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Int. J. of Appl. Math and Mech. 8 (6): 49-68, 2012.
MATHEMATICAL MODELLING OF SORET AND HALL EFFECTS
ON OSCILLATORY MHD FREE CONVECTIVE FLOW OF
RADIATING FLUID IN A ROTATING VERTICAL POROUS
CHANNEL FILLED WITH POROUS MEDIUM
R. Kumar1
and K. D. Singh2
1 Department of Mathematics, Govt. College for Girls (RKMV), Shimla-17 1001, India.
2 Department of Mathematics (ICDEOL), H.P. University, Shimla-171 005, India.
Email: [email protected]
Received 29 March 2011; accepted 28 November 2011
ABSTRACT
Heat and mass transfer in MHD (Magnetohydrodynamic) oscillatory free convective flow of a
viscous incompressible fluid through a highly porous medium bounded between two infinite
vertical porous plates has been studied. The two porous plates are subjected to a constant
injection and suction. A uniform magnetic field is applied in the direction normal to the
planes of the plates. The entire system rotates about the axis normal to the planes of the plates
with uniform angular velocity . It is also assumed that the conducting fluid is gray,
absorbing-emitting radiation and non-scattering. Using perturbation technique, the
dependence of steady and unsteady resultant velocities and their phase differences on variousparameters are obtained. The effects of thermal radiation, Soret number and Hall current have
been discussed.
Keywords: Oscillatory, Porous medium, Vertical porous channel, MHD, Hall current,
Thermal radiation, Soret number.
1 INTRODUCTION
Radiative convective flows have gained attention of many researchers in recent years. This is
justified by the fact that the radiative flows of an electrically conducting fluid with hightemperature in the presence of magnetic field plays a vital role in many engineering, industrial
and environment processes e.g. heating and cooling chambers, fossil fuel combustion energy
processes, evaporation from large open water reservoirs, astrophysical flows, solar power
technology and space vehicle re-entry. More applications and a good insight into the subject
are given by Rashad (2009), Sanyal and Adhikari (2006), Muthucumaraswamy and
Kulandaivel (2008), Prasad and Reddy (2008b), Singh and Kumar (2010) and Raptis and
Perdikis (1999). Chamkha (2000) considered the problem of steady, hydromagnetic boundary
layer flow over an accelerating semi-infinite porous surface in the presence of natural
radiation, buoyancy and heat generation or absorption. Analytical model of MHD mixed
convective radiating fluid with viscous dissipative heat have been presented by Ahmed and
Batin (2010). Soundalgekar (1984) investigated oscillatory MHD flow and heat transfer
effects on the channel. Ali et al. (1984) studied the radiation effect on free convection
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R. Kumar and K. D. Singh
Int. J. of Appl. Math and Mech. 8 (6): 49-68, 2012.
50
boundary layer flow over horizontal surfaces, using the Rosseland diffusion approximation.
Soundalgekar and Takhar (1993) have studied radiation effects on free convection flow of a
gas past a semi-infinite flat plate. Theoretical analysis of radiative effects on transient free
convection heat transfer past a hot vertical surface in porous media was presented by Ghosh
and Beg (2008). Kim and Fedorov (2003) studied transient mixed radiative convection flowof a micropolar fluid past a semi-infinite vertical porous plate. Hossain and Rees (1998)
investigated free convection from isothermal inclined plates to horizontal plates. The
interaction of free convection and radiation on boundary layer flows with fluid suction
through the porous wall was investigated by Hossain et al. (1999). Yih (1999) studied the
radiation effect on natural convection about a truncated cone. EL-Hakim and Rashad (2007)
used Rosseland diffusion approximation in studying the effect of radiation on free convection
from a vertical cylinder embedded in a fluid-saturated porous medium. Raptis and Massalas
(1998) analyzed the effects of radiation on the oscillatory flow of a gray gas, absorbing-
emitting in the presence of induced magnetic field. Beg and Ghosh (2010) investigated an
analytical study for MHD flow of radiating fluid with oscillatory surface temperature and
secondary flow effects. Prasad and Reddy (2008b) studied radiation and mass transfer effectson an unsteady MHD free convection flow past a semi-infinite plate through porous medium.
On the other hand, simultaneous heat and mass transfer from different geometries embedded
in porous media has many engineering and geophysical applications such as drying of porous
solids, thermal insulations, cooling of nuclear reactors and underground energy transport.
