Oscillatory Flow of MHD Polar Fluid with Heat and Mass Transfer

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International Journal of Mechanical Engineering and Technology (IJMET)
Volume 7, Issue 2, March-April 2016, pp. 212–231, Article ID: IJMET_07_02_024
Available online at
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ISSN Print: 0976-6340 and ISSN Online: 0976-6359
© IAEME Publication
OSCILLATORY FLOW OF MHD POLAR
FLUID WITH HEAT AND MASS TRANSFER
PAST A VERTICAL MOVING POROUS
PLATE WITH INTERNAL HEAT
GENERATION
P.H.VEENA
Dept. of Mathematics, Smt. V.G. College for Women,
Gulbarga, Karnataka, INDIA
N. RAVEENDRA
Dept. of Mathematics, RajivGandi Institute of Technology,
Cholanagar, Bangaluru, Karnataka, INDIA
V. K.PRAVIN
Dept. of Mechanical Engineering,
P.D.A. College of Engineering Gulbarga, Karnataka, INDIA
ABSTRACT
The study of unsteady two-dimensional laminar boundary layer flow of a
viscous incompressible fluid (polar fluid) through porous medium past a semi-
infinite vertical porous stretching plate in the presence of transverse magnetic
field is investigated.
The sheet makes with a constant velocity in the longitudinal direction and
the free stream velocity follows an exponentially increasing or decreasing
small perturbation law. A uniform magnetic field acts perpendicularly to the
porous sheet which absorbs the polar fluid with a suction velocity varying with
time component. The effects of all parameters encountering in the problem are
investigated for velocity and temperature fields across the boundary layer.
The present results of velocity distribution of polar fluids are compared
with the corresponding flow problems for a Newtonian fluid. For a constant
stretching velocity with prescribed magnetic and permeability parameters,
Prandtl and Grashof numbers, viscous dissipation parameter, the effects of
increasing values of suction velocity parameter increases the surface skin
friction. It is also situated that the surface skin friction decreases with
increasing values of sheet moving velocity.

Oscillatory Flow of MHD Polar Fluid with Heat and Mass Transfer Past a Vertical Moving
Porous Plate with Internal Heat Generation
Cite this Article P.H.Veena, N. Raveendra and V. K.Pravin, Study of Process
Parameters of Gravity Die Casting Defects. International Journal of
Mechanical Engineering and Technology, 7(2), 2016, pp. 212–231.
http://www.iaeme.com/currentissue.asp?JType=IJMET&VType=7&IType=2
INTRODUCTION
Unsteady flows are of importance from the practical point of view and those that are
concerned with the effects of free stream oscillation are of physical significance from
the point of view of Industrial applications. After the initiation by Sakiadis [1],
analytical solutions to such problems of flow have been presented by a number of
authors.
Polar fluids are fluids with microstructure belonging to a class of fluids with
non-symmetrical stress tensor. Physically, they represent fluids consisting of
randomly oriented particles suspended in a viscous medium. The study of heat
transfer with boundary layer flow for an electrically conducting polar fluid past a
stretching porous sheet has attracted the interest of many investigators in view of its
applications in many engineering problems such as magneto hydrodynamic
generators, plasma studies, nuclear reactors, oil exploration, geothermal energy
extractions, the boundary layer control in the field of aerodynamics and these fluids
have key importance in polymer devolatasation, bubble columns, composite
processing etc. Sakiadis [2] attempted the first problem regarding boundary layer
viscous flow over a moving surface having constant velocity. Later this problem has
been studied extensively through various aspects. Very recent investigations relevant
to this problem are dealt by [3-7]. Wang [8] was the best researcher who discussed the
viscous flow due to an oscillatory stretching surface. Although Oscillatory stretching
sheet induces the present flow but we also have a free stream velocity Oscillating
about a constant mean oscillatory flow [9-11].
A great number of Darcian porous MHD studies have been carried out examining
the effects of magnetic field on hydrodynamic flow without heat transfer in various
configuration, eg in channels and past stretching plates and wedges, etc [12, 13].
Gribban [14] considered the MHD boundary layer flow over a semi-infinite plate
with an aligned magnetic field in the presence of pressure gradient and he has
obtained the solutions for large and small magnetic Prandtl numbers using the method
of matched asymptotic expansion. Takhar and Ram[15] studied the effects of Hall
currents on hydro magnetic free convection boundary layer flow via porous medium
past a plate. Thakar and Ram [15] also studied the MHD free convective heat transfer
of water at 40
C through a porous medium.
Soundalgekar [17] obtained approximate solutions for viscous two-dimensional
flow past an infinite vertical porous plate with constant suction velocity normal to the
plate. The difference between the temperature of the plate and the free stream is
moderately large causing the free convection currents Raptis [18] studied
mathematically the case of time-varying two-dimensional natural convective heat
transfer of an incompressible MHD viscous fluid Via a highly porous medium
bounded by an infinite vertical plate. Despite recent advances in non-Newtonian and
polar fluids, it is still of interest to develop stretching flows involving polar fluids. For
example, no investigation.
In this regard Abbas et al [11] made an attempt to get analytical solution for a
second grade fluid flow due to MHD and unsteady stretching surface by HAM
technique. Further unsteady free convection MHD flow with heat transfer in a porous

