hall effects on magneto hydrodynamic flow past an
TRANSCRIPT
HALL EFFECTS ON MAGNETO HYDRODYNAMIC FLOW PAST
AN EXPONENTIALLY ACCELERATED VERTICAL PLATE IN A
ROTATING FLUID WITH MASS TRANSFER EFFECTS.
Thamizhsudar.M1, Prof (Dr.) Pandurangan.J
2 1Assistant Professor
2H.O.D
1,2Department of mathematics
1Aarupadai Veedu Institute of Technology, Paiyanoor-603104, Chennai, Tamil Nadu
1India [email protected]
ABSTRACT
The theoretical solution of flow past an
exponentially accelerated vertical plate in the presence
of Hall current and Magneto Hydrodynamic relative to
a rotating fluid with uniform temperature and variable
mass diffusion is presented. The dimensionless
equations are solved using Laplace method. The axial
and transverse velocity, temperature and concentration
fields are studied for different parameters such as Hall
parameter,Hartmann number, Rotation parameter,
Schmidt number, Prandtl number, thermal Grashof
number and mass Grashof number.It has been observed
that the temperature of the plate decreases with
increasing values of Prandtl number and the
concentration near the plate increases with decreasing
values of Schmidt number. It is also observed that Axial
velocity increases with decreasing values of Magnetic
field parameter, Hall parameter and Rotation
parameter, whereas the transverse velocity increases
with increasing values of Rotation parameter and
Magnetic parameter but the trend gets reversed with
respect to the Hall parameter. The effects of all
parameters on the axial and transverse velocity profiles
are shown graphically.
Index Terms: Hall Effect, MHD flow,
Rotation, exponentially, accelerated plate, variable mass
diffusion.
Nomenclature
aAa , , Constants
0B Applied Magnetic Field (T)
C Dimensionless concentration
c Species concentration in the fluid (mol/m3)
pc Specific heat at constant pressure (J/(kg.K)
wc Concentration of the plate
c Concentration of the fluid far away from the plate
D Mass diffusion co-efficient
erfc Complementary error function
Gc Mass Grashof number
Gr Thermal Grashof number
g Acceleration due to gravity (m/s2)
k Thermal conductivity (W/m.K)
M Hartmann number
m Hall parameter
Pr Prandtl number
Sc Schmidt number
T Temperature of the fluid near the plate (K)
wT Temperature of the plate (K)
T Temperature of the fluid far away from the plate
(K)
t Dimensionless time
t - Time (s)
0u Velocity of the plate
)( wvu
Components of Velocity Field F.
(m/s)
)( wvu
Non-Dimensional Velocity
Components
)( zyx Cartesian Co-ordinates
z Non-Dimensional co-ordinate normal to the plate.
e Magnetic Permeability (H/m)
Kinematic Viscosity (m2/s)
Component of Angular Velocity (rad/s)
Non -Dimensional Angular Velocity
Fluid Density (kg/m
3)
Electric Conductivity (Siemens/m)
Dimensionless temperature
Similarity parameter
Volumetric coefficient of thermal expansion
Volumetric coefficient of expansion with
concentration
I Introduction
Magneto Hydro Dynamics(MHD) flows of an
electrically conducting fluid are encountered in many
143 | Page September 2014, Volume-1, Special Issue-1
industrial applications such as purification of molten
metals, non-metallic intrusion, liquid metal, plasma
studies, geothermal energy extraction, nuclear reactor and
the boundary layer control in the field of aerodynamics
and aeronautics.
The rotating flow of an electrically conducting
fluid in the presence of magnetic fluid is encountered in
cosmical, geophysical fluid dynamics. Also in solar
physics involved in the sunspot development, the solar
cycle and the structure of rotating magnetic stars. The
study of MHD Viscous flows with Hall Currents has
important engineering applications in problems of MHD
Generators, Hall Accelerators as well as in Flight
Magneto Hydrodynamics.
The effect of Hall currents on hydromagneto flow
near an accelerated plate was studied by Pop.I (1971).
Rotation effects on hydromagnetic free convective flow
past an accelerated isothermal vertical plate was studied
by Raptis and Singh(1981). H.S.Takhar.et.al., (1992)
studied the Hall effects on heat and mass transfer flow
with variable suction and heat generation. Watnab and
pop(1995) studied the effect of Hall current on the steady
MHD flow over a continuously moving plate, when the
liquid is permeated by a uniform transverse magnetic
field. Takhar et. al.(2002) investigated the simultaneous
effects of Hall Current and free stream velocity on the
magneto Hydrodynamic flow over a moving plate in a
rotating fluid. Hayat and abbas (2007) studied the
fluctuating rotating flow of a second-grade fluid past a
porous plate with variable suction and Hall current.
