# UNSTEADY MHD BOUNDARY LAYER FLOW OF AN INCOMPRESSIBLE MICROPOLAR by 7HC0YF

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```									UNSTEADY MHD BOUNDARY LAYER FLOW OF AN
INCOMPRESSIBLE MICROPOLAR FLUID OVER A
STRETCHING SHEET

K.Govardhan1                    N.KISHAN2
1Department      of mathematics, CVR College of Engineering, R.R.Dist,A.P.India.
2Department      of mathematics, Osmania University, Hyderabad 500007, A.P, India,

E-mail: govardhan_kmtm@yahoo.co.in

ABSTRACT:

Aim of the paper is to investigate the MHD effects on the unsteady boundary layer flow of an
incompressible micropolar fluid over a stretching sheet when the sheet is stretched in its own plane. The
stretching velocity is assumed to vary linearly with the distance along the sheet. Two equal and opposite
forces are impulsively applied along x  axis so that the sheet is stretched, keeping the origin fixed in a
micropolar fluid. The governing non-linear equations and their associated boundary conditions are first cast
into dimensionless form by a local non-similarity transformation. The resulting equations are solved
numerically using the Adams- Predictor Corrector method for the whole transient from the initial state to
final steady- state flow. Numerical results are obtained and a representative set is diaplaced graphically to
illustrate the influence of the various physical parameters on the velocity profiles, microrotation profiles as
well as the Skin friction coefficient for various values of the material parameter K. It is found that there is a
smooth transition from the small- time solution to the large- time solution. Results for the local skin friction
coefficient are presented in table as well as in graph.
Key words: Unsteady flow, micro polar fluid, stretching surface, skin friction, MHD.

NOMENCLATURE

u, v    velocity components                                           j     micro-inertia
x, y      cartesian co-ordinates                            N        micro rotation
         dynamic viscosity                                 K        matériel parameter
      density of the fluid                             B0        applied magnetic field
        Spin gradient                                     M          magnetic field parameter
k        vortex viscocity                                           electrical conductivity
n         constant
1. INTRODUCTION