P.H.Veena, N. Raveendra And V. K.Pravin
medium has been studied extensively by several authors because of its wide spread
applications in a number of engineering and environmental applications [ref 19]- such
as fibred and granular insulations, and granular insulations, electronic system cooling,
Modern communication system, heat exchanger and geophysical studies etc.
Many works have already been done on transient free convection flow past a
vertical plate. Some of the pioneers Gephart [20]. Schetz et al [21], Gold stain and
Briggs [22], Soundalgekar [23] and Das et al [24].
However most of the previous works assume that the plate is at rest. Thus the
study reported presently here in considers the unsteady free convection flow of an
electrically conducting polar fluid in the presence of transverse magnetic field over a
semi-infinite moving porous plate with a constant velocity in the longitudinal
direction. we also consider the free stream to consist of a mean velocity and
temperature with a super imposed exponentially variation with time. Applying
perturbation technique, the solutions for velocity and temperature of the flow field are
obtained and the effects of the flow parameters are discussed with the help of graphs.
This problem is an extension of the work of Kim [16] to a porous medium with
viscous dissipation and internal heat generation/absorption in the energy transfer.
In general, the study of Darcian porous MHD is very complicated. It is necessary
to consider in detail the distribution of velocity and temperature across the boundary
layer in addition the surface skin friction. Thus the present work is an attempt made to
shed some light to these aspects.
MATHEMATICAL FORMULATION
We consider the unsteady flow of an two dimensional an incompressible laminar fluid
past a plate saturated in a porous medium and subjected to a transverse magnetic field
in the presence of a pressure gradient.
It is assumed that there is no applied voltage which implies the absence of an
electric field. The magnetic field and magnetic Reynolds number are of low
conduction and hence the induced magnetic field is negligible. Viscous and Darcy’s
resistance terms are consider with constant permeability of the porous medium.
Porous medium is considered as an assemblage of small spherical particles fixed in
space, following [31]. Due to semi infinite plane surface assumption, the flow
variables are further consider as the functions of y*
and t*
only.
Under the above assumptions the governing equations of continuity, momentum,
angular momentum and heat conservation equations can be considered in a Cartesian
frame of reference as follows
0*
*



y
v
(1)
  
















y
w
vuBu
k
TTg
y
u
vv
x
p
y
u
v
t
u
rr
*
*2
0
*
*2*
*2
*
*
*
*
*
*
*
2)(
1




(2)
2*
*2
*
*
*
*
*
*
y
w
y
w
v
t
w
j













 (3)
 













TTQ
y
T
y
T
v
t
T
2*
2
*
*
*
 (4)