Attia and Kotb (1996) investigated the two dimensional MHD flow between two porous,
parallel and infinite plates. Recently Singh and Mathew (2008) studied the injection/suction
effect on a hydromagnetic oscillatory flow in a horizontal channel in a rotating system. Singh
(2004) studied the effects of transversely applied uniform magnetic field on oscillatory flow.
When the strength of the magnetic field is strong, one cannot neglect the effects of Hall
current. The Hall current gives rise to a cross flow making the flow three-dimensional. Very
recently,Rajesh and Varma (2010) investigated heat source effects on MHD flow past an
exponentially accelerated vertical plate with variable temperature through a porous medium.
Radiation effect on convective heat transfer through a porous medium in a vertical channel
with quadratic temperature variation has been studied by Reddaiah and Prasada Rao (2010).
Alam et al. (2009) investigated the effects of variable reaction, thermophoresis and radiation
on MHD free convection flows.
If two regions in a mixture are maintained at different temperatures so that there is a flux of
heat, it has been found that a concentration gradient is set up. In a binary mixture, one kind of
a molecule tends to travel toward the hot region and the other kind toward the cold region.This is called the “Soret effect”. Eckert and Drake (1972) have pointed out that in a
convective fluid when the flow of mass is caused by a temperature difference one cannot
neglect the thermal diffusion effect (commonly known as Soret effect) due to its practical
application in engineering and science. Usually this effect has a negligible influence on mass
transfer, but it is useful in the separation of certain mixtures. Thermal diffusion effect or Soret
effect has been utilized for isotope separation and in mixtures between gases with very light
molecular weight ) He , H ( 2 and medium molecular weight ( 2 N , air) and it was found to be
of a magnitude that it cannot be neglected. More physical insight into the problem is given by
Sparrow and Cess (1962) and Renuka et. al. (2009). Reddy and Reddy (2010) investigated
Soret and Dufour effects on steady MHD free convective flow past an infinite plate. Soret
effects due to natural convection between heated inclined plates have been investigated byRaju et al. (2008).
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Mathematical Modelling of Soret and Hall Effects
Int. J. of Appl. Math and Mech. 8 (6): 49-68, 2012.
51
As the importance of radiation in the fields of aerodynamics as well as in space science
technology, the present study is motivated towards this direction. The objective of the present
paper is to analyze the effects of radiation, Soret, permeability variation and injection/suction
on heat and mass transfer of free convective flow through porous medium in the presence of
Hall current in a vertical porous channel when the entire system rotates about an axisperpendicular to the planes of the plates. In this work we assume that the temperature
differences with in the flows are sufficiently small so that4
*T may be expressed as a linear
function of temperature.
2 MATHEMATICAL MODEL
Consider an unsteady flow of an electrically conducting, viscous, incompressible fluid
through a highly porous medium bounded between two insulated infinite vertical porous
plates at distance d apart in the presence of thermal radiation. A constant injection velocity,
0w , is applied at the stationary plate 0* z and the same constant suction velocity,
0w , is
applied at the plate d z* , which is oscillating in its own plane with a velocity )t (U **
about a non-zero constant mean velocity0
U . The origin is assumed to be at the plate 0* z
and the channel is oriented vertically upward along the * x axis. The channel rotates as a
rigid body with uniform angular velocity * about the * z axis. A strong magnetic field of
uniform strength 0 H is applied along * z axis taken perpendicular to the planes of the plates.
The magnetic Reynolds number is considered to be small so that the induced magnetic field is
neglected. It is also assumed that the radiation heat flux in the * x direction is negligible as
compared to that in the
*
z direction. The temperature and concentration oscillate about aconstant mean
0T and
0C respectively at the plate 0* z and the species concentration is
assumed at low level. The fluid considered here is a gray, absorbing-emitting radiation but a
non-scattering medium. All physical quantities depend on*
z and *t for this problem of fully
developed laminar flow. The physical configuration of the problem is shown in Figure1.
