Open Journal of Fluid Dynamics, 2012, 2, 1-13
http://dx.doi.org/10.4236/ojfd.2012.21001 Published Online March 2012 (http://www.SciRP.org/journal/ojfd)
1
Effects of Hall Current and Ion-Slip on Unsteady MHD
Couette Flow
Nirmal Ghara, Sovan Lal Maji, Sanatan Das, Rabindranath Jana1, Swapan Kumar Ghosh2
1Department of Applied Mathematics, Vidyasagar University, Midnapore, India
2Department of Mathematics, Narajole Raj College, Narajole, India
Email: jana261171@yahoo.co.in, g_swapan2002@yahoo.com
Received November 26, 2011; revised December 15, 2011; accepted December 28, 2011
ABSTRACT
The unsteady MHD Couette flow of an incompressible viscous electrically conducting fluid between two infinite non-
conducting horizontal porous plates under the boundary layer approximations has been studied with the consideration of
both Hall currents and ion-slip. An analytical solution of the governing equations describing the flow is obtained by the
Laplace transform method. It is seen that the primary velocity decreases while the magnitude of secondary velocity in-
creases with increase in Hall parameter. It is also seen that both the primary velocity and the magnitude of secondary
velocity decrease with increase in ion-slip parameter. It is observed that a thin boundary layer is formed near the sta-
tionary plate for large values of squared Hartmann number and Reynolds number. The thickness of this boundary layer
increases with increase in either Hall parameter or ion-slip parameter.
Keywords: MHD; Couette Flow; Porous Plate; Hall Current; Ion-Slip; Hartmann Number and Reynolds Number
1. Introduction
The magnetohydrodynamic (MHD) flow between two
parallel plates, one in uniform motion and the other held
at rest known as MHD Couette flow, is a classical prob-
lem that has many applications in MHD power genera-
tors and pumps, accelerators, aerodynamic heating, elec-
trostatic precipitation, polymer technology, petroleum
industry, purification of crude oil and fluid droplets and
sprays. A lot of research work concerning the MHD
Couette flow has been obtained under different physical
effects. In most cases, the Hall and ion slip terms were
ignored in applying Ohm's law as they have no marked
effect for small and moderate values of the magnetic
field. However, the current trend for the application of
magnetohydrodynamics is towards a strong magnetic
field, so that the influence of electromagnetic force is
noticeable [1]. Under these conditions, the Hall currents
and ion slip are important and they have a marked effect
on the magnitude and direction of the current density and
consequently on the magnetic force term. Soundalgekar
[2] has studied the Hall and ion-slip effects in MHD
Coutte flow with heat transfer. Attia [3] has studied the
unsteady Couette flow with heat transfer considering ion-
slip. The transient Hartmann flow with heat transfer con-
sidering the ion slip has been investigated by Attia [4-6].
Attia [7] has obtained the analytical solution for flow of a
dusty fluid in a circular pipe with Hall and ion slip effect.
Seddeek [8] has studied the effects of Hall and ion-slip
currents on magneto-micropolar fluid and heat transfer
over a non-isothermal stretching sheet with suction and
blowing. The effects of Hall and ion-slip currents on free
convective heat generating flow in a rotating fluid have
studied by Ram [9]. Mittal and Bhat [10] has discussed
the forced convective heat transfer in a MHD channel
with Hall and ion slip currents. Jana and Datta [11] has
described the Couette flow and heat transfer in a rotating
system. The Hall effect on unsteady Couette flow under
boundary layer approximations has been analysed by
Kanch and Jana [12]. Attia [13] has studied the ion slip
effect on unsteady Couette flow with heat transfer under
exponential decaying pressure gradient. The combined
effect of Hall and ion-slip currents on unsteady MHD
Couette flows in a rotating system have been investigated
by Jha and Apere [14].
The present paper is devoted to study the combined
effects of Hall current and ion-slip on the unsteady MHD
Couette flow between two infinite horizontal parallel
porous plates under the boundary layer approximations.
The upper plate is moving with a uniform velocity U
while the lower plate is held at rest. The fluid is acted
upon by a constant pressure gradient and a uniform suc-
tion/injection at the plates. A uniform magnetic field 0
is applied perpendicular to the plates. It is found that the
primary velocity decreases while the magnitude of the
secondary velocity increases with increase in Hall pa-
B
C
opyright © 2012 SciRes. OJFD
N. GHARA ET AL.
2
rameter. It also is found that both the primary velocity
and the magnitude of secondary velocity decrease with
increase in ion-slip parameter. Asymptotic behavior of
the solution has been analyzed for large values of squared
Hartmann number and Reynolds number. It is observed
that a thin boundary layer is formed near the stationary
plate for large values of the magnetic parameter and
Reynolds number. The thickness of this boundary layer
increases with increases in either Hall parameter or
ion-slip parameter. Further, it is seen that the shear stress
components 0
x
and 0
z
due to the primary and sec-
ondary flows at the stationary plate 0
increase with
increase in Hall parameter for fixed value of squared
Hartmann number and ion slip parameter. It is also seen
that for fixed value of both squared Hartmann number
and Hall parameter, 0
x
increases while 0
z
decreases
with increase in ion-slip parameter.
2. Mathematical Formulation and Its
Solution
Consider the viscous incompressible electrically con-
ducting fluid bounded by two infinite horizontal parallel
porous plates separated by a distance . Choose a Car-
tesian co-ordinate system with x-axis along the lower
stationary plate in the direction of the flow, the y-axis is
normal to the plates and the z-axis perpendicular to xy-
plane (see Figure 1). Initially, at time , both the
plates are at rest. At time , the upper plate suddenly
starts to move with uniform velocity along x-axis. A
uniform magnetic field 0 is applied perpendicular to
the plates. The velocity components are
rela-
tive to a frame of reference. Since the plates are infinitely
long, all physical variables, except pressure, depend on
and only. The equation of continuity then gives
h
t
U
0
uv
0t
B
,,w
yt
Figure 1. Geometry of the problem.
0
vv
everywhere in the flow where is the suction
velocity at the plates.
0
v
Taking the effects of ion-slip, the generalised Ohm’s
law is
 