2*
2
0*
*
*
y
C
D
y
C
v
t
C








(5)
Where u*
and v*
are the components of longitudinal and transverse velocities
along the respective dimensional distances along x*
and y*
directions.
[ρ is the density and υ is the kinematic viscosity, vr is the rotational viscosity, g is
the acceleration gravity, β is the coefficient of voluaicetric thermal expansion of the
fluid, k*
is the co-efficient of permeability of the porous medium, σ is the electrical
conductivity]
The heat due to viscous dissipation term is neglected because it is of the same
order of magnitude as the viscous dissipation term but heat due to internal
generation/absorption is considered for the study. It is assumed that the porous plate
moves with constant velocity up
*
and the temperature T, the concentration C and
suction velocity v*
vary exponentially with time t*
. D is the mass diffusion co-
efficient
Under the above considerations the appropriate boundary conditions for the
velocity and temperature and concentration fields are as follows
u*
= up
*
,  

 tn
ww eCCCC
*
(6a)
2*
*2
*
*
y
u
y
w





at y*
= 0
 
 

 tn
eUUu
*
10
 
 yaswCC 0; (6b)
Where n*
is constant scalar and U0 is the scale of free stream velocity.
Solution for continuity equation yields
 
 tn
eAVv
*
10 (7)
Where V0 the non zero positive constant is the scale of suction velocity normal to
the plate and is a function of t and A is a real positive constant, ε and εA are very
small and are less than unity.
Solution of momentum, heat and mas transfer equations;
Out side the boundary layer equation (2) gives
*
0
*
*
*
*
*
1





 UBU
kdt
dU
x
p



(8)
We now set the following non-dimensional variables





2
0
*
*
000
*
0
**
0
00
,;,,,;
t
tw
VU
w
U
u
U
U
U
U
y
y
VU
u
U
p
pp  


,
parametertyPermiabili
Vk
k
VCC
CC
TT
TT
ww
2
2
0
*
22
0
*
;,,


 










.

numberandtltheis
k
C
P
D
Sc
p
r Pr


 
2
0
2
0
2
2
0
,
v
B
Mj
v
j n



 
is magnetic parameter
 
2
00VU
TTg
G wT
c



is the Grashof number
 
2
00VU
CCg
G wc
r



is modified Grashof number
Further it is deduced that












 

2
1
1
2
jj
A
(9)
Where  is the spin gradient viscosity, which is the relationship between the co-
efficient of viscosity and micro-inertia, β is the non dimensional ratio of viscosity and
is defined as
β = A/µ (10)
in which A is the co-efficient of vertex viscosity.
In view of the equations (7) to (10) the governing equations (2) to (5) reduce to
the following non-dimensional form
     
y
w
uUMGG
y
u
dt
dU
y
u
eA
t
u
rc
nt













 211 2
2
(11)
  2
2
1
1
y
w
gdt
dU
y
w
eA
t
w
c
nt







 
(12)
  
 









Q
yPy
eA
t r
nt
2
2
1
1 (13)
Where 2
0
*
V
Q
Q

  is the internal heat generation
  2
2
1
1
y
C
Sy
C
eA
t
C
c
nt








(14)
Where   11
22),( 2



 



j
MkM n
The corresponding boundary conditions (5) and (6) are reduced to their
dimensionless form as ,
 ,0Uu nt
eA1 , 



y
w
eA nt
,1 2
2
y
u


at y = 0 (15)
  yasUu ,0,0,  (16)