Muthucumaraswamy et. al.(2008) obtained the heat
transfer effects on flow past an exponentially accelerated
vertical plate with variable temperature. Magneto hydro
dynamic convective flow past an accelerated isothermal
vertical plate with variable mass diffusion was studied by
Muthucumaraswamy.R.et.al., (2011).
In all the above studies, the combined effect of
rotation and MHD flow in addition to Hall current has
not been considered simultaneously. Here we have made
an attempt to study the Hall current effects on a MHD
flow of an exponentially accelerated horizontal plate
relative to a rotating fluid with uniform temperature and
variable mass diffusion.
II Mathematical Formulation
Here we consider an electrically conducting
viscous incompressible fluid past an infinite plate
occupying the plane z =0. The x -axis is taken in the
direction of the motion of the plate and y -axis is normal
to both x and z axes. Initially, the fluid and the plate
rotate in unison with a uniform angular velocity about
the z -axis normal to the plate, also the temperature of
the plate and concentration near the plate are assumed to
be candT . At time 0t ,the plate is exponentially
accelerated with a velocity ) exp(0 taA
uu in its
own plane along x -axis and the temperature from the
plate is raised to wT and the concentration level near the
plate is raised linearly with time. Here the plate is
electrically non conducting. Also, a uniform magnetic
field 0B is applied parallel to z -axis. Also the pressure
is uniform in the flow field. If ,, wvu represent the
components of the velocity vector F, then the equation of
continuity 0 F gives w =0 everywhere in the
flow such that the boundary condition w =0 is satisfied
at the plate. Here the flow quantities depend on z and
t only and it is assumed that the flow far away from the
plate is undisturbed. Under these assumptions the
unsteady flow is governed by the following equations.
(1) )c-c(g)T-(Tg
)()1(
22
20
2
2
2
vmum
Bv
z
u
t
u e
(2) )()1(
22
20
2
2
2
vumm
Bu
z
v
t
v e
(3) c 2
2
pz
TK
t
T
(4) D t
c2
2
z
c
Where u is the axial velocity and v is the
transverse velocity. The prescribed initial and boundary
conditions are
(5) 0at
cc, TT ,0
zallfort
u
(6) 0 0
,tA ccc
, T, 0 ,
w
t 0
tallforzat
c
TveA
uu w
a
(7) , 0,0 zascc,TTvu
144 | Page September 2014, Volume-1, Special Issue-1
where ,3
1
2
0uA is a constant.
On introducing the following non-dimensional
quantities,
, ,
, , ,
3
1
20
3
1
20
3
1
20
3
1
03
1
0
ut
ut
uzz
u
vv
u
uu
,)(
,)(
, , 2
00
3
1
20
3/20
3
1
20
22
u
TTgGr
u
ccgGc
au
au
BM
ww
e
DSc
K
c
TT
TT
cc
ccC
p
ww
,
Pr
, ,
The equations (1)-(7) reduce to the following
non-dimensional form of governing equations
(8)
1
2 2
2
2
2
2
GcCGr
mvum
Mv
z
u
t
u
(9) 1
2
2
2
2
2
2
vmum
M
uz
v
t
v
(10) Pr
12
2
zt
(11) 1
2
2
z
C
Sct
C
With initial and boundary conditions
(12) 0 0C
0, 0, , 0
zallfortat
vu
(13) 0 ,0 ,1
,0 ,
ztattC
veu at
(14) 0 ,0
,0 ,0
zasC
vu
The above equations (8) - (9) and boundary
conditions (12)-(14) can be combined as
(15) C Gc Gr
11 2
2
2
2
2
2
2
m
mMi
m
Mq
z
q
t
q
(16) Pr
1
t 2
2
z
(17) 1
2
2
z
C
Sct
C
With boundary conditions
(18) 0
0C 0, , 0
zallfortat
q
(19) 0 ,0 t C
1, ,
tallforzat
eq at
(20) z as 0C
0, 0, q
Where q=u+iv.
III Solution of the problem.
