International Journal of Advancements in Research & Technology, Volume 2, Issue3, March-2013
ISSN 2278-7763
1
Unsteady Couette Flow of an Electrically Conducting Viscous, Incompressible Fluid with
Heat Transfer in Presence of Ion-Slip
Dr.Rachna Khandelwal
Department of Mathematics,
(Jaipur Institute of Technology, Group of Institutions)
khandelwalrachna@ymail.com
Abstract
Unsteady Couette flow of an electrically conducting, viscous, incompressible fluid with
heat transfer between two parallel non-conducting porous plates is investigated .A uniform
suction/injection is applied perpendicular to the plate, while the fluid motion is due to a constant
pressure gradient. An external uniform magnetic filed is applied perpendicular to the plates and
effect of Joule and viscous dissipations are taken in account. The plates are kept at different
constant temperatures. The effect of ion-slip and uniform suction and injection on both the velocity
and temperature distributions are examined, discussed and shown through graphs.
Key words : Hydromagnetic , heat transfer , Hall effect , ion-slip , Couette flow , parallel plates .
1.Introduction
The magnetohydrodynamic flow between two parallel plates, known as Hartmann flow is a
classical problem whose solution has many applications in magnetohydrodynamic (MHD) power
generators, MHD pumps, accelerators , aerodynamic heating, electrostatic precipitation ,polymer
technology , petroleum industry, purification of oil and fluid droplets and sprays etc. Hartmann and
Lazarus (1937) studied the influence of a transverse uniform magnetic field on the flow of a
conducting fluid between two infinite parallel, stationary and insulated plates . Then, a lot of
research work concerning the Hartmann flow was conducted under different physical effects . In
most of the 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 by Carmer and Pai (1973). Under these conditions, the Hall
current and ion-slip are important and have marked effect on the magnitude and direction of the
current density and consequently on magnetic force term . Tani (1962) studied the Hall effect on the
steady motion of an electrically conducting and viscous fluids in channels . Soudalgerkar et al.
(1979,1986) studied the effect of the Hall currents on the steady MHD Couette flow with heat
transfer. The temperature of the two plates were assumed either to be constant (1979) or to vary
Copyright © 2013 SciResPub.
International Journal of Advancements in Research & Technology, Volume 2, Issue3, March-2013
ISSN 2278-7763
2
linearly along the plates in the direction of the flow (1986) . Abo-El-Dahab (1993) studied the
effect of Hall current on the steady Hartmann flow subjected to a uniform suction and injection at
the bounding plates. Attia (1998) extended the problem to the unsteady state with heat transfer,
taking the Hall effect into consideration , while neglecting ion-slip,when upper plate is moving with
a variable velocity U0(1+εeiωt),while the lower plate is kept stationary. The fluid is acted upon by a
constant pressure gradient, a variable suction and injection and a uniform magnetic field
perpendicular to the plates. The induced magnetic field is neglected by assuming a very small
magnetic Reynolds number , therefore , the uniform magnetic field B0 is considered as the total
magnetic field acting on the fluid . The plates are maintained at different but constant temperatures:
T1 and T2 , respectively , where T1<T2 . Attia (2005) studied the transient flow and heat transfer of
an incompressible, viscous, electrically conducting fluid between two infinite non-conducting
horizontal porous plates with the consideration of both Hall current and ion-slip. Sharma and
Mishra (2002) investigated pulsatile MHD flow and heat transfer through a porous channel. Sharma
and Chaturvedi (2003) presented unsteady flow and heat transfer through an electrically conducting
viscous incompressible fluid between two non -conducting parallel plates under uniform transverse
magnetic field . Unsteady plane Poiseuille flow and heat transfer in the presence of oscillating
temperature of the lower plate have been discussed by Sharma, Chawla and Singh (2004) . Sharma
and Yadav (2006) investigated pulsatile MHD flow and heat transfer through a porous channel with
Ohmic effect .Sharma and Mehta (2009) presented MHD unsteady slip flow and heat transfer in a
channel with slip at the permeable boundaries .
Aim of the paper is to investigate unsteady Couette flow of an electrically conducting
viscous, incompressible fluid with heat transfer in presence of ion-slip.
2.Formulation of the Problem
The existence of the Hall term gives rise to a component of the velocity. Thus the velocity vector
of the fluid is given by

v*(y,t)  u*(y,t)iˆ Vo (t) ˆj  w*(y,t)kˆ.
…(1) The fluid
flow is governed by the momentum equation.

  
Dv *

 *  2 v *  P  J  B0
Dt
,
…(2)
where ρ and μ are the density and the coefficient of viscosity of the fluid respectively. If the
Hall and ion- slip terms are retained, the current density J is given by
Copyright © 2013 SciResPub.
International Journal of Advancements in Research & Technology, Volume 2, Issue3, March-2013
ISSN 2278-7763

J 
3
 
 
 Bi  
 * 
v

B


(
J

B
)

(
J

B
)

B
0
0
0
0

Bo
,
when
 
J  B0  
 B0 2 
{(1  BiBe)u *  Bew* }iˆ  {(1  BiBe)w*  Beu *}kˆ 
2 
(1  BiBe)
,
 is the electric conductivity of the fluid, β is the Hall factor, Bi is the ion-slip parameter and B o
= |Bo|and Be =σβB0 is the Hall parameter.
Using equation(3) into the equation (2), we get
u *
u *
dp*
 2u *
 B0 2

 V0 *   *   *2 
{1  BiBe)u*  Bew*}
2
2
t *
y
dx
y
(1  BiBe)  Be
,

w*
w*
 2 w*  B0 2 {1  BiBe) w*  Beu *}


V



.
0
t *
y*
y*2
(1  BiBe) 2  Be 2
…(3)
...(4)
The temperature distribution in the presence of rate of heat generation/absorption and
Ohmic dissipation is given by
2
 u *   w* 
T *
T *
 2T *
 C p *   C pV0 *  K *2    *    * 
t
y
y
y   y 

2
  B 2 (u *2  w*2 )
0
 S * (T *  T1 ).

2
2
(1  Bi Be)  Be

...(5)
3.Method of Solution
Introducing the following quantities
x
t *U 0
x*
y*
z*
u*
w*
p*
T *  T1
, y  , z  ,u 
,w 
,P 
,
t

,
T

h
h
h
U0
U0
U 02
h
T2  T1
λ*=V0/U0, λ*
 B02 h 2
Ec  U 0 2 / C p  T2  T1  , Ha 2 
.