The equation of continuity 0
V . , gives on integration0ww
* where )w ,v ,u(V ***
and
solenoidal relation for the magnetic field 0
H . , gives0 H H
*
z (constant) everywhere in
the flow field. The equation of conservation of electric charge 0
J . gives
*
z J constant.This constant is zero i.e. 0 J * z at the plates which are electrically non-conducting. Taking
Hall current into account the generalized Ohm’s law (Cowling [3]) is of the form is
) H V E ( H J H
J e
ee
0
, (1)
where
V is the velocity vector,
H is the magnetic field,
J is the current density,
E is the
electric field, is the electrical conductivity, e is the magnetic permeability, e is the
cyclotron frequency, and e is the electron collision time.
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R. Kumar and K. D. Singh
Int. J. of Appl. Math and Mech. 8 (6): 49-68, 2012.
52
Figure 1: Physical configuration of the problem.
For very large magnetic field, the*
x and*
y components of Ohm’s law (2.1), which includeHall current, are
)v H E ( J J *
e
*
x
*
yee
*
x 0
and
)u H E ( J J *
e
*
y
*
xee
*
y 0 .
Since the external electric field arising due to polarization of charges is negligible. Hence
0* y
* x E E . Therefore, solving for *
x J and * y J , we get
)m(
)vmu( H J
**
e*
x 2
0
1
and
)m(
)umv( H J
**
e*
y 2
0
1
. (2)
Thus within the frame work of these assumptions and making use of (2), the heat and mass
transfer flow of radiative fluid in the presence of Hall current through porous medium is
governed by the following equations:
*
***
e**
*
*
*
*
*
*
*
*
K
u
)m(
)umv( H v
z
u
x
p
z
uw
t
u
2
2
0
22
01
21
2
)C C (g)T T (g d
*
d
* , (3)
z
y
0 H
Porous Medium
x
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Mathematical Modelling of Soret and Hall Effects
Int. J. of Appl. Math and Mech. 8 (6): 49-68, 2012.
53
*
***
e**
*
*
*
*
*
*
*
*
K
v
)m(
)vmu( H u
z
v
y
p
z
vw
t
v
2
2
0
22
01
21
2, (4)
*
*
p*
*
p
*
*
*
*
zq
C zT
C k
zT w
t T
12
2
0 , (5)
22
2
1
2
0 *
*
*
*
*
*
*
*
z
T D
z
C D
z
C w
t
C
, (6)
where is the fluid density, *t is the time, *K is the permeability of the porous medium,
eem is the Hall parameter, is volumetric coefficient of thermal expansion,
is
volumetric coefficient of thermal expansion with concentration, k is thermal conductivity, pC
is specific heat at constant pressure, *q is the radiative heat flux, D is molecular diffusivity
and D1 is thermal diffusivity.
The boundary conditions for the problem are
0 0
0 0
0
0
0
1 0
* * * * *
d
*
* * *
d
* * * * * *
*
* *
d d
u v ,T T ( T T ) cos t ,
at z
C C ( C C ) cos t
u U ( t ) U ( cos t ), v ,
at z d
T T , C C
(7)
where * is the frequency of oscillations,0
U is the mean velocity,0
T is the mean
temperature,0
C is the mean concentration and is a very small positive quantity.
The quantity*
q on the right-hand side of equation (5) represents the radiative heat in the * z
direction. The local radiant for the case of an optically thin gray gas is expressed by
)T T (a z
q *
d
*44
4
, (8)
where
a is the absorption coefficient and is the Stefan-Boltzmann constant.