The fluid dynamics over a stretching surface is important in     considerd various problems in micropolar fluids. Nazar et.al
extrusion process. The production of sheeting material arises    (2004) studied the stagnation point flow of a non- Newtonian
in a number of industrial manufacturing process and includes     micropolar fluids with zero vertical velocity at the surface or
both metal and polymer sheets. Examples are numerous and         heat generation. Rajeshwari and Nath (1992) studied
they include the cooling of an infinite metallic plate in a      unsteady flow over a stretching surface in a rotating fluid,
cooling bath, the boundary layer along material handling         Noor (1992) investigated Heat transfer from a stretching
conveyers, the aerodynamic extrusion of plastic sheets, the      sheet.
boundary layer along a liquid film in condensation process,        Guram and Smith (1980) investigated the stagnation flows
paper production, glass blowing, metal spinning, and drawing     of micropolar fluids with strong and weak interactions. They
plastic films, to name just a few. The quality of the final      obtained numerical results using a fourth order Runge –
product depends on the rate of heat transfer at the stretching   Kutta method. Gorla (1983) obtained numerical results by a
surface. A comprehensive review of micropolar fluids             Runge – Kutta method for the micropolar boundary layer
mechanically has been presented by Ariman et al (1973).          flow at a stagnation point on a moving wall. Heat transfer
Since the pioneering study by Crane (1970) who presented an      over a stretching surface with variable surface heat flux in
analytical solution for the steady two – dimensional             micropolar fluids and MHD stagnation point flow towards a
stretching of a surface in a quiescent fluid, many authors       stretching vertical sheet in a micropolar fluid is studied by
have considered various aspects of this problem and obtained     Ishak et al. (2008). Recently Nazar et al (2008) studied the
similar solutions. Some mathematical results were presented      unsteady boundary layer flow of an incompressible
by many authors, and a good number of references can be          micropolar fluid over a stretching sheet. They solved
found in the papers by Magyari and Keller (1999,2000).           numerically using Keller-box method.
Sriramulu et.al (2001) studied steady flow and heat transfer     Viscous dissipation effects were consider on mhd nonlinear
of a viscous incompressible fluid through porous medium          flow and heat transfer past a stretching porous surface
over a stretching sheet.                                         embeded in a porous medium under a transverse magnetic
On the other hand, it is well known that the theory of         field is studied Anjalidevi and ganga(2010). Sharma and
micropolar fluids has generated a lot of interest and many       singh(2009) is investigated the effect of variable thermal
flow problems have been studied. The theory of micropolar        conductivity and heat source/sink on flow of a viscous
fluids was originally developed by Eringen (1964,1966) and       incompressible electrically conducting fluid in the presence
has now been applied in the investigation of various fluids.     of uniform transverse magnetic fluid and variable free stream
The theory takes into account the microscopic effects arising    near a stagnation point on a non conducting stretching sheet.
from the local structure and micro-motions of the fluid                   The purpose of the present paper is to study the
elements and provides the basis for a mathematical model for     magneto hydrodynamic effects on the unsteady boundary
non-newtonian fluids which can be used to analysis the           layer flow of an incompressible micropolar fluid over a
behavoiur of exotic lubricants, polymers, liquid crystals,       stretching sheet when the sheet is stretched in its own plane.
animal bloods and colloidal or suspension solutions, etc.        A numerical solution is obtained for the governing
Since introduced by Eringen many researchers have                momentum using the Adams predictor-corrector method.
j is microinertia per unit mass,  is spin gradient viscosity
and k is vortex viscosity.             Further, n is a constant and
2.MATHEMATICAL ANALYSIS                                            0  n  1. The case n  0 , which indicates N=0 at the
Consider the flow of an incompressible micropolar      wall represents concentrated particle flows in which the
microelements close to the wall surface are unable to rotate.
fluid in the region   y  0 driven by a plane surface located at
This case is also known as the strong concentration of
y  0 with a fixed end at x  0 . It is assumed that the
microelements.Thecase           n  1/ 2 indicates the vanishing of
surface is stretched in the x  direction such that the
anti – symmetric part of the stress tensor and denotes weak
x  component of the velocity varies linearly along it,
concentration of microelements. The case             n  1 is used for
i.e. u w ( x)  cx , where c is an arbitrary constant and
the modeling of turbulent boundary layer flows. We shall
c  0 . The simplified two - dimensional equations                 consider here both cases of       n  0 and n  1/ 2 .
governing the flow are the equations of the continuity,                  Introducing the new variables as
momentum equations under the influence of externally
  cv 1 / 2  1 / 2 xf  , , N  c / v 1 / 2  1 / 2 cxg  , ,
  c / v 1 / 2 1 / 2 y,   1  e  ,  ct ,
imposed transverse magnetic field in the boundary layer
(5)
steady laminar and incompressible micropolar fluids are;
Where  is the stream function defined in the usual way as

u v                                                                               
    0 ,(1)                                                     u       and v      , and identically satisfy (1).
x y                                                                    y           x
Substituting variables (5) in to equations (2) and (3) gives
u   u  u    k   2u k N  B 20
u v                          u ,(2)                                                  
t   x  y    y 2  y                                       1  K  f   1   
2

f    f f   f 2  Mf '    
(6)
 N    N    N    2N            u                                            f 
 j
 t u    v    
         k  2 N   ,(3)
                                  Kg    1   
       x    y    y 2
      y 
                                           

 K                     1       
Subject to the initial and boundary conditions                       1  2  g   1     2 g  2 g      fg   f  g 
                                    
g        (7)
t  0 : u  v  N  0 , for any x, y ,                                                   K  2 g  f     1   

u
t  0 : v  0, u  u w ( x)  cx, N  n        ,     at y  0 ,
K=
k
is the material parameter. Here  and j are
y                    Where