Re
=ρhU0/μ
,
=
λ
(1+εeiωt)
,
Pr
=μCp/κ
,
S=S*h2/CpU0
…(6)
into the equations (3), (4) and (5),we get
u
u
dp 1  2u Ha 2 {1  BiBe)u  Bew}
  (1   eit )
 

t
y
dx Re y 2 Re{(1  BiBe) 2  Be 2 } ,
Copyright © 2013 SciResPub.
...(7)
International Journal of Advancements in Research & Technology, Volume 2, Issue3, March-2013
ISSN 2278-7763
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w
1  2 w Ha 2 {1  BiBe)w  Beu}
it w
  (1   e )


t
y
Re y 2 Re{(1  BiBe)2  Be2 } ,
…(8)
2
2
T
1  2T Ec  u   w  
EcHa 2 (u 2  w2 )
i t T
  (1   e )




 ST


  

t
y
Re Pr y 2 Re  y   y   Re{(1  BiBe)2  Be2 }
,


where
...(9)
dP
  A(1  eit ),
dx
λ is the suction parameter, Re is the Reynolds number ,Ha is the Hartmann number, Pr is the
Prandtl number and Ec is the Eckert number.
The initial and boundary conditions are reduced to
Initial conditions
u  0, w  0, T  0, when t  0
The boundary conditions
y  1: u  0, w  0, T  0;
y  1: u  (1   eit ), w  0, T  1.
...(10)
In view of the boundary conditions, the velocity and temperature distributions are taken of the
form given below
u ( y, t )  uo ( y )   u1 ( y )eit  O ( 2 ) 

w( y, t )  wo ( y )   w1 ( y )eit  O( 2 ) 

T ( y , t )  To ( y )   T1 ( y )eit  O ( 2 ) 
…(11)
When ε=0, the problem reduces to steady free convective flow which is governed by the following
equations
1  2 u0
uo
Ha 2 {(1  BiBe)uo  Bewo }




A

Re y 2
y
Re((1  BiBe)2  Be 2 ) ,
1  2 wo
w0 Ha 2 {(1  BiBe) wo  Beuo }

 

,
Re y 2
y
Re((1  BiBe) 2  Be 2 )
Copyright © 2013 SciResPub.
...(12)
…(13)
International Journal of Advancements in Research & Technology, Volume 2, Issue3, March-2013
ISSN 2278-7763
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2
 2To
 2To
1
Ec  uo   wo  
EcHa 2






 ST0 .



 

Re Pr y 2
y
Re  y   y   Re{(1  BiBe) 2  Be 2 }


5
…(14)
The corresponding boundary conditions are reduced to
y  1: u0  0, w0  0, T0  0;
y  1: u0  1, w0  0, T0  1.
…(15)
When ε≠0 the governing equations are given by
1  2u1
u1
 2u0 Ha 2 {(1  BiBe)u1  Bew1}



i

u


A



,
1
Re y 2
y
y
Re{(1  BiBe)2  Be 2 }
…(16)
1  2 w1
w1
wo Ha 2 {(1  BiBe)u1  Beu1}



i

w




1
Re y 2
y
y
Re{(1  BiBe)2  Be 2 } ,
…(17)
1  2T1
T1
To 2 Ec  uo u1 wo w1  2 EcHa 2 (uo u1  wo w1 )



(
S

i

)
T





.
1
Re y 2
y
y
Re  y y
y y  Re{(1  BiBe) 2  Be 2 }
...(18)
The
corresponding
y  1: u1  0, w1  0, T1  0;
boundary
conditions
are
y  1: u1  1, w1  0, T1  0.
reduced
to
...(19)
Equations (12) to (14) and (16) to (18) are ordinary second order coupled differential equations
and to be solved under the boundary conditions (15) and (19), respectively.
Since the Hartmann number Ha is small therefore velocity and temperature distributions can be
expanded in the power of Ha2 as given below
F  F0  Ha 2 F1  o( Ha 4 ),
...(20)
where F stands for any u0,u1,w0,w1,T0 or T1.
Using (20) into the equations (12) to (14) and (16) to (18) , and equating the like powers of Ha 2,we
get
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International Journal of Advancements in Research & Technology, Volume 2, Issue3, March-2013
ISSN 2278-7763
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1 d 2 u00
du
  00  A
2
Re dy
dy
,
...(21)
1 d 2u01
du
{(1  BiBe)u00  Bew00 } ,
  01 
2
Re dy
dy
Re{(1  BiBe) 2  Be 2 }
...(22)
1 d 2 u10
du
du
  10  iu10   00
2
Re dy
dy
dy ,
...(23)
1 d 2 u11
d 2 u11
du
{(1  BiBe)u10  Bew10 }


 i u11   01 
2
Re dy
dy
dy
Re{(1  BiBe)}2
,
...(24)
1 d 2 w00
dw
  00  0
2
Re dy
dy
,
...(25)
1 d 2 w01
dw
{(1  BiBe) w00  Beu00 }
  01 
2
Re dy
dy
Re{(1  BiBe)2  Be 2 } ,
...(26)
1 d 2 w10
dw
dw
  10  i w10   00
2
Re dy
dy
dy ,
...(27)
1 d 2 w11
dw
{(1  BiBe)w10  Beu10 }
dw
  11  i w11 
  01
2
2
2
Re dy
dy
Re{(1  BiBe)  Be }
dy ,
...(28)
dT
1 d 2T00
Ec  u00   w00 
  00  ST00  



2
Re Pr dy
dy
Re  y   y 

2



,
...(29)
1 d 2T01
dT01
2 Ec  u00 u01 w00 w01   Ec
(u00 2  w00 2 )



ST


.