Perturbation solution: In order to reduce the set of partial differential equations (2)
to (5) to a system of ordinary differential equations in non-dimensional form, we the
perturbation technique for linear temperature as
u = v0 (y) + ε ent
u1(y) + O (ε2
)+ ……. (17)
w = w0 (y) + ε ent
w1(y) + O (ε2
)+ ……. (18)
θ = θ 0 (y) + ε ent
θ 1(y) + O (ε2
)+ ……. (19)
φ = φ 0 (y) + ε ent
φ 1(y) + O (ε2
)+ ……. (20)
substituting (17) to (20) in equations (11) to (14)
and equating the periodic and non-periodic terms and neglecting the higher order
terms of O(ε2
), we get the following pairs of equations for u0, u1, w0, w1, θ 0 , θ 1 and φ
0, φ1
        00020200 21 WGrGckMnukMnuu  (21)
        111021211 21 WGrGcAunkMnunkMnuu  (22)
000   WW (23)
  0111 WAWnWW  (24)
0PrPr 000    (25)
    0111 PrPr An  (26)
000   ScSc  (27)
    0111 ScAnSc  (28)
The corresponding boundary conditions can be written as
01,1
,1,1,,,0,
11
00110010


yat
uWuWuUpu


(29)


yas
WWuu
0,0
,0,0,0,0,1,1
11
001010


(30)
Thus the solutions for all the equations of momentum, energy and mass
concentration from (21) to (28) satisfying the respective boundary conditions (29) and
(30) are found as
  yScyyyh
eAeAeAeAyu 
 43
Pr
210
2
1 (31)
  yyScyyhyhyhyhyh
eBeBeBeBeBeBeBeByu 
 87
Pr
6543211
54321
1
(32)
  y
eDyW 
 10 (33)
  yyh
eD
n
A
eDyW  
 121
1
(34)

  y
ey Pr
0

 (35)
 
 
 yyhyh
ee
n
A
ey Pr
1
44
Pr 




 (36)
  ySc
ey 
0 (37)
 
 
 yScyhyh
eeSc
n
A
ey 


 55
1

 (38)
Where
 





 


 n
h
4
11
2
1
 
  



 1411
12
1
2 Nh
 
   



 1411
12
1
3 nNh
 





 

Pr
4
11
2
Pr
4
n
h

,
 





 

Sc
nSc
h
4
11
2
5
and
A1 = Up – 1 – A2 – A3 – A4
    NScSc
Gr
A
N
Gc
A



 2322
1
,
PrPr1 
  124
1
2
D
N
A




    2
1
2
1
1
1
1
2
D
nNhh
h
B




  1
2
2 A
n
hA
B



      
   











1
1
2
1
1
2
1
2
31
2
1
1
3
2
31
2121
2
h
k
k
h
nNhh
nNhhh
h
B




     














nNhhn
A
GB c
 4
2
4
4
1
1Pr
1
     














nNhhn
ASc
GB r
 4
2
4
5
1
1
1
    Nn
GcA
B


PrPr1
1Pr
26
     NScScn
GrScA
B



 27
1
1

 
   nN
C
n
An
AA
B














2
1
2
3
8
1
2

 
 
    2
2
2
22
22
2
21 Pr1
21
11
AhhUp
hN
N
D 









 2
33
2
111
1
2
1
hbhbk
h
D 


 
 2
6
2
7
2
5
2
44
2
221
2
1 Pr ScBBBhBhBD
A
k 


 


 8765423 1 BBBBBBk 
With respect to equations (17), (18), (19) and (20) the solutions for stream wise
and angular velocities, temperature and concentration are obtained as follows
 
 yyScyyhyhyhyhyht
yScyyyh
eBeBeBeBeBeBeBeBe
eAeAeAeAtyu


 



87
Pr
654321
43
Pr
21
54321
2
1
1,
(39)







  yyhty
eD
A
eCeeCtyw 


 121
1
),( (40)
 







  yyhyhty
ee
A
eeety PrPr 44
Pr),(

 
(41)
 







  yScyhyhtScy
eeSc
A
eeety 55
),(

 
(42)
SKIN FRICTION
Skin friction at the wall of the plate is calculated by
000 




y
w
w
y
u
VeU

 (43)
     
 
 
8765544332211
32222
Pr
Pr1
BScBBhBhBhBhBhB
eAhAhhUp t


(45)
NUSSELT NUMBER
Heat transfer co-efficient interms of Nusselt number is given by












