SIMULTANEOUS THERMAL AND
MASS DIFFUSION
IN MHD MIXED CONVECTION FLOW
Ch. V. Ramana Murthy
Department of Applied Mathematics Lakireddy Balireddy College of Engineering,
Mylavaram -521 230 (INDIA)
P. Hari Prasad
Department of Applied Mathematics K. L. University, Vaddeswaram-522502 (INDIA)
ABSTRACT
Simultaneous thermal and mass diffusion in MHD convective flow has been examined in detail in this paper. The analytical expressions for the velocity and skin friction are obtained. The influence of various parameters on concentration, velocity and skin friction are illustrated. For relatively smaller values of Schimdt number, it is noticed that the concentration decreases. Further, it is observed that as Schimdt number increases the concentration is inversely proportional. In general it is noticed that, as the Prandtl number increases the velocity. When examined in detail, is noticed that the velocity profiles are parabolic in nature and are widely dispersed as the Prandtl number increases. Also, it is noticed that the increase in Prandtl number contributes to the decrease in velocity. The velocity profiles are observed to be more dispersed as we move away from the boundary and also as the Prandtl number increases. Further, increase in Prandtl number contributes to decrease in skin friction. Also, for a constant value of magnetic field and Prandtl number - the skin friction is observed to be constant.
Key words: Magnetic field, Heat and mass transfer, Skin friction
INTRODUCTION
Convective flows often driven by temperature and concentration differences has been the objective of extensive research due to their vast applications in nature and engineering problems. The situations are often encountered in nature includes photo-synthetic mechanism, evaporation and vaporization of mist and fog, while the engineering application includes the chemical reaction in a reactor chamber consisting of rectangular ducts, chemical vapor deposition on surfaces and cooling of electronic equipment. Magneto hydrodynamics flows has applications in meteorology, solar physics, cosmic fluid dynamics, astrophysics, geophysics and in the motion of earth’s core. From the technological point of view, MHD free convection flows have significant applications in the field of stellar and planetary magnetospheres, aeronautics, chemical engineering and electronics. Due to several important applications, such flows has been studied by several authors. Heat and mass transfer on flow past a vertical plate have been studied by several authors; viz. Somess [1], Soundalgekar and Ganesan [2], Khair and Bejan [3] and Lin and Wu [4] in numerous ways to include various physical aspects. In addition to the above note worthy contributions have been made by Shercliff [5] Ferraro and Plumption [6] Cramer and Pai [7] and Elbashbeshy [8].
awareness, the effect of ohmic heating on the MHD free convection heat transfer has been examined for a Newtonian fluid by Hossain [14]. Chen [15] studied the problem of combined heat and mass transfer of an electrically conducting fluid in MHD natural convection, adjacent to a vertical surface with ohmic heating. The propagation of thermal energy through mercury and electrolytic solution in the presence of magnetic field and radiation has wide range of applications.
MATHEMATICAL FORMULATION:
We consider the mixed convection flow of an incompressible and electrically conducting viscous fluid such that X*- axis is taken along the plate in upwards direction and Y*- axis is normal to it. A transverse constant magnetic field is applied i.e. in the direction of Y* - axis. Since the motion is two dimensional and length of the plate is large therefore all the physical variables are independent of X*. Let u* and v* be the components of velocity in x* and y* directions, respectively, taken along and perpendicular to the plate. The governing equations of continuity, momentum and energy for a flow of an electrically conducting fluid along a hot, non-conducting porous vertical plate in the presence of concentration and radiation is given by
0
* *
dy
dv
(1) i.e.v
*
v
0 (constant) (2)* * *
0
p
dy
dp
is independent ofy
* (3))
(
)
(
2 * * *0 * * * 2 * * *
2
g
T
T
B
u
g
C
C
dy
u
d
dy
du
v
(4)* 2 2 2 0 * * 2 * * * * 2 * * *
u
B
y
q
dy
du
dy
T
d
dy
dT
v
C
rp
(5)2 * * 2 * * *
dy
c
d
D
dy
dc
v
(6)Here, g is the acceleration due to gravity,
T
*the temperature of the fluid near the plate,T
the free stream temperature,C
*concentration,
the coefficient of thermal expansion,
the thermal conductivity,p
*the pressure,C
pthe specific heat of constant pressure,B
0 the magnetic field coefficient,
viscosity of the fluid,*
r
q
the radiative flux,
the density,