.


01

Re Pr dy 2
dy
Re  y y
y y   Re {(1  BiBe) 2  Be 2 }
,
...(30)
1 d 2T10
dT
2 Ec  du00 u10 w00 w10  
dT00
  10  ( s  i )T10  
.

.

  
2
Re Pr dy
dy
Re  y y
y y  
dy
Copyright © 2013 SciResPub.
,
...(31)
International Journal of Advancements in Research & Technology, Volume 2, Issue3, March-2013
ISSN 2278-7763
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1 d 2T11
dT
2 Ec  du00 du11 du01 du10 dw00 dw10 dw01 dw10 
  11  (s  i )T11  



2
Re Pr dy
dy
Re  dy dy
dy dy
dy dy
dy dy 

 dT01
2 Ec {u00u10  w00 w10}

2
2
Re {(1  BeBe)  Be }
dy .
...(32)
Now, the corresponding boundary conditions are reduced to
y  1: u00  u01  u10  u11  0, T00  T01  T10  T11  0, w00  w01  w10  w11  0.
y  1: u00  1  u01  u10 , u11  0, w00  w01  w10  w11  0, T00  1, T01  0  T10  T11.
…(33)
The equations (21) to (32) are solved under the boundary conditions (33).Through straight forward
algebra,
the
solution
of
u00  y  , u01  y  , u10  y  , u11  y  ; w00  y  , w01  y  , w10  y  , w11  y  ; T00  y  , T01  y  , T10  y  , andT11  y 
are known and given by
u00  y   A3  A2 e Re y 
A
y
 ,
…(34)
u01  y   A7  A8 e  Re y  A9 ye Re y  A10 y 2  A11 y
,
u10  y   A18e R1 y  A19e R2 y 
…(35)
iA2  2 Re  Re y iA
e


,
…(36)
u11  y   A22e R1 y  A23e R2 y  (iA24  A25 )e Re y  (iA26 y  A25 )e Re y
(iA26 y  A27 )e  Re y  (iA28 y  A29 )  ( A32 y  iA33 )e RR1 y cos( IR1 y )
( A34  iA35 )e RR1 y sin( IR1 y )  ( A38  iA39 )e RR2 y cos( IR2 y )  ( A40  iA41 )e RR1 y Sin( IR2 y ) ,
…(37)
w00 ( y)  0,
…(38)
w01 ( y )  A14  A15e  Re y  A16 y 2  A17 y,
…(39)
w10 ( y )  0,
…(40)
w11 ( y )  ( A42  A48 )e R1 y  ( A43  A49 )e R2 y  iA44e  Re y  iA45 y  i (
Copyright © 2013 SciResPub.

A45  A47 )  A46 ,

…(41)
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T00 ( y )  A50 e R3 y  A51e R4 y  A52e Re y  A53 ,
…(42)
T01 ( y )  A55 eR3 y  A56 e R4 y  ( A64  A66 )e 2  Re y  A65 ye 2  Re y
 A67 ye Re y  ( A68  A69 )e Re y  A72 y 2  ( A70  A73 ) y  ( A71  A74  A75 ),
…(43)
T10 ( y )  A76 R5 y  A77 e R6 y  A83e ( R1   Re) y  A84e( R1   Re) y  iA85e2  Re y  A86e R1 y 
A87 e R2 y  A88e R3 y  A89e R4 y  A90e  Re y ,
…(44)
T11 ( y )  A91e R5 y  A92 e R6 y  ( A93  A101 )e RR1 y cos( IR1 y )  ( A94  A102 )e RR1 y sin( IR1 y )  ( A95  A103 )e RR2 y cos( IR2 y )
( A96  A104 )e RR2 y sin( IR2 y )  ( A97  A99 )e 2  Re y  A103  A104  A105 .
…(45)
where A, A2 to A105 are constants and their expressions are presented in Appendix.
4.Skin-friction
The skin-friction coefficient at the lower plate is given by
 u
u  
 u
 u 
u 
Cf   
   00  Ha 2 01     10  Ha 2 11  eit 
.
y 
y   y 1
 y  y 1
 y
 y
…(46)
The skin-friction coefficient at the upper plate is given by
 u
u  
 u
 u 
u 
C f        00  Ha 2 01     10  Ha 2 11  eit  .
y 
y   y 1
 y  y 1
 y
 y
…(47)
5.Rate of Heat Transfer
The rate of heat transfer in terms of the Nusselt number at the lower plate is given by
 T
T  
 T
T 
Nu    00  Ha 2 01     10  Ha 2 11  eit 
y 
y   y 1
 y
  y
…(48)
The rate of heat transfer in terms of the Nusselt number at the upper plate is given by
 T
T  
 T
T 
Nu    00  Ha 2 01     10  Ha 2 11  eit 
y 
y   y 1
 y
  y
…(49)
6. Tables and Graphs
CopyrightHa© 2013
Bi SciResPub.
Be
1
1
1
I
3
1
1
II
I
Steady flow
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ISSN 2278-7763
9
1.2
1
0.8
0.6
0.4
0.2
0
-1
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
-0.2
-0.4
-0.6
-0.8
-1
Fig.1. Velocity distribution versus y when Pr=5, Re=1,Ec=0.1,=0,=1,=2,
Copyright © 2013 SciResPub.
t=/4.
1
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ISSN 2278-7763
10
Unsteady Flow
2
Ha
Bi
Be

t
1
1
1
2
/4
I
3
1
1
2
/4
II
1
3
1
2
/4
III
1
1
3
2
/4
IV
1
1
1
4
/4
V
1
1
1
2
/3
VI
1.5
1
III
IV
II
0.5
VI
I
u
V
0
-1
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
1
y
-0.5
-1
Fig.2. Velocity distribution versus y when Pr=5,Re=1, S=1,Ec=0.1,=0.1,=1.
Copyright © 2013 SciResPub.
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ISSN 2278-7763
11
Steady Flow
1.2
1
II
0.8
Ha
Bi
Be
1
1
1
I
3
1
1
II
1
3
1
III
1
1
3
IV
0.6
0.4
IV
I
0.2
III
w
0
-1
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
y
-0.2
Fig.3. Velocity distribution versus y when Pr=5, Re=1,Ec=0.1,=0,=1,
t=/4.
Copyright © 2013 SciResPub.
=2,
1
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ISSN 2278-7763
12
Unsteady Flow
1.5
1
II
Ha
Bi
Be