2
00
eeee i
BB
 
 


jEqBjBjB B,
(1)
where ,
B
E
, , , qj
, e
, e
and i
are re-
spectively, the magnetic field vector, the electric field
vector, the fluid velocity vector, the current density vec-
tor, the conductivity of the fluid, the cyclotron frequency,
the electron collision time and i
the ion-slip parame-
ter.
We shall assume that the magnetic Reynolds number
for the flow is small so that the induced magnetic field
can be neglected. This assumption is justified since the
magnetic Reynolds number is generally very small for
partially ionized gases. The solenoidal relation 0

B
for the magnetic field gives constant every-
where in the flow where
0y
BB
,,
x
y
BBB
 j
z
B
0
. The equation
of the conservation of the charge gives y
j
constant. This constant is zero since y at each
plate which is electrically non-conducting. Thus
0
0
y
j
j
everywhere in the flow. Since the induced magnetic
fields are neglected, the Maxwel’s equation
t
 
B
E becomes 0

E
which gives
0
x
E
z
and 0
z
E
z
. This implies that x
E
con-
stant and z
E
constant everywhere in the flow.
In view of the above assumption, Equation (1) gives
0
1ei xezx
jjEB
 
w, (2)
0
1ei zexz
jjEB
 
u
e
, (3)
where ee

is the Hall parameter. Solving for
x
j
and
z
j, we get

 
22
00
1
1,
x
ei e
eixe z
j
EBwEBu
 
 

 
(4)

 
22
00
1
1.
z
ei e
eize x
j
EBu EBw
 
 

 
(5)
On the use of Equations (4) and (5), the equations of
motion along x- and y-directions are

 
2
0
0222
00
1
1
1
ei e
eize x
B
uu pu
v
ty x
y
EBu EBw

 
 

 



(6)

Copyright © 2012 SciRes. OJFD
N. GHARA ET AL.
Copyright © 2012 SciRes. OJFD
3
1
0p
y
 , (7)

 
EBu
 
2
0
0222
00
1
1
1
ei e
eixe z
B
wwp w
v
ty z
y
EBw


 

 




 

(8)
where



22
111
222
11
Re
1
11,
ei e
ei e
uuu M
uw

 
 






 
(15)