 TT
y
T
x
Nu
w
w
x
1
0
Re










 x
y
Nu
y
















 Pr1PrPr 4
2

  A
h
A
e t
(46)












y
e
y
t 10 

 

SHREWOOD NUMBER (Sh) Mass transfer co-efficient interms of Shrewood
number is given by
1
0


















 xy
w
eRSh
y
C
CC
y
C
xSh















 Sc
A
hSc
A
eSc t

 
15
2
(47)
0
10











 y
t
y
e
y


 
RESULTS AND DISCUSSION
The investigations of effect of magnetic field, porous media and suction velocity
varying exponentially with time about a constant mean (non-zero) on boundary layer
flow and heat transfer of an incompressible MHD polar fluid along a semi-infinite
vertical porous moving plate is studied. Numerous computations are carried out for
the velocity and temperature profiles for different numerical values of the flow and
encountered physical parameters in the study. In the present analysis y is
replaced by Ymax then velocity profile u approaches the appropriate free stream
velocity.
The distribution of velocity profiles across the boundary layer for a stationary
vertical porous plate in the presence of magnetic field and absence of magnetic field
are depicted in fig1 and 2.
In figure (3a) and (3b) the graphs of velocity and angular velocity profiles for
different values of viscosity ratio and fixed values of parameters ε, n, τ, A, Mn, k2, G,
Pr and Up.
From the figures we observe that velocity distribution is lower for β = 0. When β
takes the values greater than 0.5, the velocity distribution has a decelerating nature
near the plate. However the angular velocity profiles do not showing the consistency
with increase in viscosity ratio parameter β.
Figure 4a and 4b elucidate the behavior of variation is velocity and angular
velocity profiles for different values of plate moving velocity Up is the direction of
fluid flow. It is observed the figures that, as the plate moving velocity increase, the
peak velocity across the boundary layer decreases near the plate. However the angular
velocity distribution has an increment as an increment in plate moving velocity Up.
Figures 5a and 5b are drawn for velocity and angular velocity profiles for different
values of permeability parameter k2. It is obvious from the figures that the effect of
increasing values of permeability parameter results in a decreasing velocity
distribution and angular velocity distribution is also decreased near the plate as k2
increases.
Figure 6(a) and 6(b) are depicted to illustrate the effect of Grashof number on
velocity and angular velocity profiles respectively. From the figures it is observed that
increase in the values of G is to accelerate the velocity distribution across the
boundary layer but decelerates the angular velocity.
Figures 7 (a) and (b) shows the variation of velocity profiles against Y for
different values of Prandtl number Pr with constant values of all other parameters.

The result shows that increasing values of Prandtl number results in a decreasing
velocity and then approaches a constant value which is relevant to the free stream
velocity at the edge of the boundary layer the effect of Prandtl number decreases the
angular velocity distribution.
It has been observed that from figure 9 for a constant suction velocity parameter
A, the effect of increasing values of plate moving velocity Up results in a decreasing
skin friction on the plate. It is also evident from the figure that skin friction vanishes
near the plate velocity at Up = 1.7.
Figure 10 illustrate the graphs of surface heat transfer Nux versus suction velocity
parameter A for different values of Prandtl number. Pr. It is revealed from the figure
that as the magnitude of suction velocity A increases, the surface heat transfer from
the porous plate tends to decrease slightly.
CONCLUSIONS
We have investigated the study of an unsteady incompressible polar fluid past a semi-
infinite porous moving plate whose velocity is constant and increased in saturated
porous medium and subjected to a transverse magnetic field. The method of small
perturbation approximation technique is applied to find the solution of all governing
equations of momentum, angular momentum and heat transfer. Numerous results are
obtained and examined to illustrate the details of the flow and heat transfer
characteristics.
It is concluded from the figures that the effect of increase of magnetic parameter is
to decrease the velocity and angular velocity distribution whereas the increase in
values of permeability parameter and Grashof number increases the distribution
velocity and angular velocity across the boundary layer.
For the further understanding of the thermal behavior of the work of Kim[16], the
present work is an extension of the same and all our results are well in argument with
the published results of Kim[16].
The effect of Schmidt number Sc on the velocity and concentration are shown in
figures 8(a) and (b). from the figures it is revealed the fact that as Sc increases, the
concentration across the boundary layer decreases. This causes the concentration
Buoyancy effects to decrease Yielding a reduction in the fluid velocity.
The concentration decreases as the Schmidt number increases. The Shrewood
number decreases as the Schmidt number increases.
From figure 11 is drawn to represent Shrewood number against the Suction
velocity parameter for different values of Sc and it is noticed from the figure that as
Sc increases Shrewood number decreases.