t
1
1
1
2
/4
I
3
1
1
2
/4
II
1
3
1
2
/4
III
1
1
3
2
/4
IV
V
VI
0.5
I
w
0
-1
-0.75
-0.5
-0.25
0
-0.5
y
0.25
0.5
0.75
IV
-1
-1.5
Fig.4.Velocity distribution versus y when Pr=5,Re=1, =0.1, =1, S=1, Ec=0.1 .
Copyright © 2013 SciResPub.
1
IIII
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ISSN 2278-7763
13
Steady Flow
1.4
1.2
1
0.8
III
0.6
Ec Ha
Bi
Be
II
0.1
1
1
1
I
0.2
1
1
1
II
0.1
3
1
1
III
0.4
V
T
I
0.2
IV
0
-1
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
y
Fig.5. Temperature distribution versus y when Pr=5,Re=1,=0,=1,S=1,=2,
Copyright © 2013 SciResPub.
t=/4.
1
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ISSN 2278-7763
14
Unsteady Flow
7
6
Ec
Ha
Bi
Be

t
0.1
1
1
1
2
/4 I
0.2
1
1
1
2
/4 II
0.1
3
1
1
2
/4 III
0.1
1
3
1
2
/4 IV
0.1
1
1
3
2
/4 V
0.1
1
1
1
4
/4 VI
5
VI
4
III
3
2
I
V
T
1
IV
II
VII
0
-1
-0.75
-0.5
-0.25
0
0.25
y
-1
Fig.6. Temperature distribution versus y when Pr=5,Re=1,=0.1,=1,S=1.
Copyright © 2013 SciResPub.
0.5
0.75
1
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ISSN 2278-7763
15
Table-1(a). Numerical values of Skin-friction coefficient at lower plate for various values of
physical parameters .
Re
Ha
Bi
Be
λ
A

Cf
ε=0
ε=0.1
1
1
1
1
2
1
1
-0.8979
-0.9768
2
1
1
1
2
1
1
-1.7564
-1.9963
1
3
1
1
2
1
1
-0.8979
-0.9862
1
1
3
1
2
1
1
-0.74933
-0.8283
1
1
1
3
2
1
1
-0.6814
-0.7603
1
1
1
1
4
1
1
-1.2489
-1.3763
1
1
1
1
2
2
1
-1.8114
-1.9943
1
1
1
1
2
1
2
-3.1039
-3.1626
Table-1(b).Numerical values of Skin-friction coefficient at upper plate for various values of
physical parameters
Re
Ha
Bi
Be

A
λ
Cf
ε=0
ε=0.1
1
1
1
1
2
1
1
3.9037
4.4923
2
1
1
1
2
1
1
128.99
149.1422
1
3
1
1
2
1
1
3.9037
4.4830
1
1
3
1
2
1
1
3.8360
4.4246
1
1
1
3
2
1
1
3.8051
4.3936
1
1
1
1
4
1
1
17.2236
19.7449
1
1
1
1
2
2
1
7.1956
8.4053
1
1
1
1
2
1
2
18.0925
18.4829
Copyright © 2013 SciResPub.
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ISSN 2278-7763
16
Table-2(a). Numerical values of Nusselt number at the lower plate for various values of
physical parameters
Re
Ha
Bi
Be
λ
A

Pr
Nu
ε=0
ε=0.1
1
1
1
1
2
1
1
5
0.3361
0.2600
2
1
1
1
2
1
1
5
0.2850
0.2716
1
3
1
1
2
1
1
5
2.0241
1.1431
1
1
3
1
2
1
1
5
1.1691
0.1522
1
1
1
3
2
1
1
5
0.5531
0.5361
1
1
1
1
4
1
1
5
0.3361
-1.3127
1
1
1
1
2
2
1
5
1.0518
0.9531
1
1
1
1
2
1
2
5
-0.0838
-0.0338
1
1
1
1
2
1
1
10
0.4964
0.4118
Table-2(b). Numerical values of Nusselt number at the upper plate for various values of
physical parameters
Re
Ha
Bi
Be

