Athens Journal of Technology and Engineering June 2017 125 Numerical Analysis of Two Dimensional Unsteady MHD Free Convection and Mass Transfer of Fresh and Salt Water Flow on an Inclined Plate with Hall Current and Constant Heat Flux By Mohammad Shah Alam Mohammad Ali ‡ A two dimensional unsteady free convection and mass transfer of fresh and salt water flow on an inclined plate in the presence of Halls' current and constant heat flux has been analyzed numerically by an explicit finite difference method. The governing equations are derived using the boundary layer and Boussinesqs' approximations. These equations and boundary conditions are non- dimensionalized using usual transformation. The resulting non linear coupled partial differential equations are then solved numerically subject to the transformation boundary conditions by explicit finite difference methods. The numerical results for the dimensionless velocities; temperature and concentration profiles are examined and displayed graphically. Due to the physical interest the values of shear stress, the Nusselt number and Sherwood number are presented in tabular form for various parameters which enter into the problem to show the interesting outcomes of the solutions. Keywords: Free convection, Hall current, Heat flux, Inclined plate, Magneto hydrodynamics. Introduction The aim of this work is to make some numerical calculations on MHD free convection heat and the mass transfer flow which is of interest to the engineering community and to the researchers dealing with problems in astrophysics, renewable energy systems and also hypersonic aerodynamics. The problem of magneto hydrodynamics flow for an electrically conducting fluid past a heated surface has attracted due to many engineering applications such as plasma studies, petroleum industries, MHD power generator, cooling of nuclear reactors and boundary layer control in aerodynamics. The study of MHD incompressible viscous flows with Hall currents has grown considerably because of its engineering applications to the problems of Halls' accelerators, constructions of turbines and centrifugal mechanics, as well as flight magneto hydrodynamics. Convective mass transfer occurs when two fluids are moving together over a surface and mass transfer. In this case, it depends on the nature of fluid flow which studied by Sawhney (2010). Mass transfer which is occurred by the combination of diffusion convection effects known as a change of phase mode investigated by Venkanna (2010). Many researchers studying Associate Professor, Chittagong University of Engineering & Technology, Bangladesh. ‡ Assistant Professor, Chittagong University of Engineering & Technology, Bangladesh.
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Athens Journal of Technology and Engineering June 2017
125
Numerical Analysis of Two Dimensional Unsteady
MHD Free Convection and Mass Transfer of Fresh and
Salt Water Flow on an Inclined Plate with Hall Current
and Constant Heat Flux
By Mohammad Shah Alam
Mohammad Ali‡
A two dimensional unsteady free convection and mass transfer of fresh and salt water
flow on an inclined plate in the presence of Halls' current and constant heat flux has
been analyzed numerically by an explicit finite difference method. The governing
equations are derived using the boundary layer and Boussinesqs' approximations.
These equations and boundary conditions are non- dimensionalized using usual
transformation. The resulting non linear coupled partial differential equations are
then solved numerically subject to the transformation boundary conditions by explicit
finite difference methods. The numerical results for the dimensionless velocities;
temperature and concentration profiles are examined and displayed graphically. Due
to the physical interest the values of shear stress, the Nusselt number and Sherwood
number are presented in tabular form for various parameters which enter into the
problem to show the interesting outcomes of the solutions.
Keywords: Free convection, Hall current, Heat flux, Inclined plate, Magneto
hydrodynamics.
Introduction
The aim of this work is to make some numerical calculations on MHD free
convection heat and the mass transfer flow which is of interest to the
engineering community and to the researchers dealing with problems in
astrophysics, renewable energy systems and also hypersonic aerodynamics.
The problem of magneto hydrodynamics flow for an electrically conducting
fluid past a heated surface has attracted due to many engineering applications
such as plasma studies, petroleum industries, MHD power generator, cooling
of nuclear reactors and boundary layer control in aerodynamics. The study of
MHD incompressible viscous flows with Hall currents has grown considerably
because of its engineering applications to the problems of Halls' accelerators,
constructions of turbines and centrifugal mechanics, as well as flight magneto
hydrodynamics. Convective mass transfer occurs when two fluids are moving
together over a surface and mass transfer. In this case, it depends on the nature
of fluid flow which studied by Sawhney (2010). Mass transfer which is
occurred by the combination of diffusion convection effects known as a change
of phase mode investigated by Venkanna (2010). Many researchers studying
Associate Professor, Chittagong University of Engineering & Technology, Bangladesh.