We assume that the temperature differences with in the flow are sufficiently small such that4
*T may be expressed as a linear function of the temperature. This is accomplished by
expanding4*
T in a Taylor series about d T and neglecting higher-order terms, thus
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R. Kumar and K. D. Singh
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54
4334
4
d
*
d
* T T T T . (9)
By using equations (8) and (9), equation (5) reduces to
p
*
d d
*
*
*
p
*
*
*
*
C
)T T (T a
z
T
C
k
z
T w
t
T
3
2
2
0
16. (10)
Eliminating the modified pressure gradient under the usual boundary layer approximations,
equations (3) and (4) become
)m(
)U umv( H v
z
u
t
U
z
uw
t
u***
e**
*
*
*
*
*
*
*
*
2
2
0
22
01
22
)C C (g)T T (gK
)U u(d
*
d
*
*
**
, (11)
*
****
e***
*
*
*
*
*
*
K
v
)m(
v)U u(m H )U u(
z
v
z
vw
t
v
2
2
0
22
01
22
. (12)
Introducing the following non-dimensional quantities ,d
z*
,t t ** ,
U
uu
*
0
0
U
vv
*
,
2d *
is the rotation parameter,
2d
*
is the frequency parameter,2d
K K
*
is the
permeability parameter,
d w
0 is the injection/suction parameter,
d H M
0 is the
Hartmann number, D
Sc
is the Schmidt number,
)C C (
)T T ( DS
d
d
0
01
0
is the Soret number,
k
C Pr
p
is the Prandtl number,2
00
0
wU
)T T (gGr d
is the thermal Grashoff number,
2
00
0
wU
)C C (gGm d
is the mass Grashoff number,
d
d
*
T T
T T
0
,
d
d
*
C C
C C C
0
and
p
d
*
C
d T a R
2316
is the radiation parameter, into equations (6), (10), (11) and (12) and taking
ivuq , we get
C GmGr )U q(Sdt
dU qq
t
q 22
2
2
, (13)
R
Pr t
2
21
, (14)
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Mathematical Modelling of Soret and Hall Effects
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55
2
2
2
21
C
S
C
t
C
c
, (15)
where
K m
im M iS
1
1
12
2
2 .
The boundary conditions (7) can be written in complex notation as
0 1 1 02 2
1 0 0 12
it it it it
it it
q , ( e e ), C ( e e ) at
q U( t ) ( e e ), , C at
. (16)
3 METHOD OF SOLUTION
Now we look for a solution of equations (13) (14) and (15) under the boundary conditions
(16) of the form
it it e)(qe)(q)(q)t ,(q
210
2
, (17)
it it e)(e)()()t ,(
2102
, (18)
it it e)(C e)(C )(C )t ,(C
2102
. (19)
Substituting equations (17), (18) and (19) into equations (13), (14) and (15) and comparing
the harmonic and non-harmonic terms, we get
0
2
0
2
000 C GmGr SSqqq , (20)
1
2
1
2
111C GmGr )iS(q)iS(qq , (21)
2
2
2
2
222C GmGr )iS(q)iS(qq , (22)
0000
Pr RPr , (23)
0111 )i R(Pr Pr , (24)
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R. Kumar and K. D. Singh
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56
0222 )i R(Pr Pr , (25)
0000
cc SSC SC , (26)
10111 ccc SSC SiC SC , (27)
20222
ccc SSC SiC SC . (28)
The corresponding transformed boundary conditions are
0 1 2 0 1 2 0 1 2
0 1 2 0 1 2 0 1 2
0 1 1 0
1 0 0 1
q q q , , C C C at
q q q , , C C C at
(29)
where dashes denote differentiation with respect to ' ' .
The solutions of equations (20) to (28) under the boundary conditions (29) are obtained as:
1221
21
10
mmmm
mmee
ee)(
, (30)
3443
43
11
mmmm
mmee
ee)(
, (31)
5665
65
12
mmmm
mmee
ee)(
, (32)
21
2121210
1 mm
mm
Se Ae A
eee B B)(C c
, (33)
43
43
87
43431
1 mm
mm
mme Ae A
eee Be B)(C
, (34)
65
65
109
65652
1 mm
mm
mme Ae A
eee Be B)(C
, (35)
1 2
1 2
1 2
1 2
0 7 8
2
7 8 9 10
11
1c
n n
n n
S m m
m m
q ( ) B e B ee e
A A e A e A ee e
, (36)
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57
3 4
3 4
3 7 84
3 4
1 9 10
211 12 13 14
11
1
n n
n n
m m mmm m
q ( ) B e B ee e
A e A e A e A ee e
, (37)
5 6
5 6
5 6 9 10
5 6
2 11 12
2
15 16 17 18
11
1
n n
n n
m m m m
m m
q ( ) B e B ee e
A e A e A e A ee e
. (38)
4 RESULTS AND DISCUSSIONS
Now for the resultant velocities and shear stresses of the steady and unsteady flow, we write
)(q)(iv)(u 000
(39)
and
it it e)(qe)(q)(iv)(u
2111 . (40)
The solution (36) corresponds to the steady part which gives0
u as the primary and0
v as the
secondary velocity components. The amplitude and the phase difference due to these primary
and secondary velocities for the steady flow are given by
2
0
2
00vu R , )u / v(tan 00
1
0
. (41)
Similarly the solutions (37) and (38) together give the unsteady part of the flow. The unsteady
primary and secondary velocity components )(u 1 and )(v 1 , respectively, for the
fluctuating flow can be obtained as
t sin)(q Im)(q Imt cos)(qal Re)(qal Re)t ,(u 21211 , (42)
t cos)(q Im)(q Imt sin)(qal Re)(qal Re)t ,(v 21211 . (43)
The resultant velocity or amplitude and the phase difference of the unsteady flow are given by
2
1
2
11vu R , )u / v(tan
11
1
1
. (44)
The resultant velocities0
R ,1
R and the phase angles10
, for the steady and unsteady part of
the flow are respectively shown graphically in Figures (2) to (5). To be realistic the two
values of Prandtl number Pr as 0.71 and 7.00 are chosen to represent air and water
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Int. J. of Appl. Math and Mech. 8 (6): 49-68, 2012.