22
111
222
11
Re
1
11,
ei e
ei e
wwwM
wu

 
 






 
(16)
,
and are respectively the fluid density,
the kinematic coefficient of viscosity and the fluid pres
sure.
The boundary conditions are
cy of pressure along
y-axis. Also, the fluid flow within the channel is induced
due to uniform motion of the upper plate
fore, using boundary condition at
and (8) we get
p
-
where
1
2
0
M
Bh




is the Hartmann number and
0
Re vU
the Reynolds number.
0for 0for all
,0at for 0.
uw ty
uUwyh t
 
 
,
0at 0for 0,uwyt (9)
Equation (7) shows the constanEquations (15) and (16) can be combined into the fol-
lowing equation
=yh
in Equat
. There-
ions (6) =yh


0
22
1
0
1
1,
ei e
B
p
x
EBU E

 
 






(10)


2
2
22
2
1
Re
1
ei e
ei e
Mi
FFF
F
 

 





, (17)
where
0ei z ex
11
1Fuiw
, 1i. (18)
The initial and boundary conditions for
,F
are


0
22
0
1
0
1
1.
ei e
ei xe z
B
p
z
EEBU

 
 

 
On the use of (10) and (11), Equations (6) and (8),
under the usual boundary layer approximations become
 
,00for all ,
0,1 and 0,0for 0
F
FF

(11)

(19)
Taking Laplace transform, Equation (17) becomes

2
2
dd
Re 0
d
d
FF
asF
 , (20)



2
2
0
0222
1ei e
B
uu u
v
ty
y
1,
ei e
uU w

 





 

where
(12)
 
0
,
s
Fs eF
,d


(21)
and

2
22
1
1
ei e
ei e
M
i
a

 


. (22)



2
2
0
0222
1
1,
ei e
ei e
B
ww w
v
ty
y
wuU

 




(13)

  

Introducing the non-dimensional variables
The boundary conditions (19) become

1
0,Fs
s
and

1, 0Fs. (23)
y
h
, 1
u
uU
, 1
w
wU
, 2
t
h
,
Equations (12) and (13) become
(14)
The solution of the Equation (20) subject to the bound-
ary condition (23) is

1
22
1
2co
R
1
Re
22 2
2
1
22
Re
sh 4
e Re
,cosh sinh
44
Re
sinh 4
as
e
Fs asas
as
s









 






 

 














. (24)




N. GHARA ET AL.
4
Taking inverse transform of (24) and on using (18), we
get (see Equation (25))
where

 

1
12
22
22
2
1
1
12
22
22
1R
e





2
22
22
22
2
22
11
1Re Re
44
2
Re
44
2
1,
11
π.
ei e
ei eei e
MM
sn




  




 



 

 




 


 





 
 
(26)
On separating into real and imaginary parts one can
easily obtain the velocity components and
Equation (25). The solution given by (25) exists both
(corresponding to
1
1
u1
w
for
from
Re 000v
,
spond
foe blowing at the
nd (correing r the
Solution at Small Times
In
or
r th
to v
plates) a
suction at the
Re 0
plates).
00, fo
this case, method used by Carslaw and Jaegar [15] is
used since it converges rapidly for small times. F small
time
which correspond to large
s
, Equation (24)
s become

 
Re 22
21
,mr r
Fs e

 

 
1
2
0
m
m
e e
s


(27)
7) is The inverse transform of (2
 
2
Re Re2
2
24
2
0
Re
,4
4
n
annn
m
F
eaiT

 
 

 


,
(28)
where n
ix

erfc
lementary err
denotes the repeated integrals of the
compor function given by
 
 

1
0
erfcerfc d
x
ixi


2
1
,0,1, 2,,
erfcerfc ,
2
erfc .
π
nn
x
n
ixx
ix e

(29)
e use of On th(18), we have

*
Re
*
2
1,cosue AB


*
1,sin





 

,
(30)
 
*
Re
**
2
1,sin ,coswe AB

 




 
,
(31)
where
 






 




2
*2 *2
02 4
3
*3* *2
6
4
*4*2 *2*4
8
2
***
24
3
*3*2 *
6
4
* **2*2
8
**
,4 4
34
64
,424
34
44,
Re ,
4
A
2
TT T
T
T
BTT
T
T
  