‡ Assistant Professor, Chittagong University of Engineering & Technology, Bangladesh.
Vol. 4, No. 2 Alam et al.: Numerical Analysis of Two Dimensional Unsteady …
126
the effects of magnetic fields on mixed, natural and forced convection heat
transfer problems. Some of them are Javaherdeh et al., (2015), Ibrahim et al.,
(2014), Raptis and Singh (1983), Bejan and Khair (1985), Singh and Queeny
(1997), Alam et al., (2008), Ganesan and Palani (2004), Makinde (2011a; b),
Ahamed and Alam (2013), Alam et al., (2012), Kays and Crawford (1993).
Hall currents play a significant role in magnetic ranging solutions at realistic
plasma resistivities. Hall currents should be important in virtually all cases of
fast magnetic merging in the corona. When a current-carrying conductor is
placed into a magnetic field, a voltage will be generated perpendicular to both
the current and the field. This principle is known as the Hall effect. The basic
characteristic of the Hall effect is the Hall factor, which was discovered by Hall
in strips of a gold leaf. It is an important interaction of magnetic fields and
electric current more commonly associated with metals and a semiconductor.
Many researchers studied MHD free/forced convection boundary layer flow
heat and mass transfer in presence of Hall current some of them are Seth et al.,
(2016), Alam et al., (2014),Cowling (1957), Pop (1971), Gupta (1975), Dutta
and Jana (1976), Datta and Mazumdar (1976), Raptis and Ram (1984), Sattar
(1994), Singh et al., (1999) and Singh et al., (2000). Raptis and Kafoussias
(1982) have studied the free convective heat and mass transfer flow through a
porous medium occupying a semi-infinite region of the space bounded by an
infinite vertical plate in the presence of a transverse magnetic field. Also Alam
and Sattar (1995) have studied MHD free convection and a mass transfer flow
with Halls' current and constant heat flux. These dissertations are to make some
numerical calculations on MHD free convection heat and mass transfer flow
which is of interest to the engineering community. The main object of the
present work is to provide a boundary layer analysis for the effect of Halls'
current and viscous dissipation on free convection and mass transfer of fresh
and salt water flow with constant heat flux in presence of magnetic field. The
equations thus obtained have been solved numerically using an explicit finite
difference method. The effects of different parameters that enter into the
problem on momentum, temperature and concentration equations will be
analyzed by graph.
Mathematical Model of Flow
A two dimensional unsteady MHD free convection and Mass transfer flow
of an electrically conducting viscous incompressible fluid on a semi-infinite
inclined plate with Hall current and constant heat flux has been considered. A
semi-infinite plate is placed inclined with angle to the x-axis. The flow is
also assumed to be in an x – direction and y-axis is normal to it. The
temperature and species concentration at the plate are instantly raised from Tw
and Cw to T and C respectively and there after maintained constant. A
uniform magnetic field strength B is applied in a direction normal to the flow
that is acting along the Y – axis, which is also
Athens Journal of Technology and Engineering June 2017
127
Figure 1. Physical Configuration and Coordinate System
Electrically non-conducting (Figure 1). We assumed that the magnetic
Reynolds number of the flow should be small enough, so that the induced
magnetic field is negligible, so 00 0 ,B,B . Using the relation 0J. for
the current density )J,J,(J zyxJ we obtain yJ = constant. Since the plate is