A
λ
Pr
Nu
ε=0
ε=0.1
1
1
1
1
2
1
1
5
-5.7511
14.4582
2
1
1
1
2
1
1
5
-3707.6493
-6267.8961
1
3
1
1
2
1
1
5
-59.1991
-34.3277
1
1
3
1
2
1
1
5
-1.1531
18.6235
1
1
1
3
2
1
1
5
-9.7040
9.6742
1
1
1
1
4
1
1
5
-5.7511
188.0042
1
1
1
1
2
2
1
5
-26.628
1.4261
1
1
1
1
2
1
2
5
-59993.0905
49443.3821
1
1
1
1
2
1
1
10
1531.682
-11.8521
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ISSN 2278-7763
17
6.Results and Discussion
The computational analysis is carried out to discuss the behavior of the velocity and
temperature distributions for different variations in Physical parameters Ec, Ha, Bi, Be, ω and ωt.
Figure 1 shows the profiles of velocity component u for various values of y and independent of t
i.e. steady flow. It is observed that the velocity component u decreases with the increase of
Hartmann number, ion-slip parameter or the Hall parameter.
Figure 2 depicts the profiles of velocity component u for various values of y and depend
also on time t (i.e. unsteady part of u) . It is observed that unsteady velocity profile u is increases
with increase of ion-slip parameter or the Hall parameter and ωt, but it decreases with the increase
of the Hartmann number or frequency .
It is observed from figure 3 that the velocity component w first decreases due to increase of
Hartmann number and then increases. It decreases due to increase in ion-slip parameter or The
Hall parameter.
Figure 4 shows the unsteady velocity profile w decreases due to increase in ion-slip
parameter or The Hall parameter, while it increases due to increase in Hartmann number,
frequency or phase angle.
It is noted from figure 5 that fluid temperature decreases with the increase in ion-slip
parameter while it increases with the increase in the Hartmann number, the Eckert number or the
Hall parameter .
It is observed from figure 6 that fluid temperature increases with the increase of the
Hartmann number, the Eckert number or frequency ,while it decreases with the increase of ion-slip
parameter, Hall parameter or phase angle.
It is seen from Table-1(a) that skin-friction coefficient at the lower plate increases with the
increase in Bi or Be while it decreases with the increase in Re, Ha, ω , A or λ .The skin-friction
coefficient at the upper plate increases with the increase in Re, ω , A or λ while it decreases with
the increase in Ha,Bi or Be in unsteady case.
It is seen from Table-2(a) that Nusselt number at the lower plate increases with the increase in
Re, Ha, Be, Pr or A, while it decreases with the increases in Bi, ω or λ .The Nusselt number at the
upper plate increases with the increase in Bi, ω or λ while it decreases with the increase in Re, Ha,
Be, A or Pr in unsteady case.
Copyright © 2013 SciResPub.
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ISSN 2278-7763
18
References
1.Abo-El-Dahab,E. “The effect of Hall current on the steady Hartmann flow subjected to a uniform
suction and injection at the bounding plates.”.M.S.thesis,Department of Mathematics, Helwan
University,Egypt,1993.
2.Alpher,R. “ Heat transfer in magnetohydrodynamic flow between
and Mass Transfer, Vol.3, 1961, pp.108-112..
parallel plates.”Int. J. Heat
3.Attia,H. “The unsteady state with heat transfer, taking the Hall effect into consideration , while
neglecting ion-slip.” Canadian J.Phys., Vol.76, 1998, pp.739-760 .
4.Attia,H. “Unsteady Couette flow with heat
Phys.,Vol.29,2005,pp.379-388.
transfer considering Ion-Slip”,Turk J.
5.Cramer,K. and Pai,S. “Megnetofluiddynamics for Engineers and Applied Physicsts,” McGrawHill,New York, 1973.
6.Hartmann, J. and Lazarus, F. “The influence of a transverse
uniform magnetic field on the
flow of a conducting fluid between two infinite parallel, stationary and insulated plates.” Mat.
Fys. Medd., Vol.15, 1937, pp.6- 12 .
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8.Schilichting,H. “ Boundary Layer Theory”,McGraw-Hill,New York, 2004.
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channel” .Bull.Pure Appl. Sciences, India, Vol.21E,2002,pp.93-100.
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conducting viscous incompressible fluid between two non -conducting parallel plates under
uniform transvers magnetic field ”. ‘Ganita Sandesh’ J.Raj.Ganita Parishad ,
India,Vol.17,2003,pp.9-14.
11.Sharma,P.R.;Chawla,Seema and Singh, Ranjeet “. Unsteady plane Poiseuille flow and heat
transfer in the presence of oscillating temperature of the lower plate”. Indian J. Theo.Phys.,
India, Vol.52, 2004, pp.215-226.
12.Sharma,P.R. and Yadav,G.R. “Pulsatile MHD flow and heat transfer through a porous channel
with Ohmic effect” . Bull.Pure Appl. Sciences, India, Vol.25E,2006,pp.499-509.
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13.Sharma,P.R. and Metha,Ruchika “MHD unsteady slip flow and heat transfer in a channel with
slip at the permeable boundaries”. J. International Academy of Physical sciences ,India
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14.Soundalgerkar,V.;Vighnesam,N. and Takhar,H. “The effect of the Hall currents on the steady
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Copyright © 2013 SciResPub.
International Journal of Advancements in Research & Technology, Volume 2, Issue3, March-2013
ISSN 2278-7763
20
Appendix
A1  A3 
A3 
A
 R3 ,
A2 
2
A
 A2 e   Re

,
A7   A8 e
  Re
 A9 e
A5  A7 
  Re
A9 
B1 A2
 ,
A11 
B1 A B1 A3

 ,
3 Re
A13  A15 
A15  
 A10  A11
B2 A2
(1   Re) ,
 A17
Sin( Re) ,
BA
A
A17  2 3  3

 Re ,
1
A19 
(2 A   )
2 Sin( Re) ,
A8 
,
A10 
A12  A14 
( B1  A3 ) B1 A

2 Re  3 Re ,
A9 Sin( Re)  A11
Sin( Re)
,
B1 A
22 ,
B2 A3
A
 3 2
2
 Re  Re ,
A14   A15 e Re  A16  A17
A16 
,
A
22 ,

iA2  2 Re   Re iA 
 R2
A18  e  A19 e 
e
 



,
R1
iA2  2 Re  Re 2 R1  R e iA
(e  e
)  (1  e2 R1 )


R2
2 R1  R 2
(e  e
)
,
A22  e R1 ( A23 e R2  (iA26  A25  A27 )e   Re  ( A28  iA29 )  ( A32  iA33 )e RR1 cos ( IR1 )  ( A34  iA35 )e RR1
sin( IR1 )  ( A38  iA39 )e  RR2 cos ( IR2 )  ( A40  iA41 )e  RR2 sin( IR2 ),
A23  e  R2 [( A22 e R1  (iA24  A25 )e Re  (iA26  A27 )e  Re  ( A28i  A29 )  ( A32  iA33 )e RR1 cos ( IR1 ) 
( A34  iA35 )sin( IR1 )e RR1  ( A38  iA39 )e RR2 cos ( IR2 )  ( A40  iA41 )e RR2 sin( IR2 )]
Copyright © 2013 SciResPub.
,
International Journal of Advancements in Research & Technology, Volume 2, Issue3, March-2013
ISSN 2278-7763
21
 2 A8   A9
E1
A9 2 Re
A9 3 Re
A24 
, A25 
, A26 
, A27 



2 ,
A28 
2 A10
2 2 A10
, A29 
, A30  ( RR12  IR12   Re RR1 )