  

 



 


 
 
where
(32)
and
are given by (26) with
22
2
0
22
erfc erfc,
22
0,1, 2,3,
nn
nm
mm
Ti i
n


2
 

 
 
(33
Equations (30) and (31) show that the Hall effects be-
come important only when terms of order
)
is taken
into account.
3. Results and Discussion
To study the effects of suction/blowing, Hall para
ion-slip parameter and time on the velocity distrib
we have presented the non-dimensional velocity compo-
nents and against
meter,
utions
1
u1
w
in Figures 2-6 for various
values the sred-Hartmnn number ofquaa2
M
, Reynolds
num eber Re , Hall paramter e
, ion-slip eter parami
and time
. It is
1
u
se
seen frFigure 2both the
and the mde ndary
om that
of seco
pri-
ve-mary velocity
1
wagnitu
locity increa with increase in 2
M
. Figure 3
veals tary velocity incwhereas
maf ndaryases
increase es numed
Figure 4ary ases
th
re-
the
while
h
gnitude
in
at the
o
R
th
prim
seco
ynold
at the prim
1
u
locity
Re
velocity
r
1
w
t
1
u
eases
decre
is observ
decre
ve
ber
I
with
from .
e magnitude of secondary velocity 1
w increases with
increase in Hall parameter e
. Figure 5 displays that
both the primary velocity u and the magnitude of sec-
ondary veloci
1
ty w1 decrease with increase in ion-slip
parameter i
. It is also seen from
mary velocity u increases whe
e
Figure 6
s the
that the pri-
gnitude
1
secondary velocity 1
w decreases with increase in time
ra ma of
.
 

 
 
1
Re
11
2
11 11112
22
1
11 11
cosh2 π1
1coshsinhsin π1
sinh π
n
s
n
in
uiweiie n
ini

 
 
 

  
 

 
 
, (25)
Copyright © 2012 SciRes. OJFD
N. GHARA ET AL. 5
Figure 2. Velocity profiles for M2 when Re = 5, βe = 0.5, βi = 0.5 and τ = 0.02.
Figure 3. Velocity profiles for Re when M2 = 5, βe = 0.5, βi = 0.5 and τ = 0.02.
For small values of time, we have drawn the velocity
components and on using the exact solution
given by Equation (25) and the series solution given by
Equations (30) and (31) in Figures 7, 8. It is seen that the
series solution given by (30) and (31) converge more
quickly than the exact solution given by (25) for small
time. Hence we conclude that for small time, the nu-
merical values of the velocities can be calculated from
the Equations (30) and (
an
1
u1
w
31) instead of Equation (25).
The non-dimensional shear stresses due to the primary
d the secondary velocities at the stationary plate
0
are given by
0
2
22
111
1
11
0
Re
π
4
2
2
122 2
2
22
2 sinh2sin2
d1Re
d2 cosh2cos2
Re
π
1
Re
4
x
n
n
u
e
n

4
πcossin ,n
 


 














 


(34)
Copyright © 2012 SciRes. OJFD
N. GHARA ET AL.
6
Figure 4. Velocity profiles for βe when M2 = 5, Re = 5, βi = 0.5 and τ = 0.02.
Figure 5. Velocity profiles forβi when M2 = 5, Re = 5, βe = 0.5 and τ = 0.02.

0
2
22
1111
1
11
0
Re
π
4
2
2
122 2
2
22
πsin ,
4
n
2 sinh2sin2
d
dcosh2cos2
Re
π
4
Re
cos
z
n
n
w
e
n