Graphs
U0x
t*
U0
*
, T0
y
Porous
media
Tw
U0
Fig I physical configuration of the problem
|
1
|
2
|
3
|
4
| 5
|
6
0.5 -
1.0 -
1.5 -
2.0 -
u
0
y
k2= 0.5
Gr= 2
sc= 5
0
7.0Pr2
20
10
00





Up
M
GA
t
c


Fig 1a Velocity profiles across the boundary layer for a stationary vertical plate in
presence of magnetic field when viscosity ratio is zero

|
1
|
2
|
3 |
4
|
5
|
6
0.5 -
1.0 -
1.5 -
2.0 -
u
0
y
k2= 0.5
Fig 2 Velocity distribution of across the boundary layer for a
stationary vertical porous plate in the absence of magnetic field.
ϵ=0 η=0
β=0 M=0
A=0
Fig 3a Velocity profiles against y-c0-ordinate for different
values of viscosity ratio β.
| 1 | 2 | 3 |
4
|
5
|
6
0.0 -
5.0 -
10.0 -
15.0 -
u
0
Y
β= 0.5
5,0
3Pr,0
5.0,0
1,0,0
2




ScUp
Mn
kA
t
= 0.0
= 3.0
= 1.0
= 0.7

| 1 | 2 | 3 | 4 | 5 | 6
-40.0 -
-20.0 -
20.0 -
W
0 y
β = 0.7
Fig 3b Angular velocity profiles versus Y-co-ordinate for different
values of viscosity ratio β
-60.0 -
40.0 -
=1.0
= 3.0
= 0.7
= 0.7
ϵ=0 k2=0.5 η=0 t=1
A=0 Sc=5 Mn=0
Pr=3 Up=0
| 7
2.5 -
= 1.0
| 1 |
2
| 3
|
4 | 5
|
6
0.5 -
1.0 -
1.5 -
2.0 -
u
0
y
Up= 0.25
= 0.0
=0. 5
Fig 4a Velocity profiles versus co-ordinate Y for various
values of plate moving velocity Up.

|
7
-15.0 -
15.0 -
= 1.0
|
1
|
2
|
3
|
4
|
5
|
6
-10.0 -
-5.0 -
5.0 -
10.0 -
w
0
y
Up= 1.5
= 0.0
=0. 5
Fig 4b Graph of angular Velocity profiles versus Y -
co-ordinate for different values of Up.
|
7
2.5 -
= 8.0
|
1 |
2
|
3
|
4
|
5
|
6
0.5 -
1.0 -
1.5 -
2.0 -
u
0
y
k 2 = 0.0
= 2.0
= 4.0
Fig 5a Distribution of velocity and angular velocity profiles versus
y- co-ordinate for various values of permeability parameter k2

|
7
-6.0 -
-10.0
-
4.0 -
= 6.0
|
1
|
2
|
3
|
4
|
5
|
6
-8.0
-
-4.0 -
-2.0
-
2.0 -
w
0.0
y
K2= 8.0
= 2.0
=4.0
Fig 5b
2.5 -
= -2.0
|
1
|
2
|
3
|
4
|
5
|
6
0.5 -
1.0 -
1.5 -
2.0 -
u
0
y
Gc = 4.0
= 2.0
= 0.0
Fig 6a Distribution of velocity and angular velocity profiles versus
y- co-ordinate for various values of permeability parameter k2