2
,
A32 
A31  ( IR1 RR1   Re IR1   Re) ,
A33 
 B1 Re( RA18 A31  IA 18 A30 )
A302  A312
,
A35 
 B1 Re( IA18 A37  RA18 A39 )
A302  A312
,
B1 Re(RA18 A30  IA18 A31
A302  A312
,
A34 
A36  ( RR22  IR22   Re RR1 )
,
B1 Re( RA19 A36  IA19 A37 )
A38 
A362  A372
,
B Re( RA19 A37  IA 19 A36 )
A40  1
A362  A372
,
A37  ( IP2 RP2   Re IP2   Re) ,
A39 
 B1 Re( RA19 A37  IA 19 A36 )
A362  A372
,
A41 
 B1 Re( IA19 A37  RA19 A36 )
A362  A372
,
B1 Re( RA18 A31  IA 18 A30 )
A302  A312
,
A42  A48  [( A43  A49 )e R1  R2  iA44e ( R1   Re)  {iA45  i(

A45  A47 )  A46 }e R1 ]

,

A  A47 )  A46 }(1  e 2 R1 )}
 45
(e(2 R1  R2 )  e R2 )
,
iA44 (e Re  e (2 R1  Re) )  iA45 (1  e 2 R1 )  {i (
A43  A49 
A44 
A47 
( A15 2 Re  B 2 E1 )
2A16
 2 2 A16 
, A45 
, A46 
 A45



2
,
iA17   B2 E2
BA
BA
, A48  2 2 18
, A49  2 2 19
R1
R2

  R1  i
  R2  i
Re
Re
,
A50  e  R3 ( A51e R4  A52e Re  A53 )
A51 
e2 R3  A52 (e 2 R3  Re  e Re )  A53 (e2 R3  1)
(e 2 R3  R4  e R4 )
,
Copyright © 2013 SciResPub.
A52 
Ec 2 A 22 Pr Re
2 2 Re(2  Pr)  S Pr ,
International Journal of Advancements in Research & Technology, Volume 2, Issue3, March-2013
ISSN 2278-7763
A53 
22
EcA
Re S  ,
A54   A56 e R3  R4  ( A64  A66 )e R3  2  Re  A65 e R3  2  Re  A67 e R3  Re
 ( A68  A69 )e R3  Re  ( A70  A71  A72  A73  A74  A75 )e R3
( A64  A66 )(e 2 R3 2  Re  e2  Re )  A65 (e 2 R3 2  Re  e 2  Re )  A67 (e2 R3   Re  e Re )  ( A68  A69 )(e 2 R3   Re  e Re )
A56 
(e 2 R3  R4  e R4 )
 ( A70  A73 )(e 2 R3  1)  ( A71  A72  A74  A75 )(e 2 R3  1)

A2 B 
A57  Ec  2 A2 A8  2 Re  2 A2 A9  2 2 
Be 

,
A58  Ec  2 A2 A9  2 Re 
4 EcAA10
,
 Re
A62 
 2 AA11 A B2 
A63  Ec 


Be 
 Re 
,
A58
(4 Re  2 R22 Pr  S Re Pr)
2
A66 
2
,
A59 (2 / Pr   )
A67 
2
(2 Re  2 2 Re 2 Pr  S Re Pr) 2 ,
A69 
A60
 1

 2 Re   1  S
 Pr 
,
A71 
A61
S2
A73 
2 A62
S2 ,
,
Copyright © 2013 SciResPub.
4EcA2 B2
2
,
A64 
2
3
A65 
,
AA9 A2 A3 B2 

A60  2 Ec  A2 A11  AA8 

 Re
Re  ,

A AB 

A59  2 Ec  2 A2 A10  AA9  22 2 
 Be  ,

A61 
,
A74 
A57
 2

2 2 Re   1  S
Pr


,
A58 Re(4  Pr)
(4 Re 2 2 Re 2 Pr  S Re Pr)2 ,
2
 2

A59 

 Pr

A68 
2
2
2
(2 Re   Re Pr  S Re Pr)2 ,
A70 
A61
S ,
A72 
A62
S ,
2 A62
S 2  Re Pr ,
International Journal of Advancements in Research & Technology, Volume 2, Issue3, March-2013
ISSN 2278-7763
A75 
A63
S
A77 
p1  q1e 2 R5
(e R6  e 2 R5  R6 ) ,
A76   A77 e R6  R5  p1e  R5
,
A80  2 EcE1 2 Re A2
2 EcAA18 R1
 Re
,
A84  A50 R3 ,
A86  A52 2 R3 ,
A85  A51 R4 ,
A89 
A91 
A78
2
( R1   Re)
  ( R1   Re)  ( S  i )
Re Pr
,
A82
R
  R2  ( S  i )
Re Pr
,
R42
  R4  ( S  i )
Re Pr
,

A79
2
( R2   Re)
  ( R2   Re)  (S  i )
Re Pr
,
A50  R3
2
3
R
  Re ( S  i )
Re Pr
,
A52  2 Re  iA83
A92 
 1

 2 Re   1  ( S  i )
 Pr 
,
A51 R4
A93   A92 e R5  R6  p2
A88 
A90 
2
2

,
A82  2EcAE1 ,
A83  E1 EcA ,
A87 
,
A78  2 EcA2 A18 R1 ,
A79  2 EcA2 A19 ,
A81 
23
( p2 e 2 R5  q2 )
A94  2 R5  R6
(e
 e R6 ) ,
,
 2 RR2

{( RD61  RD69 )C3  ( ID61  ID69 )C4 } 
   IR2
 Re Pr

RA95 
2
2
C3  C4
,
 2 RR2

{( RD61  RD69 )C4  ( ID61  ID69 )C3 }
   IR2
 Re Pr

IA95  
2
2
C3  C4
,
{( RD61  RD69 )C4  ( ID61  ID69 )C3 }(S 
RA96 
Copyright © 2013 SciResPub.
C32  C42
RR22
IR 22
  RR2 
)
Re Pr
Re Pr
International Journal of Advancements in Research & Technology, Volume 2, Issue3, March-2013
ISSN 2278-7763
24
 ( RD61  RD69 )C3  ( ID61  ID69 )C4 }  {RD61  RD69 )C4  ( ID61  ID69 )C3 }
,
( RD61  RD69 )C4  ( ID61  ID69 )C3 }  {RD61  RD69 )C4  ( ID61  ID69 )C3}{S 
IA96 
A97 
A98 
A100 
A102 
C32  C42
D62
A98 
2
 4 Re