(4)
         k               K
u  0 , , as y   , N  0                                         assumed to be given by             j   1   j
    2          2
Where u and v are the velocity components along the
v
x  and y  axes, respectively, t is time, N is the                and   j  , respectively. The boundary conditions equation
c
microrotation or angular velocity whose direction of rotation      (4) becomes
is in the   xy  plane,  is dynamic viscosity,  is density,
f  ,0  0, f  ,0  1, g  ,0  n f   ,0 ,                   K
1   g   f g   f  g  K 2 g  f   0 (16)
   2
f  ,    0, g  ,    0.                               (8)
Subject to the boundary conditions (14).
The physical quantity of interest in this problem is the

skin friction coefficient C f , which is defined as
3.METHODOFSOLUTION
To solve the equations (6) and (7), we have convert
into a system of five first order equations, we have at
   ,
w
Cf                   ,                                       (9)         y1       f , y 4  g , y1  y 2 , y 2  y3
           
 uw 2 / 2
                                                   
 w is the skin friction, given by                                            1    y2      y2    1    2 y3  
Where                                                                            y3                                                       
                       
 y1 y3  y 2 2  M * y2  ky5
                                                    

              u                                                        / 1  k 
 w     k   kN                       .                 (10)
        y                   y 0                               ,

y 4  y 5 and

Using variables (5) in (9) and (10), we obtain                                            y4      y  5             1        
 1                            1     y4  y5  

y5                                              2        2 
                                     y1 y5  y2 y4    ,
C f Re x1/2   1/2 1  1  n  K  f   ,0  .
                                       (11)                                                                       
 k
Further, we can obtain some particular cases of this problem.                                                          / 1  
 2
A. Early Unsteady Flow                                              Early unsteady flow is obtained by solving these equations
For early unsteady flow 0    1, we have   0 , so                with   0 . For   0 , the above equations reflect a fully
implicit scheme with respect to     . In both cases, assuming
(6) and (7) reduce in the leading order approximation to
y 3  ,0    and y 5  ,0    , the above system is
1  K  f       f   Kg   0,                        (12)       solved up to  m ax    .
2
To solve  and  by Newton –Raphson method. We need
   K            1                                                         y 2 y 4 y 2     y 4
1   g   g   g  0,                                    (13)               ,    ,     and      at   m ax , these quantities are
   2       2     2                                                                      
obtained by solutions
and the boundary conditions (8) become                                                                           
y1  y 2 , y 2  y3 ,

f  0   0, f   0   1, g  0   n f   0  ,
           y2                        
 1                    1    y3 
(14)
f      0, g     0.                                                  
y3                                    2         / 1  k 
   y1Y3  Y1 y3  2 y2Y2  M * y2   ky5 
B. Final steady- state Flow                                                                                                    
For this case,          1 and     (6) and (7) take the following
forms:                                                                         
y 4  y5                    and
1 K  f   f f   f   Mf   Kg  0 (15)
2
           y4                1                       state flow   1 is represented in the figures 5&6 for the
 1                  1     y4  y5     k 
 
y5                                    2    2   / 1          cases n  0 and n  1/ 2 , respectively. These figures show
  y Y  Y y  y Y  Y y                         2        that the velocity profiles corresponding to increasing of
                                                 
  0    1
1 5    1 5     2 1    2 1
approach     the   final    steady    profile
Once                                                                corresponding to   1 . It has seen that there is a smooth
with   y1  0  y2  0  0, y3  0  0, y4  0  n ,           transition from small time solution       0    to large time
y 5 0  0 and another time with                                       solution   1 .
The effect of magnetic parameter M on velocity
y1 0  y 2 0  0, y 2 0  0, y 3 0  0, y 4 1  0, y 5 0  distribution f ( ) is shown in figure 7. The magnetic field
1.
parameter m, effect is shown in fig 7 for the velocity
distribution f ( ) of final steady flow   1 with K  1
This procedure converging in about three iterations
giving correct values of  and  . The system of Ordinary               and n  0 . The velocity distribution of final steady state
differential equation is solved by Adams predictor- corrector           flow (  1) for various M values with K  1, n  0 is
methods of fourth order. Accuracy is ensured by solving with            shown in figure 7. It is obvious that existence of Magnetic
different  , m ax ,  .                                             field M decelaretes the velocity profiles.