non-conducting, 0J y at the plate and hence zero everywhere. The
generalized Ohm’s law in the absence of the electric field for weakly ionized
gas is of the form .B)qEBJJ e
e
e μ(σ)(m
τeμ Also under usual
assumptions the thermo electric pressure and ion slip are negligible. Then from
Ohm’s law it is found that w),(mu)mρ(1
μBσJ
2
e
2
0x
.mw)(u)m(1ρ
μBσJ
2
e
2
0z
Under the above assumption and Boussinesq
approximation, the basic equations relevant to the problem are:
Continuity equation:
0y
v
x
u
(1)
α
X
Y
Tw
T
wC
C
w
u
v
0B
Vol. 4, No. 2 Alam et al.: Numerical Analysis of Two Dimensional Unsteady …
128
Momentum equation in x-direction:
mw)(u)mρ(1
μBσ
cosα)C(Cgβcosα)T(Tgβy
uν
y
uv
x
uu
t
u
2
e
2
0
2
2
(2)
Momentum equation in z-direction:
)w(mu
m1ρ
μBσ
y
wν
y
wv
x
wu
t
w2
e
2
0
2
2
(3)
Energy equation:
22
p
2
2
p y
w
y
u
ρC
ν
y
T
ρC
κ
y
Tν
x
Tu
t
T (4)
Concentration equation:
2
2
my
CD
y
Cν
x
Cu
t
C
(5)
Corresponding boundary conditions:
0yatCC,κ
Q
y
T0,w0,v0.u w
(6)
yasCC,TT0,w0,u
Where u, v and w are velocity components in x, y and z- direction
respectively, B0 is the constant magnetic field, eμ is the magnetic permeability,
Q is the constant heat flux per unit area, ν is the kinematic viscosity, g is the
acceleration due to gravity, ρ is the density, β is the coefficient of volume
expansion, *β is the volumetric coefficient with concentration, T is the
temperature of the fluid inside the thermal boundary layer, T is the
temperature in the free stream, C is the concentration in the boundary layer,
C is the concentration outside the boundary layer, pC is the specific heat
with constant pressure, κ is the thermal conductivity, is the electron collision
time. Dm is the coefficient of mass diffusion and remaining symbols have their
usual meaning.
Mathematical Formulation
To make the non-dimensional of the system of equations (1) to (5) with
boundary conditions (6) we adopt the well-defined usual transformation
Athens Journal of Technology and Engineering June 2017
129
technique. For this purpose the following non-dimensional variables are
introduced: 000
00
U
wW,
U
vV,
U
uU,
ν
UyY,
ν
UxX
CC
CCC,
νQ
)T(TUκT,
ν
Utη
w
0
2
0
Here, 0U is the constant velocity, its scale value greater or equal to 0.01.
The equations (1) – (5) become in terms of dimensionless variables as;
0Y
V
X
U
(7)
)m(Um1
MCcosαGTcosαG
Y
U
Y
UV
X
UU
η
U2mr2
2
w
(8)
w)(mUm1
M
Y
W
Y
WV
X
WU
η
W22
2
(9)
22
C2
2
r Y
W
Y
UE
Y
T
P
1
Y
TV
X
TU
η
T (10)
2
2
C Y
C
S
1
Y
CV
X
CU
η
C
(11)
Boundary conditions:
U = 0, V = 0, W = 0 1C1Y
T
at Y = 0 (12)
0C0,T,0W0.U as Y
where, 3
0
Wr
U
)Tgββ(νG
, 3
0
w
*
mU
)C(CgβνG
3
0
e
2
0
ρU
μBσνM ,
κ
CρνP
p
r ,QCν
UκE
P
3
0C ,
m
CD
νS
Vol. 4, No. 2 Alam et al.: Numerical Analysis of Two Dimensional Unsteady …
130
Numerical Procedure
Figure 2. Explicit Finite Difference System Grid
To obtain the different equations the region of the flow is divided into a
grid or mesh of lines parallel to the X-and Y- axes, where the X- axis is taken
along the plate and Y- axis is normal to the plate (Figure 2). Here the plate of
height X max (= 100) is considered i.e X varies from 0 to 100 and assumed Y
max (= 25) as corresponding to Y i.e. Y varies from 0 to 25. There are m' =
250 and n' = 250 grid space in the X and Y directions respectively and taken as
follows )1000(4.0 xx and )250(1.0 YY with the smaller time
step .005.0 Let, U', W',
T and
C denote the values of U, W, T and C
at the end of a time – step respectively. Using the explicit finite difference
approximation in to the partial differential equations (7) - (11) we get an
appropriate set of finite difference equations.