To solve the dimensionless governing equations
(15) to (17), subject to the initial and boundary conditions
(18)-(20) Laplace –Transform technique is used. The
solutions are in terms of exponential and complementary
error function:
scsc
scerfc
tc2
2
exp2
)( sc)21(
(21)
)( prerfc
(22)
145 | Page September 2014, Volume-1, Special Issue-1
))(()(2exp(
))(()(2exp(
2
tbaerfctba
tbaerfctbaeq
at
)()2exp(
)()2exp(
2
1
1
1
22
2
bterfcbt
bterfcbt
e
d
e
d
)()2exp(
)()2exp(
2_
2
2
bterfcbt
bterfcbtt
e
d
)()2exp(
)()2exp(
2
2
2
bterfcbt
bterfcbt
b
t
e
d
))(())(2exp(
))(())(2exp(
2
)exp(
11
111
1
1
teberfcteb
teberfctebte
e
d
))(())(2exp(
))(())(2exp(
2
)exp(
22
222
22
2
teberfcteb
teberfctebte
e
d
)() 2exp(
)() 2exp(
2
)exp(
11
111
1
1
teprerfctepr
teprerfcteprte
e
d
)( 1
1 prerfce
d
scsc
e
d
scerfcscte
d
2
2
2
2
2
2
exp2
_
21
)( 2
2
2 Scerfce
d
)()2exp(
)()2exp(
2
)exp(_
22
222
22
2
teScerfctSce
teScerfctScete
e
d
(23)
Where
,1Pr
e ,1Pr
,)1
(1
2 112
2
2
2 bGrd
m
mMi
m
Mb
tzSc
be
Sc
Gcd 2/ ,
1 ,
122
In order to get a clear understanding of the flow
field, we have separated q into real and imaginary parts to
obtain axial and transverse components u and v.
IV Results and Discussion
To interpret the results for a better understanding of
the problem, numerical calculations are carried out for
different physical parameters M,m, ,Gr,Gc,Pr and Sc.
The value of Prandtl number is chosen to be 7.0 which
correspond to water.
Figure 1 illustrates the effect of Schmidt number
(Sc=0.16, 0.3, 0.6), M=m=0.5, =0.1, a=2.0, t=0.2 on
the concentration field. It is observed that, as the Schmidt
number increases, the concentration of the fluid medium
decreases.
The effect of Prandtl number (Pr) on the
temperature field is shown in Figure2. It is noticed that an
increase in the prandtl number leads to a decrease in the
temperature.
“F
ig.1.” concentration profiles for different values of Sc.
“Fig.2.”Temperature profiles for different values of Pr.
Figure 3.illustrates that effects of the Magnetic field parameter (M=1.0, 3.0, 5.0), Gr = Gc =5.0, m=0.5, a=2.0, t=0.2) on axial velocity. It is observed that, the axial velocity increases with the decreasing value of M . This agrees with the expectations, since the magnetic field exerts a retarding force
on the free convective flow.
0 0.5 1 1.5 2 2.5 3 0
0.2
0.4
0.6
0.8
1
z
θ
0.71
7.0
pr
146 | Page September 2014, Volume-1, Special Issue-1
“Fig.3.”Axial velocity profiles for different values of M.
The effect of Rotation parameter on axial velocity is
shown in Figure.4. It is observed that the velocity
increases with decreasing values of (3.0, 5.0).
“Fig.4.”Axial velocity profiles for different values of Ω.
Fig.5 demonstrates the effect of Hall parameter m on
axial velocity. It has been noticed that the velocity
decreases with increasing values of Hall parameter.
“Fig.5.”Axial velocity profiles for different values of m
Figures 6 and 7 show the effects of thermal Grashof
number Gr and mass Grashof number Gc. It has been
noticed that the axial velocity increases with increasing
values of both Gr and Gc,
“Fig.6.”Axial velocity profiles for different values of Gr.
“Fig.7.”Axial velocity profiles for different
values of Gc.
Figure.8 illustrates the effects of Magnetic field
parameter M on transverse velocity. It is observed that the
transverse velocity increases with increasing values of M
and it is also observed that the transverse velocity peaks
closer to the wall as M is increased.
z 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
0.5
1
1.5
u
1.0
3.0
5.0
M
u
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 -0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
z
u
10.
5.0
2.0
Gr
z
u
0 0. 1 1.5 2 -0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.0
5.0
10.0
Gc
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0
0.5
1
1.5
z
u
3.0
5.0
Ω
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 -0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
z
3.0
0.5
m
5.0
147 | Page September 2014, Volume-1, Special Issue-1
“Fig.8.” Transverse velocity profiles for different
values of M.
The Transverse velocity profiles for different values
of Rotaion parameter are shown in Figure.9. It is
observed that the velocity increases with decreasing
values of .