4.DISSCUSSION OF THE                                                 Figures.8-13 represents microrotation distribution for
various values of K , n for steady state and unsteady state
RESULTS                                                             flow. The microrotation distribution of final steady state flow
The transformed equations (6) and (7) satisfying the
  1 is increases with the increase of the material
boundary conditions (8) were solved numerically using the
parameter   K is observed from Figure.8 when n  0 .The
Adams predictor-corrector method for several values of the          microrotation distribution of final steady state flow  1
material parameter K . Numerical results for Skin friction          with n  1/ 2 is shown in fig.9, from which the
coefficients, the velocity distribution and microrotation           microrotation decreases as K increases in the vicinity of the
plate where as it increases as one moves away from it. Fig.
distribution are shown graphically.                                 10 represents the microrotation distribution of fully
To validate our method we have compared the Skin friction          developed unsteady flow for n  0 and K  1
coefficients   C f Re1/ 2 x values with Rosilinda nazar (2008) is   for 0    1 . It is noticed that the microrotation distribution
shown in Table, there is very good agreement between the            as parabolic distribution and increases with the increase of  .
results when we solved fully unsteady boundary layer                Fig. 11 shows that microrotation distribution of early
equations and final steady state equations.                         unsteady flow       0       1 for various K values
Though computations have been carried out for various              with n  1/ 2 . The microrotation distribution decreases as
values of the n and the material parameter K are presented.         K increases near the plate but reverse phenomena is
The velocity distribution of initial flow (   0 ) and             observed as one moves away from the plate. The
microrotation distribution of fully developed unsteady flow
unsteady flow (0    1) for various values of         K with      when K  1 and n  1/ 2 and final steady state flow
n  0 and n  1/ 2 is shown graphically in figures 1& 2               1 is shown in fig.12. The microrotation distribution
respectively. From both the figures it is observed that the
increases near the plate while, the reverse happens far away
velocity boundary layer thickness increases with the
the plate with the increase of  is observed. The magnetic
increasing values of K , for both the cases n  0
and n  1/ 2 . The figures 3&4 represent for final steady           field effect on the microrotation distribution is ploted in
state flow   1 for the cases n  0 and n  1/ 2 ,
figure 13. It can be seen from the figure that the magnetic
field effect occelerates the microrotation distribution near the
respectively. It is observed from the figures that the velocity     plate, where as it decelerates the microrotation distribution
increases with the increase of K . The velocity distribution of     far away from the plate.The magnetic field effect is more on
fully developed unsteady flow 0    1 and final steady          microrotation distribution when far away from the plate.
C f Re x with  for various                impulsive motion, the skin friction coefficients as large
1/ 2
The Skin friction coefficient
magnitude (absolute value) for small time (   0 or
values of K is drawn when n  0 in fig.14 and n  1/ 2
in     fig.15.   It   can   be   seen   that     the   values   of
C f Re1/ 2 x decreases as K increases.

  0)    after the start of the motion, and decreases
CONCLUSION.                                   monotonically and reaches the steady state value at     1
It is clear from the figs that the microrotation effects are           .
more pronounced for n  1/ 2 when compared to those
of n  0 . The microrotation profile for n  0 is different as
compared to n  1/ 2 it has a parabolic distribution                 The magnetic field effects is to decelerates the velocity
distribution f ( ) . The microrotation distribution of final
when n  0 , where as it has continuously decreasing when
n  1/ 2 .                                                           steady flow   1 is to increase near the plate, where as it
decreases far away from the plate with the effect of magnetic
The values of the Skin friction coefficients for the final      field is observed.
steady flow are shown in Table1. It is noticed that due to

TABLE.1 VALUES OF THE SKIN FRICTION COEFFICIENT C f Re1/ 2 x                                     FOR
VARIOUS VALUES OF K AND n WHEN   1 .

K /n                               0                                       1/2
0                               -1.0043                                  -1.0043
1                               -1.3952                                  -1.24005
2                               -1.6635                                  -1.4532
4                               -2.0092                                  -1.8105
ACKNOWLEDGEMENT

I thank you for referees, their valuable suggestions for making my paper more lucid
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1

0.9

0.8

0.7

0.6

0.5
K= 0, 1, 2, 3
0.4
f       0.3

0.2

0.1

0
0       1       2       3   4       5               6        7       8       9    10


Fig.1. Velocity distribution of initial flow (   0) and early unsteady flow ( 0    1) for various K with n  0

1

0.9

0.8

0.7

0.6

K= 0, 1, 2, 4
f  
0.5

0.4

0.3

0.2

0.1

0
0       1       2       3           5               6        7       8       9       10