 2 2 Re S   i

 Pr

,
D63
2
 4 Re

 2 2 Re S   i

 Pr

 4

D63 

 Pr

A99 
2
2
 4 Re

2
 2 Re  S  i 

 Pr

D63
2
 4 Re

 2 2 Re S   i

 Pr

,
D64
A101 
  2 Re

  2 Re  S   i

 Pr

,
D65
 R12

  R1  S   i

 Re Pr

D66
2
2
 R

  R2  S   i

 Re Pr

,
A103 
D71
( S  i )
A103 
D72
( S  i ) ,
A105 
D72
(S  i )
A105 
 D72
(S  i ) ,
B1 
(1  BiBe)
Re{(1  BiBe)2  Be2 } ,
B2 
Be
Re{(1  BiBe)2  Be 2}
C1   ( A25  A27 )(e 2 RR1  Re cos( IR1 )  e Re )  A24e 2 RR1  Re sin( IR1 ) 
C2   A 26 e 2 R1  Re Sin( IR1 )
,
C3  { A 32 e RR1 (cos (2 IR1 )  1)  A33e RR1 Sin(2 IR1 )}cos ( IR1 )
C4  ( A 34 (cos (2 IR1 )  1)  A35 sin(2 IR1 ))e RR1 sin( IR1 )
Copyright © 2013 SciResPub.
,
,
,
RR22
IR22
  RR2 
}
Re Pr
Re Pr ,
International Journal of Advancements in Research & Technology, Volume 2, Issue3, March-2013
ISSN 2278-7763
C5  A 38 (e 2 RR1  RR2 cos (2 IR1 )  e RR2 )  A39e 2 RR1  RR2 sin(2 IR1 )
,
C6  { A 40 (e2 RR1  RR2 cos (2 IR1 )  e RR2 )  A41e2 RR1  RR2 sin(2 IR1 )}sin( IR2 )
C7  e 2 RR1  RR2 cos (2 IR1  IR2 )  e RR2 cos ( IR2 )
,
C8  (e 2 RR1  RR2 sin(2 IR1  IR2 )  e RR2 sin( IR2 ))
,
C9   A24 (e 2 RR1  Re cos(2 IR1 )  e  Re )  ( A25  A27 )(e 2 RR1  Re sin( IR2 ) 
C10  A26 (e  Re  e RR1  Re cos (2 IR1 ))
25
,
,
,
C11  ( A32 sin(2 IR1 )  A33 (cos (2 IR1 )  1))e RR1 cos ( IR1 )
C12  ( A35 (cos (2 IR1 )  1)  A34 sin(2 IR1 ))e RR1 sin( IR1 )
,
,
C13  (( A38e 2 RR1  RR2 sin(2 IR1 )  A39 (e 2 RR1  RR2 cos(2 IR1 )  e RR2 ))cos ( IR2 )
C14  ( A41 (e 2 RR1  RR2 cos (2 IR2 )  e RR2 )  A40e 2 RR1  RR2 sin(2 IR1 )) sin( IR2 )
,
,
C15  ( A44 (e Re  e2 RR1  Re cos (2 IR2 )  A45e 2 RR1 sin(2 IR1 )  A46 (e 2 RR1 cos ( IR1 )  1)
 A47 (1  e 2 RR1 cos (2 IR1 )))
,
C16   A44 (e 2 RR1  Re sin(2 IR2 ))  A45 (e 2 RR1 cos (2 IR1 )  1)  A46 e2 RR1 sin(2 IR1 )
 A47 (1  e2 RR1 cos (2 IR1 ))
,
C17  (e RR2 cos ( IR2 )  e 2 RR1  RR2 cos (2 IR1  IR2 ))
C18  (e RR2 sin( IR2 )  e 2 RR1  RR2 sin(2 IR1  IR2 ))
,
RD1   Re A2 ( RA22 RR1  IA22 IR1 ) ,
ID1   Re A2 ( RA22 IR1  IA22 RR1 ) ,
RD2   Re A2 ( RA23 RR2  IA23 IR2 ) ,
ID2   Re A2 ( RA23 IR2  IA23 RR2 ) ,
RD 3  A2  2 Re 2 ( A25  A27 ) ,
ID 3  A2  2 Re 2 ( A26  A24 ) ,
RD 4   A2  2 Re 2 IA26 ,
ID 4   A2 RA26  2 Re 2 ,
RD5   A2  Re RA28 ,
ID5  A2  Re IA28 ,
Copyright © 2013 SciResPub.
International Journal of Advancements in Research & Technology, Volume 2, Issue3, March-2013
ISSN 2278-7763
RD6  A2  Re( RA32  IA33 ) ,
ID6  A2  Re( IA32  RA33 ) ,
RD7  A2  Re( RA34  IA35 ) ,
ID7  A2  Re( IA34  RA35 ) ,
RD8  A2  Re( RA38  IA39 ) ,
ID8  A2  Re( IA38  RA39 ) ,
RD9  A2  Re( RA40  IA41 ) ,
ID9  A2  Re( IA40  RA41 ) ,
RD10 
A
( RA22 RR1  IA22 IR1 )

,
RD12  A Re( A25  A27 ) ,
ID11 
A
( RA23 RR2  IA23 IR2 )