0ΔY
UV
ΔX
UU 1ji,ji,1ji,ji,
(13)
Y
3-i
0j
2j 1j j 1j 2j 3j 4j nj 0i
i
1-i
2-i
i +1
i +2
i +3
I=m’
ΔX
ΔY
Athens Journal of Technology and Engineering June 2017
131
ΔY
UUV
ΔX
UUU
Δη
UU ji,1ji,
ji,
j1,iji,
ji,
ji,ji,
)W j,im(U
m1
MCcosαGTcosαG
ΔY
U2UUji,2
ji,mji,r2
1ji,ji,1ji,
(14)
ΔY
WWV
ΔX
WWU
Δη
WW ji,1ji,
ji,
j1,iji,
ji,
ji,ji,
)j,iW(mU
m1
M
ΔY
W2WWji,22
1ji,ji,1ji,
(15)
ΔY
TTV
ΔX
TTU
Δη
TT ji,1ji,
ji,
j1,iji,
ji,
ji,ji,
2
ji,1ji,
2
ji,1ji,
c2
1ji,ji,1ji,
r ΔY
WW
ΔY
UUE
ΔY
TT2T
P
1 (16)
2
1ji,ji,1ji,
C
ji,1ji,
ji,
j1,iji,
ji,
ji,ji,
ΔY
CCC
S
1
ΔY
CCV
ΔX
CCU
Δη
CC
(17)
With boundary conditions:
1CΔY,TT0,W0,V0,Un
i,0
n
i,1
n
i,0n
i,
n
i,0
n
i,0 (18)
Lwhere0,C0,T0,W0,Un
Li,
n
Li,n
Li,
n
Li,
Here the subscript i and j denote the grid points with X and Y- coordinates
respectively and the superscript n indicates a value of time, = ,n where n
= 0,1,2,..... . The primary velocity (U), Secondary velocity (W), temperature
T and concentration C distributions at all interior nodal points may be
computed by successive applications of the above finite difference equations.
The numerical values of the local shear stresses and local Sherwood number
are evaluated by a five – point approximate formula for the derivation and then
the average shear stress and Sherwood number are calculated by use of the
Simpson’s 3
1 integration formula.
Hence the stability conditions are:
1
ΔY
2Δ
ΔY
ΔηV
Δx
UΔΔ2
(19)
1
ΔY
2Δ
P
1
ΔY
ΔηV
Δx
UΔΔ2
r
(20)
Vol. 4, No. 2 Alam et al.: Numerical Analysis of Two Dimensional Unsteady …
132
1
ΔY
2Δ
S
1
ΔY
ΔηV
Δx
UΔΔ2
c
(21)
1
ΔY
2Δ
P
1
ΔY
ΔηV
Δx
UΔΔ2
r
(22)
and convergence criteria of the method are .1Pr and 750.5Sc .
Skin – Friction Co-efficient, Nusselt and Sherwood Numbers
The quantities of the chief physical interest are the local Skin friction
coefficient, Nusselt and Sherwood numbers. The shearing stress at the plate is
generally known as the Skin friction, the following relations represent the local
and average shear stress at the plate. Local shear stress in x- direction,
0y
0xLy
uμτ
and average shear stress in x – direction
dx
y
uμτ 0xA
which are proportional to 0YY
U
and dX
Y
U100
0
respectively. The local
shear stress in z – direction
0y
0zLy
wμτ
and average shear stress in z –
direction
dxy
wμτ 0zA which are proportional to
0YY
W
and
dXY
W100
0
respectively. From the temperature field, the effects of various
parameters on the local and average heat transfer coefficients have been
studied. The following relations represent the local and average heat transfer
rate that is a well known Nusselt number. Local Nusselt number
0y
0uLy
TμN
and average Nusselt number, dx
y
TμN 0uA
, which
are proportional to 0Y
Y
T
and
100
0
dXY
Trespectively. From the concentration
field, the effects of various parameters on the local and average mass transfer
coefficients have been analyzed. The following relations represent the local
and average mass transfer rate that is well known Sherwood number, Local
Sherwood number, 0y0hLy
CμS
and average Sherwood number
Athens Journal of Technology and Engineering June 2017
133
dxy
CμS 0hA
which are proportion to
0YY
C
and dX
Y
C100
0
respectively.