58
respectively. The values of Schmidt number cS are taken for water-vapor ( cS =0.60) and
Ammonia ( cS =0.78). Figures (2) and (3) for the steady and unsteady resultant velocities,0
R
and1
R respectively show that both these resultants increase with the increase of Grashoff
number Gr or mass thermal Grashoff number Gm or Hartmann number M or Hall parameterm or injections/suction parameter or permeability of the porous medium K or Soret
number0
S or Schmidt number cS , while these resultants decrease with the increase of
radiation parameter R or Prandtl number Pr . It is interesting to note that with the increase of
the rotation of the channel , both these resultants increase near the stationary plate and then
decrease near the oscillating plate. Figure (3) shows clearly that1
R decreases with the
increase of frequency of oscillation .
It is also found from Figures (4) and (5) that the phases differences0
and1
for the steady
and unsteady parts of the flow respectively decrease with increasing Gr or Gm or M or or
or 0S or cS . However, 0 and 1 both increase with increasing m or R or Pr or K . Figure
(5) shows clearly that1
increases with the increase of frequency of oscillations .
For the steady flow the amplitude and the phase difference of shear stresses at the stationary
plate 0 can be obtained as
) / (tan , x yr y xr 00
1
0
2
0
2
00 , (45)
Where
0
0
00
q
i y x.
Here x0 and y0
are, respectively, the shear stresses at the stationary plate due to the primary
and secondary velocity components. The numerical values for the resultant shear stress r 0
and the phase angle r 0 are listed in Table 1. This Table shows that r 0
and r 0 goes on
increasing with increasing rotation of the channel. The amplitude of shear stress r 0 also
increases with the increase of Gr or Gm or M or or 0S or cS and decreases with the
increase of m or K . The increase of these flow parameters has opposite effect on r 0 . It is
interesting to note that both x0 and r 0 increase with the increase of radiation parameter R
and Prandtl number Pr .
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59
Figure 2: Resultant velocity0
R due to0
u and0
v .
Figure 3: Resultant velocity1
R due to1
u and1
v at4
t .
0 R
1 R
Gr Gm M m R Pr K S0 Sc
5 4 2 1 2 0.2 10 0.71 1 1 0.60 5 I
15 4 2 1 2 0.2 10 0.71 1 1 0.60 5 II5 12 2 1 2 0.2 10 0.71 1 1 0.60 5 III
5 4 4 1 2 0.2 10 0.71 1 1 0.60 5 IV
5 4 2 3 2 0.2 10 0.71 1 1 0.60 5 V5 4 2 1 10 0.2 10 0.71 1 1 0.60 5 VI
5 4 2 1 2 1.0 10 0.71 1 1 0.60 5 VII
5 4 2 1 2 0.2 20 0.71 1 1 0.60 5 VIII5 4 2 1 2 0.2 10 7.00 1 1 0.60 5 IX
5 4 2 1 2 0.2 10 0.71 5 1 0.60 5 X
5 4 2 1 2 0.2 10 0.71 1 5 0.60 5 XI5 4 2 1 2 0.2 10 0.71 1 1 0.78 5 XII
5 4 2 1 2 0.2 10 0.71 1 1 0.60 15 XIII
5 4 2 1 2 0.2 40 0.71 1 1 0.60 5 XIV5 4 2 1 2 0.2 80 0.71 1 1 0.60 5 XV
Gr Gm M m R Pr K S0 Sc
5 4 2 1 2 0.2 10 0.71 1 1 0.60 I
15 4 2 1 2 0.2 10 0.71 1 1 0.60 II
5 12 2 1 2 0.2 10 0.71 1 1 0.60 III
5 4 4 1 2 0.2 10 0.71 1 1 0.60 IV
5 4 2 3 2 0.2 10 0.71 1 1 0.60 V
5 4 2 1 10 0.2 10 0.71 1 1 0.60 VI5 4 2 1 2 1.0 10 0.71 1 1 0.60 VII
5 4 2 1 2 0.2 20 0.71 1 1 0.60 VIII
5 4 2 1 2 0.2 10 7.00 1 1 0.60 IX5 4 2 1 2 0.2 10 0.71 5 1 0.60 X
5 4 2 1 2 0.2 10 0.71 1 5 0.60 XI
5 4 2 1 2 0.2 10 0.71 1 1 0.78 XII5 4 2 1 2 0.2 40 0.71 1 1 0.60 XII
5 4 2 1 2 0.2 80 0.71 1 1 0.60 XIV
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60
Figure 4: Phase angle0
due to0
u and0
v .