(35)
where
 



,
, 1
and 1
are given by (26).
Numsults he non-dimensional shear
stresses
erical reof t
and 0
z
0
x
due to the primary and secondary
flows at plate the0
2
are shown graphically in Fig-
ures 9-ains
12 agt
M
for different values of e
, i
,
Re and
. Figure ows that the shear st 9 shresses 0
x
and 0
z
due to thary and the secondary f
plate
e prim
0
lows at
the stationary
increase withcrease in in Hall
parameter e
for fixed value of 2
M
,
an
of
, i
d . Re
2
It is seen from Figure 10 that for fixed value
M
,
Copyright © 2012 SciRes. OJFD
N. GHARA ET AL. 7
Figure 6. Velocity profiles for time τ wh en M2 = 5, Re = 5, βi = 0.5 and βe = 0.5.
Figure 7. Velocity profiles for general solution and small time solution when M2 = 5, Re = 5, βe = 0.5 and βi = 0.5.
e
,
and . Re0
x
increases while 0
z
decrease
parameter with increase in ion-slipi
. shows
that for fixed value of
Figure 11 2
M
, 0
x
increases while 0
z
. Fur- decreases with increase ds number
in Reynol Re
Copyright © 2012 SciRes. OJFD
N. GHARA ET AL.
8
Figure 8. Velocity profiles for general solution and small time solution when M2 = 5, Re = 5, βe = 0.5 and βi = 0.5.
for βe when Re = 5, βi = 0.5 and τ = 0.02. Figure 9. Shear stresses and z0
x0
Copyright © 2012 SciRes. OJFD
N. GHARA ET AL. 9
for βi when Re = 5, βe = 0.5 and τ = 0.02. Figure 10. Shear stresses and zx
0 0
for Re when βe = 0.5, βi = 0.5 and τ = 0.02. Figure 11. Shear stresses and z0
x0
Copyright © 2012 SciRes. OJFD
N. GHARA ET AL.
Copyright © 2012 SciRes. OJFD
10
Figure 12. Shear stresses x and z0
0 .
ther, it is also found from Figure 12 that both
for time τ when βe = 0.5, βi = 0.5 and Re = 5
0
x
and
0
z
decrease with increase in time
for fixe of d value
2
M
, i
, e
and
timehear plate
Re .
s, the sFor small stress at the 0
due to the primary and the secondary flows can be ob-
tained as

 
0
***
0,
1
0,cos0,sin,
2
x
u
U
eC D

 



 



*1
2
2
** 3
4
3
*2 **35
6
,4Re
24Re
34Re,
Y
DT
Y
T
Y
T
 
 
 









 


(39)
with
(36)
22
d
d2
nn
TY
1


 
0
***
0,
1
0,sin0,cos,
2
z
w
U
eC D

 



(40)
and
, (37)
where
 




*
11
02
2
*2 *23
4
3
*3* *25
6
,Re 4Re
4Re
34Re,
YY
CT T
Y
T
Y
T


 


 



 



 


2121
21
0
22 2
erfcerfc ,
22
0,1, 2,3,
nn
nm
mm
Yi i
n








(41)
For small time, the numerical values of the shear stress
components calculated from Equations (34)-(37) are
given in Tables 1 and 2 for different values of
(38) e
and
. It is observed that for small times the shear stresses
calculated from the Equations (36) and (37) give better
result than that calculated from Equations (34) an(35). d
N. GHARA ET AL. 11
Table 1. Shear stresse M =
1.
s due to primary flow for2 = 5, S
x
(For General solution)
x
(Solution for small times)
e
0.005 0.010 0.015 0.005 0.010 0.015
0.0 5.896093 3.644242 2.6733815.896093 3.6442382.673358
0.5 6.032860 3.765285 2.7827676.032872 3.7653582.782959
1.0 6.290388 3.998794 2.9982426.290404 3.9988862.998493
1.5 6.515985 4.208713 3.1962426.515995 4.2087623.196378
Table 2. Shear stresses due to secondary flow for M2 = 5, S
= 1.
y
(For General Solution)
y
(Solution for small times)
e
0.005 0.010 0.015 0.005 0.010 0.015
0.0 0.000000 0.000000 0.0000000.000000 0.0000000.000000
0.5 0.393570 0.361540 0.3374860.393615 0.3617900.338174
1.0 0.595888 0.555407 0.5247760.595908 0.5555170.525087
1.5 0.660252 0.622620 0.5940050.660258 0.6226500.594089
4. Steady state solution
The steady state velocity components are obtained from
(25) by letting
 as (see Equation (42))
Now, we shall discuss the following cases:
Case 1): When 21
M
and .
In this case, Equation (42) gives
Re 1
1
Re
2
1cose

11
u




 , (4) 3
1
Re
2
11
sinwe





 , (44)
re whe




1
2
1
1,
ei
M




22
11
22
2
1
.
11
ei e
e
ei eei
M

 