Results and Discussion
Unsteady MHD free convection and mass transfer of fresh and salt water
flow on an inclined plate with Hall current and constant heat flux have been
investigated using the explicit finite difference method. To study the physical
situation of this problem, the numerical values of the primary velocity,
secondary velocity, temperature and concentration within the boundary layer
have been computed, and also the Shear stress, Nusselts' and Sherwoods'
numbers at the plate have been found. The velocity in the x – direction is
called primary velocity and that in the z – direction is called secondary
velocity. For the purpose of discussing the effects of various parameters on the
flow behaviors near or on the plate, numerical calculations have been carried
out for different values of Magnetic parameter M , Hall parameter m, angle of
inclination α , Eckert number cE and Schmidt number cS , for fixed values of
rP (in the case of fresh and Salt water; the value of rP is taken to be 1(one) for
salt water and 7(seven) for fresh water). These computed results have been
shown graphically. The importance of a cooling problem in nuclear
engineering in connection with the cooling of the reactors, the value of Gr and
Gm are taken to be positive (Gr > 0, Gm > 0).The values 0.78, 0.97 and 1.00 are
also considered for the entering parameter Schmidt number Sc, which
represents specific conditions of the flow (in particular, 0.78 for Ammonia,
0.97 for Formic acid and 1 for CH4 in air). The values of other parameters are
however chosen arbitrary. The results are obtained to illustrate the velocities,
temperature and concentration profiles, while the values of the parameters are
fixed at real constant with
.o
ccmr 53α0.01,E0.78,S2,G4.00,G0.85,m2.25,M
From Figure 3(a) – 3(d) investigate the effect of magnetic parameter on
primary velocity, secondary velocity, temperature and concentration for fresh
water respectively.
The presence of the transverse magnetic field produces a resistive force
like Lorentz force on the fluid flow, which leads to slow down the motion of
electrically conducting fluid. Thus for the increasing effect of the magnetic
parameter (M) the primary velocity decreases in case of both fresh water;
which is shown in Figure 3(a). The same case arises for secondary velocity as
depicted in Figure 3(b). The effect of the magnetic parameter (M) on the
temperature profile is shown in Figure 3 (c) for fresh water. From this figure it
is observed that the temperature decreases within the interval Y0 1
Vol. 4, No. 2 Alam et al.: Numerical Analysis of Two Dimensional Unsteady …
134
(Approx.) and then increases. It is evident that the temperature increases with
increasing M, which implies that the applied magnetic field normal to the flow
of the fluid tends to heat the fluid and thus reduces the heat transfer from the
wall therefore the temperature increases. From Figure 3(d) it is observed that
the concentration profiles increased for increasing values of magnetic
parameter (M) for fresh water.
Figure 3(a). Primary Velocity for Various
Values of M
Figure 3 (b). Secondary Velocity for
Various Values of M
0 5 10 150
0.1
0.2
0.3
0.4
0 5 10 150
0.1
0.2
Figure 3(c). Temperature for Various
Values of M
Figure 3(d). Concentration for Various
Values of M
0 2 4 60
0.5
1
1.5
2
2.5
3
0 5 10 150
0.2
0.4
0.6
0.8
1
Figures 4(a) – 4(d) depict the effect of the magnetic parameter on
velocities, temperature and concentration profile for salt water. From Figure
4(a) it is observed that the primary velocity decreases for increasing values
of M , because magnetic field produces a resistive force like Lorentz force on
the fluid flow, which leads to slow down the motion of electrically conducting
fluid. Similar results have been found in the case of secondary velocity shown
in Figure 4 (b). Figure 4(c) displays the effect of magnetic parameter (M) on
007.Pr
Y
U
2.25,2.5,2.75M
W
Y
00.7
75,2,50.2,25.2
rP
M
007
752502252
.P
.,.,.M
r
Y
T
M=2.25, 2.50, 2.75
007.Pr
C
Y
Athens Journal of Technology and Engineering June 2017
135
temperature profile for salt water. From this figure it is observed that the
temperature increases. It is evident that the temperature increases with the
increasing M, which implies that the applied magnetic field normal to the flow
of the fluid tends to heat the fluid and thus reduces the heat transfer from the
wall therefore the temperature is increased. From Figure 4(d) it is observed that
the concentration profiles increased for the increasing values of magnetic
parameter (M) for salt water.