“Fig.9.”Transverse velocity profiles for different values
of Ω.
Figure.10 shows the effect of Hall parameter m on
transverse velocity. It is found that the velocity increases
with increasing values of m.
“Fig.10.”Transverse velocity profiles for
different values of m.
Figure.11. demonstrates the effect of Thermal
Grashof number Gr on transverse velocity. It is observed
that there is an increase in Transverse velocity as there is
an increase in Gr
“Fig.11.” Transverse velocity profiles for different values
of Gr.
The effect of Mass Grashof number on
transverse velocity is shown in Figure.12. Numerical
calculations were carried out for different values of Gc
namely 3.0, 5.0, 10.0. From the Figure it has been
noticed that with decreasing values of Gc the tansverse
velocity is increased.
z
v
0 0.5 1 1.5 2 -0.45
-0.4
-0.35
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
z
Ω
3.0
5.0
0 0.5 1 1.5 2 -0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
z
v
8.0
5.0
2.0
Gr
0 0.
2
0.
4
0.
6
0.
8
1 1.
2
1.
4
1.
6
1.
8
2 -
0.02
0
0.0
2
0.0
4
0.0
6
0.0
8
0.
1
0.1
2
0.1
4
v
M
1.
0 3.
0 5.
0
0 0.
5
1 1.
5
2 -
0.8
-
0.7
-
0.6
-
0.5
-
0.4
-
0.3
-
0.2
-
0.1
0
0.
1
z
v
5.
0
1.
0
3.
0
m
148 | Page September 2014, Volume-1, Special Issue-1
“Fig.12.”Transverse velocity profiles for different values
of Gc.
V Conclusion
In this paper we have studied the effects of Hall
current, Rotation effect on MHD flow through an
exponentially accelerated vertical plate with uniform
temperature and variable mass diffusion. In the analysis
of the flow the following conclusions are made.
(1) The concentration near the plate increases with
decreasing values of the Schmidt number.
(2) The temperature of the plate decreases with increasing
values of prandtl number
(3) Both Axial velocity and Transverse velocity increase
with decreasing values of Magnetic field parameter or
Rotation parameter. Also the Axial velocity increase
with decreasing values of Hall parameter but the trend
is reversed in Transverse velocity.
(4) Both Axial velocity and Transverse velocity increase
with increasing values of thermal Grashof number.
(5) Axial velocity increases with increasing values of
mass Grashof number whereas the transverse velocity
increase with decreasing values of mass Grashof
number.
References
[1] Pop I(1971): The effect of Hall Currents on
hydromagnetic flow near an accelerated plate. -
J.Math.Phys.Sci. vol.,5, pp.375-379.
[2] Raptis A., and Singh A.K., (1981): MHD free
convection flow past an accelerated vertical plate -
Letters in Heat and Mass Transfer Vol.8, pp.137-143.
[3] H.S.Takhar., P.C.Ram., S.S.Singh (1992): Hall
effects on heat and mass transfer flow with variable
suction and heat generation.- Astrophysics and space
science Volume.191,issue1.pp.101-106.
[4] Watanabe T.,and Pop I (1995): Hall effects on
magnetohydrodynamic boundary layer flow over a
continuous moving flat plate. : - Acta Mechanica vol.,
108, pp.35-47.
[5] Takhar H.S., Chamkha A.J., and Nath G.(2002) :
MHD flow over a moving plate in a rotating fluid with
a magnetic field, Hall currents and free stream
velocity.:- Intl.J.Engng.Sci. vol,40(13).pp.1511-1527.
[6] Hayat T., and Abbas Z. (2007): Effects of hall Current
and Heat Transfer on the flow in a porous Medium
with slip conditions:- Journal of Porous Media,
vol.,10(1), pp.35-50.
[7] Muthucumaraswamy R., Sathappan KE and Natarajan
R., (2008): Heat transfer effects on flow past an
exponentially accelerated vertical plate with variable
temperature,:- Theoretical Applied Mechanics,
Vol.35, pp.323-331.
[8] Muthucumaraswamy R., Sundar R.M., and
Subramanian V.S.A.,(2011): Magneto hydro dynamic
convective flow past an accelerated isothermal vertical
plate with variable mass diffusion :-International
Journal of Applied Mechanics and
Engineering,Vol.16,pp.885-891.
z 0 0.5 1 1.5 2
-1.6
-1.4
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
v
3.0 Gc
5.0
10.0
149 | Page September 2014, Volume-1, Special Issue-1