Fig.2.Velocity distribution of initial flow (   0) and early unsteady flow ( 0    1) for various K
with n  1/ 2
1

0.9

0.8

0.7

0.6

0.5
K= 0, 1, 2, 4
0.4

f      0.3

0.2

0.1

0
0       1       2       3       4       5          6       7       8       9   10


Fig.3. Velocity distribution of final steady- state flow (   1) for various K with n  0

1

0.9

0.8

0.7

0.6

0.5                                                 K= 0, 1, 2, 3
f      0.4

0.3

0.2

0.1

0
0       1       2       3       4          5       6       7       8       9    10


Fig.4. Velocity distribution of final steady-state flow (   1) for various K with n  1/ 2
1
— Unsteady-state flow 0    1
--- Final steady- state flow   1
0.9

0.8

0.7

0.6                                                          0,0.2,0.4,0.6,0.8,1
f  ,     0.5

0.4

0.3

0.2

0.1

0
0       1       2       3       4         5           6       7       8        9   10


Fig.5. Velocity distribution of fully developed unsteady flow for K=1 when n  0

1

0.9                                               — Unsteady-state flow 0    1
0.8                                               ….. Final steady – state flow   1
0.7

0.6                                                 0, 0.2, 0.4, 0.6, 0.8,1
f  ,  0.5
0.4

0.3

0.2

0.1

0
0       1       2       3       4   5        6        7       8       9       10


Fig.6. Velocity distribution of fully developed unsteady flow for K=1 when n  1/ 2
1.2

1

0.8
f ( )                                                          M  0,5,10
0.6

0.4

0.2

0
0           2           4         6             8           10        12

Fig.7.Velocity distribution of final steady-state flow (   1 ) for various M with n  0 and K=1

0.14

0.12
K=1, 2, 3
0.1

0.08

0.06

g  
0.04

0.02

0
0       1   2       3       4   5      6        7        8   9        10


Fig.8.Micro rotation distribution of final steady – state flow (   1) for various K when n  0
0.45

0.4

0.35

0.3

0.25

0.2
K=1, 2, 3
g               0.15

0.1

0.05

0
0       1       2   3       4       5        6        7         8         9      10



Fig.9. Micro rotation distribution of final steady- state flow (   1) for various K when n  1/ 2

0.1
Unsteady state flow ( 0    1)
0.09
Final steady-state flow (   1)
0.08

0.07                                                                       0.1, 0.2, 0.4, 0.6, 0.8,1
0.06

g  ,    0.05

0.04

0.03

0.02

0.01

0
0           1       2       3       4           5         6         7         8         9         10


Fig.10.Micro rotation distribution of fully developed unsteady flow for n  0 and K=1
0.3

0.25

0.2

g        0.15                                                        K=1, 2, 4
0.1

0.05

0
0       1       2   3
   4        5           6       7       8       9          10

Fig.11. Micro rotation distribution of early unsteady flow ( 0    1) for various K with n  1/ 2

— Unsteady- state flow 0    1
0.45

0.4                                          ….. Final steady-state flow   1

0.35

0.3

0.25

g  ,                                                      0,0.2,0.4,0.6,0.8,1
0.2

0.15

0.1

0.05

0
0       1   2   3       4       5        6       7       8       9         10


Fig.12.Microrotation distribution of fully developed unsteady flow for n  1/ 2 and K  1
0.09

0.08                                                                     M  0,5,10

0.07

0.06

g ( )       0.05

0.04

0.03

0.02

0.01

0
0           2             4                 6         8             10           12


Fig.13. Micro rotation distribution of final steady-state flow (   1 ) for various M when K=1, n  0 .


0     0.2         0.4           0.6       0.8           1
1

-1

-3

C f Re1/ 2 x         -5                  K= 0, 1, 2, 4                        Final steady- state flow

-7

-9

-11

Fig.14. Variation with  of the skin friction coefficient for various K with n  0

0   0.2         0.4          0.6           0.8           1
1

-1

-3
1/ 2
C f Re      x
-5                  K= 0, 1, 2, 4
Final steady- state flow
-7

-9

-11

Fig.15. Variation with  of the skin friction coefficient for various K with n  1/ 2

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