Figure 4(a). Primary Velocity for
Various Values of M
Figure 4(b). Secondary Velocity for
Various Values of M
0 5 100
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0 5 100
0.1
0.2
0.3
Figure 4(c). Temperature for Various
Values of M
Figure 4(d). Concentration for
Various Values of M
0 2 4 6 80
0.5
1
1.5
2
2.5
0 2 4 6 8 10 120
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Figure 5(a) – 5(d) show the effect of the Hall parameter on velocities,
temperature and concentration profile for fresh water. When the Hall parameter
is increased, the induced current along x-axis increases, as a result the primary
velocity increases which is shown in Figure 5(a). It is observed from Figure
752502252 .,.,.M
U
Y
001.Pr
00.1
75,2,50.2,25.2
rP
M
Y
W
T
Y
001
752502252
.P
.,.,.M
r
001
752502252
.P
.,.,.M
r
Y
C
Vol. 4, No. 2 Alam et al.: Numerical Analysis of Two Dimensional Unsteady …
136
5(a) that the primary velocity is increased within the interval
Y0 4(Approx.) and then decreased for fresh water. Figure (b) depicts the
secondary velocity for various values of m for fresh water. From this figure it
is observed that the secondary velocity increases for the increasing values of
the Hall parameter ( m ).The effect of the Hall parameter on temperature
profiles are shown in the Figure 5(c) for fresh water. From this figure it is
observed that the temperature profile is increased for the increasing values
of m . It is noted that, for the increasing values of the Hall parameter, the ion
collisions will be increased which translates to a more thermal generation,
hence increased the temperature profile. Figure 5 (d) shows the effect of the
Hall parameter ( m ) on the concentration profile for fresh water. It is observed
from this figure that the Hall parameter ( m ) has a decreasing effect on
concentration.
Figure 5(a). Primary Velocity for
Various Values of m
Figure 5(b). Secondary Velocity for
Various Values of m
0 5 10 150
0.1
0.2
0.3
0.4
0.5
0 5 10 150
0.1
0.2
0.3
0.4
Figure 5(c). Temperature for Various
Values of m
Figure 5(d). Concentration for
Various Values of m
0 2 40
0.5
1
1.5
2
2.5
3
3.5
4
0 5 10 150
0.2
0.4
0.6
0.8
1
0.85,1.0,1.5m
007.Pr U
Y
0.85,1.0,1.5m
007.Pr
W
Y
007.Pr
Y
T
0.85,1.0,1.5m
C
Y
007
5101850
.P
.,.,,m
r
Athens Journal of Technology and Engineering June 2017
137
Figures 6(a) – 6(d) show the effect of the Hall parameter on velocities,
temperature and the concentration profile for salt water. From Figure 6(a) it is
observed that the primary velocity is increased within the interval Y0 3
(Approx.) and decreased while Y 3 for salt water. By analyzing the Figure
6(b), it is concluded that the secondary velocity increases for salt water. The
effect of the Hall parameter on the temperature profiles is shown in Figure 6(c)
for salt water. From this figure it is observed that the temperature profile is
increased for the increasing values of m . It is noted that, for increasing values
of the Hall parameter, the ion collisions can be increased which translates to
more thermal generation, hence it increased the temperature profiles. It is
observed from Figure 6(d) that the Hall parameter ( m ) has a decreasing effect
on the concentration profile.
Figure 6(a). Primary Velocity for
Various Values of m
Figure 6(b). Secondary Velocity for
Various Values of m
0 2 4 6 8 100
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0 5 100
0.1
0.2
0.3
0.4
0.5
0.6
Figure 6(c). Temperature for Various
Values of m
Figure 6(d). Concentration for
Various Values of m
0 2 4 6 80
0.5
1
1.5
2
2.5
0 2 4 6 8 10 120
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
In Figures 7(a) – 7(b) the effect of the Schmidt number cS on velocities,
and concentration profiles has been depicted for fresh water. From Figure 7(a)
U
Y
01
501001850
.P
.,.,.m
r
01
5101850
.P
.,.,.m
r
Y
W
T
Y
01
5101850
.P
.,.,.m
r
001
5101850
.P
.,.,,m
r
Y
C
Vol. 4, No. 2 Alam et al.: Numerical Analysis of Two Dimensional Unsteady …
138
it is shown that the primary velocity field decreases as the Schmidt number
increases. For the increasing effects of Sc the velocity boundary effects
decrease, thus the fluid primary velocity as well as the primary velocity
boundary layer decease. The concentration profiles decrease as the Schmidt
number ( cS ) increase, which is observed from the Figure 7(b) for fresh water.