Figure 5: Phase angle1
due to1
u and1
v at4
t .
0
1
Gr Gm M m R Pr K S0 Sc
5 4 2 1 2 0.2 10 0.71 1 1 0.60 5 I
15 4 2 1 2 0.2 10 0.71 1 1 0.60 5 II
5 12 2 1 2 0.2 10 0.71 1 1 0.60 5 III5 4 4 1 2 0.2 10 0.71 1 1 0.60 5 IV
5 4 2 3 2 0.2 10 0.71 1 1 0.60 5 V
5 4 2 1 10 0.2 10 0.71 1 1 0.60 5 VI5 4 2 1 2 1.0 10 0.71 1 1 0.60 5 VII
5 4 2 1 2 0.2 20 0.71 1 1 0.60 5 VIII
5 4 2 1 2 0.2 10 7.00 1 1 0.60 5 IX5 4 2 1 2 0.2 10 0.71 5 1 0.60 5 X
5 4 2 1 2 0.2 10 0.71 1 5 0.60 5 XI
5 4 2 1 2 0.2 10 0.71 1 1 0.78 5 XII5 4 2 1 2 0.2 10 0.71 1 1 0.60 15 XIII
5 4 2 1 2 0.2 40 0.71 1 1 0.60 5 XIV
5 4 2 1 2 0.2 80 0.71 1 1 0.60 5 XV
Gr Gm M m R Pr K S0 Sc
5 4 2 1 2 0.2 10 0.71 1 1 0.60 I
15 4 2 1 2 0.2 10 0.71 1 1 0.60 II
5 12 2 1 2 0.2 10 0.71 1 1 0.60 III5 4 4 1 2 0.2 10 0.71 1 1 0.60 IV
5 4 2 3 2 0.2 10 0.71 1 1 0.60 V5 4 2 1 10 0.2 10 0.71 1 1 0.60 VI5 4 2 1 2 1.0 10 0.71 1 1 0.60 VII
5 4 2 1 2 0.2 20 0.71 1 1 0.60 VIII
5 4 2 1 2 0.2 10 7.00 1 1 0.60 IX5 4 2 1 2 0.2 10 0.71 5 1 0.60 X
5 4 2 1 2 0.2 10 0.71 1 5 0.60 XI
5 4 2 1 2 0.2 10 0.71 1 1 0.78 XII5 4 2 1 2 0.2 40 0.71 1 1 0.60 XII
5 4 2 1 2 0.2 80 0.71 1 1 0.60 XIV
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Table 1: Amplitude r 0 and the phase angle r 0
due to0
u and0
v .