,
ID12  A Re( A26  A24 ) ,
D14 
D13  A Re A26 ,
26
A
A28

,
RD15 
A
A32

,
ID15 
A
A33

,
RD16 
A
A34

,
ID16 
A
A35

,
RD17 
A
A38

,
ID17 
A
A39

,
RD18 
A
A40

,
ID18 
A
A41

,
RD19  A8 Re( RR1 RA18  IR1 IA18 ) ,
ID19  A8 Re( RR1 IA18  IR1 RA18 ) ,
RD 20  A8  Re( RR2 RA19  IR2 IA19 ) ,
ID20  A8 Re( RR2 IA19  IR2 RA19 ) ,
D21  E1 2 Re A8
,
RD22  A9 ( RA18 RR1  IA18 IR1 ) ,
RD23   Re A9 ( RR2 RA19  IR2 IA19 ) ,
D24  E1 A9  Re ,
ID25   Re A9 ( RR1IA18  IR1RA18 )
,
ID26   Re A9 ( RR2 RA19  IR2 IA19 ) ,
Copyright © 2013 SciResPub.
ID22  A9 ( RA18 IR1  IA18 RR1 ) ,
ID23   Re A9 ( RR2 IA19  IR2 RA19 ) ,
RD25   Re A9 ( RR1 RA18  IR1 IA18 )
,
RD26   Re A9 ( RR1 RA19  IR2 IA19 ) ,
ID27  E1 A9 2 Re 2
International Journal of Advancements in Research & Technology, Volume 2, Issue3, March-2013
ISSN 2278-7763
27
RD28  2 A10 ( RR1 RA18  IR1IA18 ) ,
ID28  2 A10 ( RR1IA18  IR1 RA18 ) ,
RD29  2 A10 ( RR2 RA19  IR2 IA19 ) ,
ID29  2 A10 ( RR2 IA19  IR2 RA19 ) ,
D30  2 A10 E1 Re ,
ID31  A11 ( RR1RA18  IR1 RA18 ) ,
ID32  A11 ( RR2 IA19  IR2 RA19 ) ,
ID33  E1  Re A11 ,
RD35  A3 IA19 ,
ID35  A3 IA19 , D36  A3 E1 ,
ID38  A2 IA18 , RD39  A2 RA19 ,
ID39  A2 IA19 ,
RD31  A11 ( RR1 RA18  IR1IA18 )
,
RD32  A11 ( RR2 RA19  IR2 IA19 )
,
RD34  A3 RA18
D37  E 2 A3 ,
D40  A3 E1 ,
,
ID34  A3 IA18 ,
RD38  A2 RA18 ,
D41  A3 E 2
,
RD42  A2 RA18 , ID42  A2 IA18 ,
RD43  A2 RA19 ,
ID 43  A2 IA19 ,
D44  A2 E1 ,
AE1
A
A
A
RD46  RA18
RD47  RA19
ID47  IA19
D48 
D45  E 2 A2 ,




,
,
,
,
E A
D49  2
D51  R 4 A56 , D52  22 Re( A64  A66 ) , D53  A65  ,

, D50   Re 3 A55 ,
D54  2 A65 2 Re , D55  A67  ,
D56  22 Re A67 , D57  2 Re( A68  A69 ) , D58  2A72 ,
D59   ( A70  A73 ) ,
2 Ec
2 Ec
 2 Ec

RD60   
RD1 
RD19 
RD22  B3 RD38  B3 RD42 
Re
Re
 Re
 ,
2 Ec
2 Ec
 2 Ec

ID60   
ID1 
ID19 
ID22  B3 ID38  B3 ID42 
Re
Re
 Re
,
2 Ec
2 Ec
 2 Ec

RD61   
RD2 
RD20 
RD23  B3 RD39  B3 RD43 
Re
Re
 Re
 ,
2 Ec
2 Ec
 2 Ec

ID61   
ID2 
ID20 
ID23  B3 ID39  B3 ID43 
Re
Re
 Re
 ,
2 Ec
 2 Ec

RD62   
RD3 
RD21  D52  RD53 
Re
 Re
 ,
2 Ec
 2 Ec

ID62   
ID3 
ID21  B3 D44 
Re
 Re
 ,
2 EC
2 Ec
 2 Ec

RD63   
RD4 
RD27 
D54 
Re
Re
 Re
 ,
Copyright © 2013 SciResPub.
International Journal of Advancements in Research & Technology, Volume 2, Issue3, March-2013
ISSN 2278-7763
28
2 EC
 2 Ec

ID63   
ID4 
ID27 
Re
 Re
 ,
2 Ec
2 Ec
2 Ec
2 Ec
 2 Ec

RD64   
RD5 
RD12 
RD13 
D24 
ID33  D55  D57 
Re
Re
Re
Re
 Re
,
2 Ec
2 Ec
2 Ec
2 Ec
 2 Ec

RD64   
ID5 
ID12 
ID13 
D30 
D33  B3 D36  B3 D40  B3 D 45  B3 D48 
Re
Re
Re
Re
 Re
 ,
2 Ec
 2 Ec

RD65   
RD10 
RD31  B3 RD34 
Re
 Re
,
2 EC
 2 Ec

ID65   
ID10 
ID31  B3 ID34 
Re
 Re
,
2 EC
 2 Ec

RD66   
RD11 
RD32  B3 RD35 
Re
 Re
 ,
2 Ec
 2 Ec

ID66   
ID11 
ID32  B3 ID35 
Re
 Re
 ,
RD67  
2 Ec
( RD15 RR1  RD16 IR1 )
Re
,
ID67  
2 Ec
( ID15 RR1  ID16 IR1 )
Re
,
RD68  
2 Ec
( RD15 IR1  RD16 RR1 )
Re
,
ID68  
2 Ec
( ID15 IR1  RD16 RR1 )
Re
,
RD69  
2 Ec
( RD17 RR2  RD18 IR1 )
Re
,
ID69  
2 Ec
( ID17 RR1  ID18 IR2 )
Re
,
RD70  
2 Ec
( RD17 IR2  RD18 RR2 )
Re
,
ID70  
2 Ec
( ID17 IR2  ID18 RR2 )
Re
,
RD71  
2 Ec
D14  D 59
Re
, ID71  B3 D41 ,
Copyright © 2013 SciResPub.
E1 
A2 Re

,
E2  A /  .