Figure 7(a). Primary Velocity for
Different Values of cS
Figure 7(b). Concentration for
Different Values of cS
0 2 4 6 8 10 12 14 160
0.1
0.2
0.3
0.4
0 5 10 150
0.2
0.4
0.6
0.8
1
In Figure 8(a) – 8(b) the effect of the Schmidt number cS on velocity,
and concentration profiles has been depicted for salt water. From Figure 8(a) it
is shown that the primary velocity field decreases as the Schmidt number
increase. For the increasing effects of Sc the velocity boundary effects
decrease, thus fluid primary velocity as well as the primary velocity boundary
layer decease. The concentration profiles decrease as the Schmidt number ( cS )
increases, which is observed from Figure 8(b) for salt water. The reduction in
the concentration profiles are accompanied by reduction in concentration
boundary layer. This causes the concentration buoyancy effects to decrease
compliant to a reduction in the fluid velocity. Thus the simultaneous reduction
effects of velocity and concentration reduce the velocity and concentration
boundary layer thickness. From these figures it is found that the bounder layer
thickness is thicker for fresh water than that of salt water.
007.Pr
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0.78,0.97,1.00cS
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Y
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.,.,.S
r
c
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Figure 8(a). Primary Velocity for
Different Values of cS
Figure 8(b). Concentration for
Different Values of cS
0 5 100
0.1
0.2
0.3
0.4
0.5
0.6
0 2 4 6 8 10 120
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
From Figure 9(a) - 9(d) investigated the effect of the Eckert number cE
on velocities, temperature and concentration profiles for fresh water. From
Figure 9(a) it is observed that the primary velocity increased in the interval
Y0 3.1 (Approximately) and then decreased for the increasing values of Ec
in case of fresh water. For the increasing values of Ec the heat energy is stored
in the vicinity of the plate in liquid due to frictional heating. Thus for the
increasing effect of Ec the primary velocity is increased. Similar behavior is
observed from Figure 9(b) in the secondary velocity profile for the increasing
values of Ec. Increasing the values of the Eckert number causes the fluid to
become warmer and therefore increase the temperature of it. This is put
forward as cause to the viscous dissipation. Increasing the values of Ec can lead
to a situation that the viscous dissipation becomes significant hence the
temperature is increased which is observed from Figure 9(c) for fresh water.
Figure 9(d) displays the concentration profile for different values of Ec. From
this figure it is observed that for various values of Ec, the concentration profile
is decreased for fresh water.
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001
001970780
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001
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.,.,.S
r
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Vol. 4, No. 2 Alam et al.: Numerical Analysis of Two Dimensional Unsteady …
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Figure 9(a). Primary Velocity for
Different Values of cE
Figure 9(b). Secondary Velocity for
Different Values of cE
0 5 10 150
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 5 10 150
0.1
0.2
0.3
Figure 9(c). Temperature for
Different Values of cE
Figure 9(d). Concentration for
Different Values of cE
0 1 2 3 40
2
4
6
8
10
0 5 10 150
0.2
0.4
0.6
0.8
1
In Figure 10(a) – 10(d) the effect of the Eckert number cE on velocities,
temperature and concentration profiles has been depicted for salt water. From
Figure 10(a) it is observed that the primary velocity increased for the
increasing values of Ec in the interval Y0 5 (Approx.) and then decreased
in the flow of salt water. For the increasing values of Ec the heat energy is
stored in the vicinity of the plate in the liquid due to the frictional heating. Thus
for the increasing effect of Ec the primary velocity is increased. Similar
behavior is observed from Figure 10(b) in secondary velocity profile for
increasing the values of Ec. Increasing the values of the Eckert number causes
the fluid to become warmer and therefore increase the temperature of it. This is
put forward as a cause to the viscous dissipation. Increasing the values of Ec
can lead to a situation that the viscous dissipation becomes significant hence
the temperature is increased which is observed from Figure 10(c) for salt water.
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.,.,.E
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Y
007
030020010
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.,.,.E
r
c
C
Y
007
030020010
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.,.,.E
r
c
Athens Journal of Technology and Engineering June 2017
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Figure 10 (d) displays the concentration profile for different values of Ec. From
this figure it is observed that for various values of Ec, the concentration profile
is decreased.
