1
Ha2/Re=0,1,3,5,8
f'

0 2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Fig.3.Variation of velocity profiles with  for various values of a.
a=1,3,4,5,6,7
Gr=2.0,Pr=1.0,Ec=0.1,=0.1,Ha2/Re=0.5,X=1.5
f'


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0 2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
1.0
Fig.4.Variations of velocity profiles with  for different values of X.
Ec=0.1,=0.1,Ha2/Re=0.5,pr=1.0,Gr=2.0,a=2
X=0.1,1.0,2.0,4.0,6.0
f'

0 2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Fig.5.Temperature profiles vs. for various values of a.
a=-5,-4,-2,-1,0,1,2,5,0
Ec=.001,Gr=0.0,Ha2/Re=5.0,=0.01,X=0.5,Pr=1.0



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0 5 10 15 20
0.0
0.5
1.0
1.5
Fig.6.Temperature profiles vs. for various values of Ha2/Re and Gr.
Ha2/Re=0,3,8
Ec=0.001,=0.01,X=0.5,Pr=1.0,a=-2 ---Gr=0
___Gr=0.5


0 5 10 15 20
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
Fig7 temperature profile for various values of Gr
X=0.5,Pr=1.0,a=-2,Ha2/Re=3,Ec=0.001,=0.01
Gr=-1.0,-0.5,0.0,0.5,2.0,5.0



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0 1 2 3 4 5 6 7 8 9
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
Fig.8.Temperature profiles vs. for various values of a and X.
X=1.5,0.5,0.0
Pr=1.0,Gr=1.0,Ha2/Re=3,Ec=0.001,=0.01
---a=-1.0
___a-2.0


0 1 2 3 4 5
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Fig.9.Temperature profiles vs. for various values of Ha2/Re and X when a=5.
Ha2/Re=0,6,8
Ec=0.001,=0.01,Pr=1.0,Gr=1.0,a=5
-----X=0.1
_____X=0.7



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0 2 4 6 8 10 12 14 16 18
0.0
0.2
0.4
0.6
0.8
1.0
Fig10.Variation of temperature with  for different values of Pr
Pr=1.0,2.0,3.0,4.0
Gr=0.0,Ec=0.001,X=0.5,a=-1.5,Ha2/Re=5.0
N
1
=1.0,N
2
=1.5


E
G
H
I
0 1 2 3 4 5 6 7
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Fig.11.Temperature profile vs. for various values of N1.
N1=0.0,1.0,3.0,5.0,8.0
Gr=2.0,Pr=1.0,Ec=0.1,W=-1.5,
X=1.5,l=0.1,Z=1.0,N2=1.5
f'()


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0 1 2 3 4 5 6 7
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Fig.12.Temperature profile vs.  various values of N2.
N2=0.0,1.0,3.0,5.0,8.0
Gr=2.0,Pr=1.0,Ec=0.1,W=-1.5,
X=1.5.L=0.1,Z=1.0,N1=1.0
f'()

0 2 4 6 8 10 12 14 16 18
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Fig13 Variation of temperature with  for different values of 

Gr=0.0,Ec=0.001,X=0.5,a=-1.5,Ha
2
/Re=5.0,Pr=1.0
N
1
=1.0,N
2
=1.5


E
G
H
I

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0 2 4 6 8 10 12 14 16
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
N1=1.0,N2=0.5
K =1.0,2.0,3.0
Ec=0.001,Gr=0.0,Ha2/Re=5.0,=0.001,X=0.5,pr=1.0,a=0.0
()

E
G
H
Fig14: Effects of K on the temperature profiles ( ), Where K=
* 3
*
4 T
k k
 
Radiation number
0 2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
1.0
N1=0.0,1.0,3.0,5.0,8.0
Pr=1.0.Gr=0.0,Ec=0.001,
X=0.5,=0.01,a=0.0,Z=5.0,N2=0.5k=1.0,


E
G
H
I
J
Fig 15: Effect of N1 on the temperature profiles ( )

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0 2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
1.0
N2=0.0,1.0,3.0,5.0,8.0
Pr=1.0,Gr=0.0,Ec=0.001,=0.5,a=0.0
Z=5.0,k=1.0,N1=1.0
 

E
G
H
I
J
Fig 16: Effect of N2 on the temperature profiles ( )
5. CONCLUSIONS
Due to the presence of porous parameter,the thermal boundary
layer thickness increases.Effect of drag coefficient also
enhance the thermal boundary layer thickness.
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