0.0
-
-6.0 -
= 0.0
|
1
|
2
|
3
|
4
|
5
|
6
-8.0 -
-4.0 -
-2.0 -
w
0.0
y
Gc = 4.0
= - 2.0
= 2.0
Fig 6a and 6b Graphs of velocity and angular velocity profiles against Y
various values of Grashof number Gc with ε = 0.1, β = 0.2, n = 1, t
= 1, A = 0.5, M = 0.5, k2 = 2, Pr = 3, Up = 0.5, Gr = 2, Sc = 5
Fig 7 Variation of velocity and temperature distribution versus Y for various values of Prndtl
number Pr with ε = 0.1, β = 0.2, n = 1, t = 1, A = 0, k2 = 0.5, Gr = 2, Up = 0.5, Gc = 1, Mn =
0. 5
2.5
-
= 5.0
|
1
|
2
|
3
|
4
|
5
| 6
0.5 -
1.0 -
1.5 -
2.0
-
u
0
y
Pr = 0.7
= 1.0
= 3.0
2.5
-
= 5.0
|
1
|
2
|
3
|
4
|
5
| 6
0.5 -
1.0 -
1.5 -
2.0 -
T
0
y
Pr = 0.7
= 1.0
= 3.0

|
7
3.5 -
3.0 -
2.5 -
= 0.5
|
1
|
2
|
3
|
4
|
5
|
6
0.5 -
1.0 -
1.5 -
2.0 -
U
0
y
= 2. 0
= 1.5
= 1.0
1.2 -
1.0 -
= 0.5
|
1
|
2 |
3
|
4 |
5
|
6
0.2 -
0.4 -
0.6 -
0.8 -
C
0
y
Sc = 2.0
= 1.5
= 1.0
Fig 8a and 8b Variation of velocity concentration distribution versus Y co-ordinate for various values of
Schimdt number Sc with ε = 0.1, β = 0.2, n = 1, t = 1, A = 0, k2 = 0.5, Gr = 2, Up = 0.5, Gc = 1, Mn = 0. 5
-15.0 -
-5.0 -
0
A = 0.5
|1 |2
|
3
|
4 |5
|
6
-10.0 -
0.0 -
5.0 -
τ
10.0 -
Up
A = 1.0
A= - 0.5
A= 0.0
Fig 9 Graph of skin friction co-efficient versus plate
moving velocity for various values of suction parameter A and with ε = 0.1,
β = 0.2, n = 1, t = 1, A = 0.5, M = 0.5, k2 = 0.5, Pr = 1, Up = 0.5, Gr =
2, Sc = 3

-8.0 -
-6.0 -
0
-2.0 -
-1.0 -
Pr = 1
|1 |2 |3 |4 |5 |6
-7.0 -
-5.0 -
-4.0 -
NUx
-3.0 -
A
Pr = 0.7
Pr = 5
Pr= 3
Fig 10 Graph of surface heat transfer Vs suction velocity
parameter for various values of Prandtl number Pr A and
with ε = 0.1, β = 0.2, n = 1, t = 1, Mn = 0.5, k2 = 0.5,
Up = 0.5, Gr = 2, Gc = 1
-8.0
0
-6.0
-
Sc = 1.5
-2.0
-
-1.0 -
|
1
|
2
|
3
|
4
|
5
|
6
-7.0
-
-5.0
-
-4.0 -
Sh Rex
-1
-3.0
-
A
Sc = 2. 0
Sc =
0.5
Sc = 1.0
Fig 11 Graph of surface mass transfer Vs suction velocity parameter A for various
values of Schmidt number Sc with ε = 0.1, β = 0.2, n = 1, t = 1, Mn = 0.5, k2 = 0.5,
Up = 0.5, Gr = 2, Gc = 1
|
7

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Oscillatory Flow of MHD Polar Fluid with Heat and Mass Transfer

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