Figure 10(a). Primary Velocity for
Different Values of cE
Figure 10(b). Secondary Velocity for
Different Values of cE
0 5 100
0.1
0.2
0.3
0.4
0.5
0 2 4 6 8 10 120
0.1
0.2
Figure 10(c). Temperature for
Different Values of cE
Figure 10(d). Concentration for
Different Values of cE
0 2 4 6 80
1
2
3
4
5
0 2 4 6 8 10 120
0.2
0.4
0.6
0.8
1
In Figure 11(a) – 11(d) depict the effect of angle of inclination on
velocities, temperature and concentration profiles for fresh water. From Figure
11(a) it is observed that the primary velocity is decreased in the interval
Y0 5.12(Approx) and then increased for the increasing values of in
fresh water. It is obvious that, since the angle of inclination increases, the effect
of the buoyancy force due to the thermal diffusion decreases by the factor of
cosα ; as a result the driving force to the fluid decreases, thus the primary
velocity decreases. Figure 11(b) displays the secondary velocity for various
values of angle of inclination (α ). From this figure it is observed that the
secondary velocity decreased in the interval Y0 5.2 (Approx.) and then
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T
001
030020010
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.,.,.E
r
c
Y
001
030020010
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.,.,.E
r
c
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C
Vol. 4, No. 2 Alam et al.: Numerical Analysis of Two Dimensional Unsteady …
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increased for fresh water. Both primary and secondary velocity distribution
attain a distinctive maximum value in the vicinity of the plate and then
decrease properly to approach the free stream value. From Figure 11(c) it is
observed that the temperature is decreased in the interval Y0 1 (Approx.)
and then increased, for this reason the thermal boundary layer thickness
increases in the vicinity of the wall and away from the wall the thermal
boundary layer thickness decreases as the angle of inclination increase for fresh
water. By analyzing the Figure 11(d) it is revealed that the concentration
profiles decreased for the increasing values of the angle of inclination (α ) for
fresh water.
Figure 11(a). Primary Velocity for
Different Values of
Figure 11(b). Secondary Velocity for
Different Values of
0 5 10 150
0.2
0.4
0 5 10 150
0.1
0.2
Figure 11(c). Temperature for
Different Values of
Figure 11(d). Concentration for
Different Values of
0 1 2 3 4 50
0.5
1
1.5
2
2.5
3
3.5
0 5 100
1
2
3
4
In Figure 12(a) – 12(d) depict the effect of angle of inclination on
velocities, temperature and concentration profiles for salt water. From Figure
12(a) it is observed that the primary velocity is decreased in the interval
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,,
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Athens Journal of Technology and Engineering June 2017
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Y0 5.1(Approx) and then increased for the increasing values of α in salt
water. It is obvious that, since the angle of inclination increases, the effect of
the buoyancy force due to the thermal diffusion decreases by the factor of
cosα ; as a result the driving force to the fluid decreases, thus the primary
velocity decreases. Figure 12(b) displays the secondary velocity for various
values of angle of inclination (α ). From these figures it is observed that the
secondary velocity decreased in the interval Y0 5.2 (Approx.) and then
increased for fresh and salt water. From Figure 12(c) it is observed that the
temperature profiles increase for increasing values of for salt water. By
analyzing Figure 12(d) it is revealed that the concentration profiles decreased
for the increasing values of the angle of inclination (α ) for salt water.
Figure 12(a). Primary Velocity for
Different Values of
Figure 12(b). Secondary Velocity for
Different Values of
0 2 4 6 8 10 12 14 160
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 5 10 150
0.1
0.2
0.3
0.4
Figure 12(c). Temperature for
Different Values of
Figure 12(d). Concentration for
Different Values of
0 5 100
1
2
3
4
0 5 10 150
0.2
0.4
0.6
0.8
1
In Table 1 the values of the local Skin friction coefficient are represented,
Nusselt and Sherwood numbers for various values of magnetic parameter M ,
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Hall parameter m , Grashof number rG , Eckert number rE , Schmidt
number cS angle of inclination in case of fresh water and In Table 2
represents the values of local Skin friction coefficient, Nusselt and Sherwood
numbers for various values of magnetic parameter M , Hall parameter m ,
Eckert number rE , angle of inclination , in case of fresh water, Grashof
number rG .
Table 1. Values of Shear Stress, Nusselt Number and Sherwood Number for
Different Values of Entering Parameters while 07.0Pr (For Fresh Water-