International Journal of Engineering Trends and Technology (IJETT) – Volume 9 Number 8 - Mar 2014
Effect of Radiative Heat Flux of a Magneto hydrodynamics Micropolar Fluid Flow
towards a Stretching or Shrinking Vertical Surface in Presence of a Heat Source or
Sink
A.K.Maiti
Assistant professor, Department of Mathematics,
Shyampur Siddheswari Mahavidyalaya, Ajodhya, Howrah-711312
Abstract—
A
steady
two-dimensional
incompressible
magnetohydrodynamics (MHD) micropolar fluid flow towards a
stretching or shrinking vertical sheet under suction or blowing
with prescribed surface heat flux is studied in this paper. The
transport equations employed in the analysis include the effect of
radiative heat flux under mixed convection in presence of heat
generation or absorption. Similarity transformation is used to
convert the governing non-linear boundary-layer equations to
coupled higher order nonlinear ordinary differential equation.
These transformed differential equations are solved numerically
by a finite difference scheme, known as Keller-box method.
Numerical results are obtained for the velocity, microrotation
and temperature distributions, as well as the skin friction
coefficient and local Nesselt number for varius parameters and
then these are shown graphically. The present results are
compared with available results in literature and found a good
agreement with them.
Keywords— Heat generation/absorption, Keller-box method,
micropolar fluid, Radiative flow, similarity transformation.
Subject classification: 76R10, 76D10
I. INTRODUCTION
The theory of micro rotation fluids, first studied by
Eringen[1], displays the effects of local rotary inertia and
couple stresses, can explain the flow behaviour due to the
microscopic effects arising from the local structure and micro
motions of the fluid elements in which the classical
Newtonian fluids theory is inadequate. These fluids contain
dilute suspensions of rigid micro molecules with individual
motions which support stress and body moments and are
influenced by spin-inertia. The theory of micropolar fluids can
be used to analyse the behaviours of exotic lubricants
(Khonsari[2]), polymetic fluids (Hadimoto[3]), liquid crystals
(Lee et al. [4]), paints, animal blood (Ariman et al. [5]),
colloidal suspensions, ferroliquids etc.
In non-Newtonian fluids, the stagnation point flow of a
micropolar fluid towards a stretching sheet was studied by
Nazar et al. [6]. Similarly Ishak et al.[7] investigated
stagnation-point flow over a shrinking sheet in a micropolar
fluid. Their results indicated that the solutions are different
from a stretching sheet, and it was found that the solutions for
a shrinking sheet are not unique. Laminar mixed convection in
two-dimensional stagnation flows around heated surfaces in
ISSN: 2231-5381
the case of arbitrary surface temperature and heat flux
variations was examined by Ramachandran et al. [8]. They
established a reverse flow developed in the buoyancy
opposing flow region and dual solutions are found to exist for
a certain range of the buoyancy parameter. Heat transfer
effects on MHD viscous flow over a stretching sheet with
prescribed surface heat flux is studied by Adhikari and Sanyal
[9]. Bachok and Ishak[10] studied MHD stagnation-point flow
of a micropolar fluid with prescribed wall heat flux towards a
vertical plate.
The present paper deals with a two-dimensional steady
MHD mixed convection stagnation point flow of an
incompressible micropolar fluid towards a stretching vertical
surface with prescribed surface heat flux under uniform
transverse magnetic field. The transport equations employed
in the analysis include the effect of radiative heat flux under
mixed convection in presence of heat generation or absorption.
2. MATHEMATICAL FORMULATION
Consider a steady, two-dimensional flow of an
incompressible electrically conducting micropolar fluid
towards a stagnation point past a vertical plate with prescribed
surface heat flux. The frame of reference(x,y) is chosen such
that the x-axis is along the direction of the surface and the yaxis is normal to the surface. It is assumed that the velocity of
the flow external to the boundary layer U( = ue(x) = ax) and
the surface heat flux qw(x) (=bx), temperature Tw(x) of the
plate are proportional to the distance x from the stagnation
point, where a, b are constants. A uniform magnetic field of
strength B0 is assumed to be applied in the positive y-direction,
normal to the vertical plate. The assisting flow situation
occurs if the upper half of the flat surface is heated while the
lower half of the flat surface is cooled. In this case the flow
near the heated flat surface tends to move upward and the
flow near the cooled flat surface tends to move downward. So
this behaviour acts to assists the flow field. The opposing flow
situation occurs if the upper half of the flat surface is cooled
while the lower half of the flat surface is heated.
The magnetic Reynolds number of the flow is taken to be
small enough so that the induced magnetic field is negligible.
Under the Boussinesq and the boundary layer approximation
the governing equations are given by
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International Journal of Engineering Trends and Technology (IJETT) – Volume 9 Number 8 - Mar 2014
u v