the magnetic permeability of fluid,v
0 constant suction velocity,
the kinematic viscosity and D molecular diffusivity.Introducing the following non- dimensional quantities
v
y
v
y
* 0
, 0 *v
u
u
,
2 0 2 2 0 2v
v
B
M
,
pr
C
P
,
T
T
T
T
w *
C
C
C
C
C
* ,
3 0 2)
(
v
T
T
v
g
G
w r
,D
v
S
c
, 30 *
)
(
v
C
C
g
G
m
,2 0 '
4
v
C
vI
F
p
(7)In the equations (4), (5) and (6), we get
du
u
d
2
0
2 2
dy
dc
S
dy
C
d
c (10)Where,
G
r= Grashoff number, F = Radiation parameterP
r= Prandtl numberS
c= Schmidt number M = Magnetic parameterThe corresponding boundary conditions in dimensionless form are reduced to
0
y
:u
0
,
1
,C
1
y :
u
0
,
0
(11)The physical variables u,
andC
can be expanded in the power of Prandtl number (Pr). This can be possible physically as Pr for the flow of an incompressible fluid is always less than unity. It can be interpreted physically as the flow due to the Joules dissipation is super imposed on the main flow. In the fitness of the situation, it can be assumed as:)
(
)
(
)
(
)
(
y
u
0y
P
ru
1y
o
P
r2u
)
(
)
(
)
(
)
(
y
0y
P
r
1y
o
P
r2
)
(
)
(
)
(
)
(
y
C
0y
P
rC
1y
o
P
r2C
(12)Using equation (12) in equation (8) – (10) and equating the coefficient of like powers of
P
r, the following set of equations are obtained0 0
0 2 0
0
u
M
u
G
G
C
u
r
m (13)0
0 0
0
P
rFP
r (14)0
0 0
C
S
C
c (15)1 1
1 2 1
1
u
M
u
G
G
C
u
r
m (16)0
1 11
P
rFP
r (17)0
1 1
C
S
C
c (18)Together with the modified boundary conditions as
0
,
1
,
0
,
1
,
0
,
0
;
0
0
1
0
1
0
1
u
u
C
C
y
0
,
0
,
0
,
0
,
0
,
0
;
0
1
0
1
0
1
u
u
C
C
y
(19)Solving equations (13) to (18) with the help of (19), we get
y Sc
e
C
0
y A
e
2 0
)
(
)
(
4 2 46 5 0 y S y A y A y
A
e
A
e
e
ce
A
u
Where
2
4
2 2 r rr
P
FP
P
A
2 2 2 2 5
M
A
A
G
A
r
6 2 2M
S
S
G
A
c c m
The Skin- friction coefficient on the lower plate is given by
)
(
)
(
4 2 6 45 0 c y
S
A
A
A
A
A
y
u
Results & Conclusions:
1. Fig.1 shows the consolidated influence of Schimdt number along with Prandtl number and applied magnetic intensity. For Prandtl number = 0.25 and applied magnetic intensity = 0.5 and for relatively smaller values of Schimdt number it is noticed that the concentration decreases. Further, in the same illustrations it is noticed that as Schimdt number increases the concentration is observed to be inversely proportional.
2. In fig.2 for the Prandtl number = 0.25 and applied magnetic intensity = 0.50 and for sufficiently improved values of Schimdt number the concentration profiles are plotted. It is seen that as Schimdt number increases the concentration of the fluid decreases gradually. Further, it is noticed that the profiles are linear with respect to the distance from the boundary but with a negative slope.
3. The influence of applied magnetic intensity with respect to the Schimdt number is illustrated in fig.3, fig.4, fig.5 and fig.6. In all these illustrations in general it is noticed that as the Prandtl number increases the velocity in general decreases. When examined in each of these illustrations it is noticed that the velocity profiles are parabolic in nature and are widely dispersed. In addition to the above it is noticed that as the Schimdt number increases (irrespective of Prandtl number applied magnetic intensity) the velocity decreases.
4. The influence of the Schimdt number with respect to the magnetic and Prandtl number on the velocity field is depicted in fig.7, fig.8, fig.9 and fig.10. In general it is noticed that the increase in Prandtl number contributes to the decrease in velocity. When the situation is examined in detail it is noticed that as Schimdt number increases (irrespective of Prandtl number applied magnetic intensity) the velocity decreases. The velocity profiles are observed to be more dispersed as we move away from the boundary and also as the Prandtl number increases.
5. The effect of Schmidt number with respect to the Prandtl number and applied magnetic intensity on the skin friction is shown in fig.11 and fig.12. From both these figures it is noticed that as Schmidt number increases the skin friction decreases. Further, increase in Prandtl number contributes to decrease in skin friction. For a constant value of magnetic field and Prandtl number the skin friction is observed to be constant.
Fig. 1 Effect of Schmidt number on Concentration field.
Fig. 3 Contribution of Schmidt number on velocity field
Fig. 5: Influence of Schmidt number on velocity field
Fig. 7 Effect of Schmidt number on velocity field
Fig. 9: Contribution of Schmidt number on velocity field
Fig. 11: Influence of Schmidt number on Skin friction lower plate
Fig. 13: Effect of Schmidt number on Skin friction lower plate
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