Gr Gm M m R Pr K 0S cS r 0
r 0
5
155
5
5
5
5
5
5
5
5
5
55
4
412
4
4
4
4
4
4
4
4
4
44
2
22
4
2
2
2
2
2
2
2
2
22
1
11
1
3
1
1
1
1
1
1
1
11
2
22
2
2
10
2
2
2
2
2
2
22
0.2
0.20.2
0.2
0.2
0.2
1.0
0.2
0.2
0.2
0.2
0.2
0.20.2
10
1010
10
10
10
10
20
10
10
10
10
4080
0.71
0.710.71
0.71
0.71
0.71
0.71
0.71
7.00
0.71
0.71
0.71
0.710.71
1
11
1
1
1
1
1
1
5
1
1
11
1
11
1
1
1
1
1
1
1
5
1
11
0.60
0.600.60
0.60
0.60
0.60
0.60
0.60
0.60
0.60
0.60
0.78
0.600.60
4.6641
4.68634.6814
5.3684
4.5637
4.6742
4.9791
6.4247
4.6693
4.6544
4.6638
4.6671
8.991212.660
0.7186
0.70490.7065
0.6313
0.7533
0.7193
0.4863
0.7527
0.7195
0.7365
0.7179
0.7186
0.77070.7795
For the unsteady part of the flow, the amplitude and the phase difference of shear stresses at
the stationary plate 0 can be obtained, for4
t , as
x
y
r y xr tan ,1
11
1
2
1
2
11
, (46)
Where
0
2
0
1
11
v
iu
i y x.
The amplitude r 1 and phase difference r 1
of the unsteady shear stresses are shown in
Figures (6) and (7). It is clear from Figures (6) and (7) that r 1 increases with the increasing
Gr or Gm or M or or or0
S . However, these flow parameters have opposite effect on
r 1 . The r 1 decreases and r 1 increases with the increase of Hall parameter m. It is interesting
to note that an increase in K or cS leads to an increase in r 1 and r 1
. It is also observed from
Figures (6) and (7) that with the increase of radiation parameter R and Prandtl number Pr , r 1
increase for smaller values of oscillations and decreases for larger values of oscillations ,
while r 1 increases with the increase of radiation parameter R or Prandtl number Pr . There
always remains a phase lead.
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62
Figure 6: The amplitude r 1 of unsteady shear stress at
4
t .
5 PARTICULAR CASES
(I) Our results reduce to the results of Singh and Kumar (2010) in the absence of free
convection, heat and mass transfer, radiation and Soret effect
(II) In the absence of porous medium, mass transfer and radiation and Soret effect, our
solutions are similar to those of Singh and Kumar (2009).
(III) Our results are found in good agreement in the absence of Hall currents ( m = 0) and
heat and mass transfer for an ordinary medium )( K as our solution reduces to the
one obtained by Singh and Mathew (2008).
r 1
Gr Gm M m R Pr K S0 Sc
5 4 2 1 2 0.2 10 0.71 1 1 0.60 I
15 4 2 1 2 0.2 10 0.71 1 1 0.60 II
5 12 2 1 2 0.2 10 0.71 1 1 0.60 III5 4 4 1 2 0.2 10 0.71 1 1 0.60 IV
5 4 2 3 2 0.2 10 0.71 1 1 0.60 V5 4 2 1 10 0.2 10 0.71 1 1 0.60 VI5 4 2 1 2 1.0 10 0.71 1 1 0.60 VII
5 4 2 1 2 0.2 20 0.71 1 1 0.60 VIII
5 4 2 1 2 0.2 10 7.00 1 1 0.60 IX
5 4 2 1 2 0.2 10 0.71 5 1 0.60 X
5 4 2 1 2 0.2 10 0.71 1 5 0.60 XI
5 4 2 1 2 0.2 10 0.71 1 1 0.78 XII5 4 2 1 2 0.2 40 0.71 1 1 0.60 XII
5 4 2 1 2 0.2 80 0.71 1 1 0.60 XIV
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63
Figure 7: The phase r 1 of unsteady shear stress at
4
t .
6 CONCLUDING REMARKS
Both0
R and1
R increase rapidly from zero near the stationary plate and then approach unity
in the form of damped oscillations. For large values of rotations and injection/suction at the
plates, a phase lag is also observed for both steady and unsteady phase angles0
and1
. The
resultant velocities0
R and1
R both increase with the increase of thermal Grashoff number,
mass Grashoff number, Hartmann number, Hall current, injection/suction parameter, Soret
number and Schmidt number, while these resultants decrease with the increase of radiation
and Prandtl number. The resultant r 0 of the steady part of the shear stress and the phase
angle r 0 both go on increasing with increasing rotation of the channel. The resultant r 1
of
the unsteady shear stress goes on increasing with the increase of thermal Grashoff number,
mass Grashoff number, Hartmann number, injection/suction parameter, Soret number and
rotation of the channel; however these flow parameters have opposite effect on unsteady
phase angle r 1 .