 0,
...(1)
x y
u
u
dU    k   2 u k N
u
v
U




x
y
dx    y 2  y
Bo2
U  u   g T  T ,
...(2)

 N

N 
2N
u 
  
j  u
v
 k  2 N  , ...(3)
2
y 
y 
y
 x

T
T
 2T
1 q r Q0
u
v
 2 

T  T ,
x
y
c p y c p
y
k

 K
      j   1   j,
2
2


k
is the micropolar or material parameter,

K  0 for micropolar fluid and K  0 for the classical
where
K
Newtonian fluid. This assumption is invoked to allow the field
of equations that predicts the correct behaviour in the limiting
case when the microstructure effects become negligible and
the total spin N reduces to the angular velocity (Ahmadi [11],
Yucel[12]).
By using the Rosseland approximation the radiative heat flux
q r in y-direction is given by( Brewster [13])
qr  
...(4)
subject to the boundary conditions
at y  0 : u  u w  x   cx,
v  v w  x ,
q
u
T
,
  w , ...(5)
y
y
k
at y   : u  U  u e  x   ax,
N  0,
T  T ,
...(6)
where u and v are the velocity components along the x
and y  axis respectively, u w  x  the wall shrinking or
stretching velocity ( c  0 for stretching, c  0 for shrinking
and c  0 for static wall ), v w  x  the wall mass flux
velocity, N is the microrotation or angular velocity whose
direction of rotation is in the xy plane,  is the dynamic
viscosity,  is the density of the fluid,  is the electrical
conductivity, j is the micro-inertia per unit mass, i.e., microinertia density,  is the spin gradient viscosity, k is the
vortex viscosity or micro-rotation viscosity, T is the fluid
temperature in the boundary layer, T is the uniform ambient
temperature,  is the thermal expansion coefficient,  is the
thermal diffusivity, k is he thermal conductivity, q w is the
N  n

1
3
1

wall heat flux, Q0 Js m K
is the dimensional heat
generation or absorption coefficient. Note that n is a constant
such that 0  n  1 . When n  0 then N  0 at the wall
represents concentrated particle flows in which the
microelements close to the wall surface are unable to rotate.
This case is also known as the strong concentration of
microelements. When n=1/2 , we have the vanishing of antisymmetric part of the stress tensor and denotes weak
concentration of micro elements, the case n  1 is used for
the modelling of turbulent boundary layer flows. We shall
consider here both cases of n=0 and n=1/2.
Assume
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4 s T 4
,
3k e y
...(7)
where  s is the Stefan-Bolzmann constant and k e the mean
absorption coefficient. It should be noted that by using
Rosseland approximation, the present study is limited to
optically thick fluids.
Expanding
T 4 in a Taylor series about T as:
2
T 4  T4  4T3  T  T   6T 2  T  T   ...,
...(8)
and then neglecting higher order terms beyond the first degree
in T  T  , we get
T 4  4T3T  3T4
...(9)
In view of the equation (7) and (8), the equation (4) becomes
u
T
T
 2T 16 sT3  2T
v
 2 
x
y
3k e c p y 2
y
Q0
T  T ,
c p
Introduce a Stream function  as follows


u
, v
,
y
x

...(10)
...(11)
The momentum, angular momentum and energy equations can
be transformed into the corresponding ordinary differential
equations by the following transformation:

a
y,
v
f   
θ η  

, p   
x av
k T  T  a
,
qw
v
N
a
ax
v
,
...(12)
Where  the independent dimensionless similarity variable,
thus u and v are given by
u  axf  ,
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v   av f   .
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International Journal of Engineering Trends and Technology (IJETT) – Volume 9 Number 8 - Mar 2014
Substituting variables (12) into equations (2) to (4), we get the
following ordinary differential equations:
1  K  f   f f   1  f 2  Kp  M 1  f   λθ  0
...(13)
 K
1   p  fp  pf   K 2 p  f   0 ,
2

1
4 
1 
θ   fθ   θf   H c θ  0 ,
Pr  3F 
...(14)
...(15)
Subject to the boundary conditions (5) and (6) which
become
f 0  s ,
f 0  e ,
w
xqw
...(18)
, Nu x 
2
U / 2
k Tw  T 
where the wall shear stress  and the heat flux  are
Cf 
given by


 T 
u
 w    k   kN  , q w  k   , ...(19)
y

 y 0
 y  y 0
with k being the thermal conductivity. Using the similarity
variables (12), we get
1
Nu
1
 1  n K  
,
C f Re1x/ 2  1 
f 0, 1/x2 