(45)
It is seen from Equations (43) and (44) that there exists
a single-deck boundary layer of thickness of the order
1
1
Re
O2



where 1
is given by (45). It i
that the thickness of this boundary layer increases with
in either Hall parameter
s seen
increase e
or ion-slip pa-
rameter i
but it decreases withcrease in either
Hartmaber
in
nn num
M
or Reynoldsber.
Case 2): When
num Re
.
ions given by (42)
become
Re 1, 21
M
In this case, the velocity distribut
1
11
1cosue

 , (46)
1
11
sinwe

, (47)
where




2
122
2
2
12
22
1
Re 1,
Re 1
.
Re 1
ei
ei e
e
ei e
M
M

 





(48)
Equations (46) and (47) show the existence of single-
deck boundary layer of thickness of order
1
1
O
where 1
creases with in
is given by (48). The thickness of
increase in either Hall parameter
this layer
e
or
ion-sliprameter pai
as 1
decreases with in
either
crease in
e
or i
.
5. Single Plate Motion
As , the velocity distribution given by (42) be-
comes
h
1
2
11
1co
S
ue

s




 , 9) (4
1
2
S

11
sinwe
, (50)
where




1
12
22
22
2
1
1
12
22
22
2
1
2
2
00
2
1,
44
2
1
44
2
, ,
SS
SS
vB
Uz SM
UU













 



 
 



(51)
and ,
are given by (26). It is clear from above
Equations (49) and (50) that the flow exhibits a boundary
layer behavior with boundary layer thickness of order of
1
1
O2
S


. Since 1
increases with increase in either
or
S2
M
it means that increase in either suction pa-


 
Re
112
111111
11
cosh
1sinhcosh
sinh
i
uiwii e
i
 


 



(42)
Copyright © 2012 SciRes. OJFD
N. GHARA ET AL.
12
rameter or magnetic parameter
S 2
M
causes thinning
of the ary layer. Further, fod and boundr fixeS2
M
and 0
z
,
1
decrwith increase in eithrameter eases er Hall pae
or ion-parameter slip i
. Hence, nclude that t
boundary layer thickness near the
we co
plate
he
0
increases
with increase in either e
or i
. The solutions given
by (49) and (50) are also valid for the blowing (<0S) at
the plate.
In thsence of ion e ab-slip (0
i
), the above Equa-
tions (49)d (50) be ancome
2
11cosue
S

 , (52)




2
1sin
S
we





, (53)
where
1
2
2
22
22
22
4
411
2
e
e
e






i
22

1
2
2
2
2,
1
22
2
2
22
1M
SM

  

1
2
2
2
M
2.
1SM


11
2
41
41
eM


e
e
e

e
SM
S




(54)
Equations (52) and (53) coincide with Equatons (36)
and (37) of Gupta [16] when 0
i
(abse ce of
on th
tte flow n two iite ho
boun layer
imas have been studied. It is founat th
mary vcity decreases while th
nda velo increases with increase in
e
n
dary
d th
ion-
e un-
rizon-
ap-
e pri-
Hall
slip).
6. Conclusion
Co
ste
tal
prox
seco
param
mbined effects of Hall current and ion-slip
ady MHD Couebetweenfin
parallel porous plates under the
tion
elo
ry
ter
1
u
city
e magnitude of
1
w
e
. It is
magni
in
also
tude
ion-slip
found that bot
of secondary velcity decrease
parameter
h the primary ve-
locity and
with incr
t
ea
he
se
o
i
. obse
at a th form
e fouof magnetic r
It is
ed near the stationa
ete
rved
ry
th
plat
in
r la
boun
rge
da
val
ry layer is
es param2
M
and
Reynolds. The thickne these boary
layers increases with increases in either Hall parameter
or ion-slip parameter. Further, it is seen that the shear
stresses
number Ress ofund
0
x
due to the primary and secondary
flows at plate the stationary 0
increase with in-
crease in Hall paramter ee
for fixed value of 2
M
. It
is also seen thfixeat for d value of 2
M
, 0
x
while
increases
0
z
decrease with increase in ion-slip pater rame
i
.
REFE
nd P
ineers and Appli
h
o p
/TPS.1979.4317226
RE
S.-I.
e
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d
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