r 1
Gr Gm M m R Pr K S0 Sc
5 4 2 1 2 0.2 10 0.71 1 1 0.60 I
15 4 2 1 2 0.2 10 0.71 1 1 0.60 II
5 12 2 1 2 0.2 10 0.71 1 1 0.60 III5 4 4 1 2 0.2 10 0.71 1 1 0.60 IV
5 4 2 3 2 0.2 10 0.71 1 1 0.60 V
5 4 2 1 10 0.2 10 0.71 1 1 0.60 VI5 4 2 1 2 1.0 10 0.71 1 1 0.60 VII
5 4 2 1 2 0.2 20 0.71 1 1 0.60 VIII
5 4 2 1 2 0.2 10 7.00 1 1 0.60 IX5 4 2 1 2 0.2 10 0.71 5 1 0.60 X
5 4 2 1 2 0.2 10 0.71 1 5 0.60 XI
5 4 2 1 2 0.2 10 0.71 1 1 0.78 XII
5 4 2 1 2 0.2 40 0.71 1 1 0.60 XII
5 4 2 1 2 0.2 80 0.71 1 1 0.60 XIV
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64
APPENDIX
2
422
1
Pr RPr Pr m
,
2
422
2
Pr RPr Pr m
,
2
422
3
)i RPr(Pr Pr m
,
2
422
4
)i RPr(Pr Pr m
,
2
422
5
)i RPr(Pr Pr m
,
2
422
6
)i RPr(Pr Pr m
,
2
422
7
cccSiSS
m
,
2
422
8
ccc SiSSm
,
2
422
9
ccc SiSSm
,
2
422
10
ccc SiSSm
,
2
42
1
Sn
,
2
42
2
Sn
,
2
42
3
)iS(n
,
2
42
4
)iS(n
,
2
42
5
)iS(n
,
2
42
6
)iS(n
,
c
m
c
Sm
emSS A
1
10
1
2
,c
m
c
Sm
emSS A
2
20
2
1
,cc
m
c
SimSm
emSS A
3
2
3
2
30
3
4
,
cc
m
c
SimSm
emSS A
4
2
4
2
40
4
3
,cc
m
c
SimSm
emSS A
5
2
5
2
50
5
6
,
cc
m
c
SimSm
emSS A
6
2
6
2
60
6
5
,
S
BGm A 1
7 ,
S)S(S
BGm A
cc
12
2
8
,
1
1
2
1
92
1 AGmeGr
Smm A
m
, 2
1
2
1
101
1 AGmeGr
Smm A
m
,
3
3
2
3
114
1 AGmeGr
)iS(mm A
m
, 4
4
2
4
123
1 AGmeGr
)iS(mm A
m
,
)iS(mm
BGm A
7
2
7
3
13 ,)iS(mm
BGm A
8
2
8
4
14 ,
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65
5
5
2
5
156
1 AGmeGr
)iS(mm A
m
,
6
6
2
6
165
1
AGmeGr )iS(mm Am
,
)iS(mm
BGm A
9
2
9
5
17 ,
)iS(mm
BGm A
10
2
10
6
18 ,
21
1211 11
1 mms
S ee Ae A Ae Bc
c
,
)e( A)e( Ae
Bmm
Sc
21 1111
1212
,
)ee( A)ee( Aeee
Bmmmmm
mm
84388
87433
1
,
)ee( A)ee( Aeee
Bmmmmm
mm
47737
87434
1
,
)ee( A)ee( Aeee
Bmmmmm
mm
10651010
109655
1
,
)ee( A)ee( Aeee
Bmmmmm
mm
69959
109656
1
,
)ee( A)ee( A)ee( A)e( Ae BnmmnnSnn c 2212222
109877 1 ,
)ee( A)ee( A)ee( A)e( Ae B
mnnmSnnn c 2111111
109878 1
,
)ee( A)ee( A)ee( A)ee( Ae Bnmnmnmmnn 484744344
141312119 ,
)ee( A)ee( A)ee( A)ee( Ae Bmnmnmnnmn 837343333
1413121110 ,
)ee( A)ee( A)ee( A)ee( Ae Bnmnmnmmnn 6106966566
1817161511 ,
)ee( A)ee( A)ee( A)ee( Ae Bmnmnmnnmn 1059565555
1817161512 .
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66
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