2
2 
Re x
θ 0

...(20)
p0  nf 0,
 (0)  1 ,
...(16)
II. NUMERICAL RESULT AND DISCUSSION
f    1, p    0 ,     0 as    ....(17)
With the help of the implicit finite-difference scheme
Here f   , p   and    give (dimensionless) the known as the Keller-box method [15, 16] the equations (13) –
velocity, the angular velocity and temperature respectively. In
the above equations, primes denote differentiation with
v
the characteristic length (Rees and
a
B02
v
Bassom [14]), Pr 
the Prandtl number, M 
the

a
c
magnetic parameter, e 
the velocity ratio parameter,
a
v x 
S w
the constant mass flux with s  0 for suction
av
Grx
and s  0 for injection,  
the buoyancy or mixed
Re 5x / 2
respect to
;
j 
convection parameter, Grx 
Ux
is the local Raynolds number,
v
k k
xQ0
F  e 3 the radiation parameter and H c 
is
c pU
4 sT
the heat source or sink parameter. Here  is a constant and
the negative and positive values of  correspond to the
opposing and assisting flows respectively. When,   0 i.e.
when Tw  T is for pure forced convection flow.
Ramchandran [8] considered the present problem with
M  0 and K  0 .
The skin friction coefficient C f and the local Nusselt
ISSN: 2231-5381
dimensionless boundary layer   had to be adjusted for
different values of parameters to maintain accuracy within the
interval 0      . We execute the programme in
MATLAB upto the desired level of accuracy. The validity of
the numerical results has been compared with the results of
Bachok and Ishak [10] and they are found to be in a very good
agreement,
as
when
,
 1
K  0 , n  0.5 , M  0 , e  0 , F  7000 , H c  0 ,
s  0,   0.02 we get
f 0 = 1.8339,
1
 0.7776 for Pr  0.7 . The choice of
 0 
 max  15 ensured that all numerical solution approached the
g Tw  T x 3
the local far field asymptotic values correctly. This is an important
v2
point that is often overlooked in the publications on boundary
Grashoff number, Re x 
number Nu x are defined as
(15) subject to the boundary conditions (16) – (17) are solved
numerically. The step size  of  and the edge of the
layer flows (Pantokratotars [17]).
The variation of skin friction coefficient f 0  and the
local Nusselt number
1
with λ for different values of the
θ 0
suction parameter s are given by figures 1 and 2 respectively.
The dual solutions were obtained by setting two different
values of   , which produce two different velocity and
temperature profiles both satisfy the boundary conditions. It is
seen that for the opposing flow   0 dual solutions are
found to exist for the values of s considered. For a particular
value of s the solution is present up to a critical value of  ,
say  c , outside which the boundary layer separates from the
surface and the solution based upon the boundary-layer
approximations are not feasible. It is clear from the figures 1
and 2 that larger values of s enhance the range of λ for which
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Page 391
International Journal of Engineering Trends and Technology (IJETT) – Volume 9 Number 8 - Mar 2014
the solution exists. In this study the critical values of λ
(i.e., c ) are given by this table.
the opposing flow (first solution) but increase for the opposing
flow (second solution).
REFERENCES
s
c
-1
-0.14
0
-0.18
1
-0.22
[1]
A. C. Eringen, Theory of Micropolar Fluids, J. Math. Mech. 16 (1966)
1-18.
M. M. Khonsari, On the self-excited whirl orbits of a journal in a
sleave bearing lubricated with micropolar fluids, Acta Mech. 81 (1990)
235-244.
B. Hadimoto and T. Tokioka, Two-dimensional shear flows of linear
micropolar fluids, Int. J. Eng. Sci., 7(1969)515-522.
J. D. Lee and A. C. Eringen Boundary effects of orientation of numatic
liquid crystals, J. Chem. Phys., 55(1971) 4509-4512.
T. Ariman, M.A.
Turk and N.D. Sylvester, Application of
Microcontinuum fluid mechanics, Int. J. Eng. Sci., 12 (1974) 273-293.
R. Nazar, N. Amine, D. Filip and I. Pop, Stagnation point flow of a
micropolar fluid towards a stretching sheet, Int. J. Non-Linear Mech.,
39(2004) 1227-1235.
A. Ishak, Y.Y. Lok and I. Pop, Stagnation point flow over a shrinking
sheet in a micropolar fluid, Chem. Eng. Commun., 197(2010)14171427.
N. Ramachandram, T.S. Chen and B.F. Armaly, Mixed convection in
stagnation flows adjacent to a vertical surface, ASME Jr. Heat
Transfer 110(1988) 373-377
A. Adhikari and D.C. Sanyal, Heat transfer on MHD viscous flow over
a stretching sheet with prescribed heat flux, Bull. Int. Math. Virtual
Instt. 3 (2013) 35-47.
N. Bachok and A. Ishak, MHD stagnationpoint flow of a Micropolar
Fluid with Prescribed Wall Heat Flux, European J. of Sci. Research,
35(3) (2009) 436-443.
G. Ahmadi, Self-similar solution of incompressible micropolar
boundary layer flow over a semi-infinite flat plate, Int. J. Eng. Sci., 14
(1976) 639-646.
Yucel A., Mixed convection in micropolar fluid flow over a horizontal
plate with surface mass transfer, Int. J. Eng. Sci. 27 (1989) 1593-1602.
M.Q. Brewster, Thermal Radiative Transfer Properties, John Wiley
and Sons, Canada, 1992.
D.A.S. Rees and A.P. Basson, The Blasius boundary-layer flow of a
micropolar fluid, Int. J. Eng. Sci. 34 (1996) 113-124.
T. Cebeci and P. Bradshaw, Physical and Computational Aspects
Convective Heat Transfer, (1988) (Springer, New York).
T. Cebeci and J. Cousteix, Modeling and Computing of Boudary-Layer
Flows: Laminar, Turbulent and Transitional Boundary Layers in
Incompressible and Compressible Flows, (2005) (Springer).
A. Pantokratoras, Study of MHD boundary layer flow over a heated
stretching sheet with variable viscosity: a numerical reinvestigation,
Int. J. Heat Mass Transfer 51 (2008) 104-110.
[2]
Table 1
Hence the boundary-layer separation is delayed with
increase of s . So suction holds up the boundary layer
separation respectively compared to the no-suction
[3]
s  0 case. Figure 1 depicts that the value of f 0 
decreases as s increases, this suction shows drag reduction
[5]
compared to the no-suction case.
Figure 3 displays the dual solutions for the opposing flow
for different values of s where the first solutions are stable
with the most physically relevance while the second solutions
are not. The region of reversed flow exists for the case of the
second solutions from figure 3 and this would unacceptable as
possible asymptotic solution to which a fully forward flow
developing near the stagnation point could grow.
The velocity, angular velocity and temperature profiles for
both assisting   0  and opposing flow   0  are given
in the figures 3 to 5 for different values of the suction
parameter s . Here Pr  0.7 , n  0.5 , e  0.5 , M  0.5 ,
[4]
[6]
[7]
[8]
[9]
[10]
[11]
K  1.0 , H c  0.5 and F  0.05 .
[12]
Figure 3 depicts that the velocity profiles decrease for the
assisting flow and for opposing flow (second solution) but the
profiles increase for the opposing flow (first solution) with the
increase of s .
For the assisting and opposing flows (second solution)
angular velocity profiles increase near boundary but after a
certain point the profiles decrease with the increasing of s and
for the opposing flows (first solution) the profiles show the
reverse nature (fig. 4).
From the figure 5 it is clear that the temperature profiles
decrease with the increase of s for the assisting flow and for
[13]
[14]
[15]
[16]
[17]
0.2
0.8
0.6
P =0.7, M=0.5, F=0.05, K=1,
r
n=0.5,e=0.5, H =0.5, =0.02
0.18
s=1, 0, -1
c
0.4
0.14
0.2
c
for s=1
 (0)
//
f (0)
 = -0.22
0
s=1, 0, -1
0.16
 =-0.14 for s = -1
c
-0.2
0.12
0.1
0.08
-0.4
 = -0.18
c
0.06
for s=0
-0.6
0.04
-0.8
-1
-0.25
0.02
-0.25
-0.2
-0.15
-0.1
-0.05
0

Fig1: Skin friction coefficient
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-0.2
-0.15
-0.1
-0.05
0

Fig.2: Nusselt number
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International Journal of Engineering Trends and Technology (IJETT) – Volume 9 Number 8 - Mar 2014
1.6
s=-1,0,1 for the assisting flow
(=0.5)
1.4
1.2
1
f /()
0.8
s=-1, 0, 1 for the opposing flow 1st solution (=-0.1)
0.6
0.4
s=1,0,-1 for the opposing flow2nd solution
P =0.7, M=0.5, F=0.05, K=1,
r
(=-0.1)
0.2
0
n=0.5, e=0.5, H =0.5, =0.02
c
-0.2
0
2
4
6
8
10
12
14

Fig.3: Velocity distribution
0.08
s=-1,0,1 for the assisting flow (=0.5)
h()
0
s=-1,0,1 for the opposing flow 2nd solution (=-0.1)
s=1, 0, -1 for the opposing flow 2nd solution
(=-0.1)
-0.1
0
4
8
12

Fig.4: Angular velocity distribution
25
20
()
15
s=1,0,-1 for opposing flow 2nd solution
(=-0.1)
10
s=-1,0,1 for opposing flow 1st solution (=-0.1)
5
0
s=1,0,-1for assisting flow (=0.5)
-5
0
2
4
6
8
10
12
14

Fig.5: Temperature distribution
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