Post on 05-Dec-2020
International Journal of Theoretical and Applied Mechanics.
ISSN 0973-6085 Volume 12, Number 1 (2017) pp. 71-81
© Research India Publications
http://www.ripublication.com
Magneto Hydro Dynamics Convective Flow Past a
Vertical Porous Surface in Slip-Flow Regime
G.Dharmaiah1, Uday Kumar.Y.2 and N.Vedavathi3
1Department Of Mathematics,NarasaraoPeta Engineering College, Guntur, A.P, India.
2Department Of Mathematics,Hindu College,Guntur, A.P, India.
3Department Of Mathematics,K.L.University, Guntur, A.P, India.
Abstract
This paper deals with the influences of heat and mass transfer on Two
dimentional MHD free convection flow, laminar and boundary layer of
viscous fluid along a semi vertical permeable moving plate, a uniform
transverse magnetic field, thermal and concentration bouyancy effects.The
governing nonlinear partial difference equations have been decreased to the
coupled nonlinear ordinary differential equations by small perturbation
technique. Numerical evalution of the analytical results is performed and some
graphical results for the velocity, temperature and concentration profiles with
in the boundary layer.
Keywords: MHD, Heat Transfer, Mass Transfer, Slip-flow, vertical plate.
1. INTRODUCTION
Combined heat and mass transfer from different processes with porous media has a
wide range Engineering and Industrial applications such as enhanced oil recovery,
underground energy transport, geothermal reservoirs, cooling of nuclear reactors,
drying of porous solids, packed-bed catalytic reactors and thermal insulation. The
process of heat and mass transfer is encountered in aeronautics, fluid fuel nuclear
reactor, chemical process industries and many engineering applications in which the
72 G.Dharmaiah, Uday Kumar.Y. and N.Vedavathi
fluid is the working medium. The applications are often found in situation the such as
fiber and granules insulation, geothermal systems in the heating and cooling chamber,
fossil fuel combustion, energy processes and Astro-physical flows. Further, the
magneto convection place an important role in the control of mountain iron flow in
the steady industrial liquid metal cooling in nuclear reactors and magnetic separation
of molecular semi conducting materials. In certain porous media applications such as
those involing heat removal from nuclear fuel debris, underground disposal of
radioactive waste material, storage of food stuffs, and exothermic and /or endothermic
chemical reactions and dissociationg fluid in packed-bed reactors, the working fluid
heat generation (source) or absorption (sink) effects are important.
Convection in porous media can be applied to underground ground water hydrology,
iron blast furnaces, cooling of nuclear reactors, solar power collectors, energy
efficient drying processes, cooling of nuclear fuel in shipping flasks, cooling of
electronic equipment’s, coal gasification, and wall cooled catalytic reactors, and
natural convection in earth’s crust. More examinations of the applications related to
convective flows in porous media in Nield and Bejan [1] can be found. The
fundamental problem of flow through porous media has been investigated
extensively [2, 3]. A survey of Magneto Hydro Dynamics revises in the technological
spheres in Moreau [6] can be found. When heat and mass transfer occur
instantaneously between the fluxes, the driving potentials are of more intricate nature.
The uses of magnetic field to control the flow and heat transfer processes in fluids
near different types of boundaries are familiar now. It has led to significant interest in
the study of boundary layer. Hossain [5] analysed on MHD free convective heat
transfer for a Newtonian fluid. The natural convection flows adjacent to both vertical
and horizontal surface, which result from the combined buoyancy effects of thermal
and mass diffusion, was first investigated by Gebhart et al. [7] and Pera et al. [8].
While Soundalgekar [9] investigated the situation of unsteady free convective flows
wherein the effects of viscous dissipation on the flow past an infinite vertical porous
plate was highlighted. In the course of analysis, it was assumed that the plate
temperature oscillates in such a way that its amplitude is small. Later, Chen et al [10]
studied the combined effect of buoyancy forces from thermal and mass diffusion on
forced convection.
The present analysis discussed here in, is based on the study as referred and suggested
by Soundalgekar [11]. Under the assumptions made by Sharma et al. [12] have also
discussed the free convection flow past a vertical plate in slip-flow regime. They had
quoted several applications that occur in several engineering applications wherein
heat and mass transfer occurs at high degree of temperature differences.
In all above presentations, the plate was assumed to be maintained at a constant
temperature, which is also the temperature of the surrounding stationary fluid.
However, in many applications that occur in industrial situations are not those simple
and at high temperatures, quite often the plate temperature starts oscillating about a
non-zero mean temperature. In many practical applications, the particle adjacent to a
solid surface no longer takes the velocity of the surface. The particles at the surface
always possess a finite tangential velocity and it "slips" almost along the surface.
Magneto Hydro Dynamics Convective Flow Past a Vertical Porous Surface … 73
Therefore, the flow regime is called the slip - flow regime and such an effect cannot
be neglected. Hence, in the fitness of the industrial and scientific applications and to
be more realistic the effect of periodic heat and mass transfer on unsteady free
convection flow past a vertical flat porous plate under the influence of applied
transverse magnetic field has been examined. The slip flow regime it is assumed that
the suction velocity oscillates in time about a non-zero constant mean because in
actual practice temperature, species concentration and suction velocity may not
always be uniform.
2. FORMULATION OF THE PROBLEM
An unsteady free MHD convective flow of a viscous incompressible fluid past an
infinite vertical porous flat plate in slip-flow regime, with periodic temperature and
concentration when variable suction velocity distribution * *
0 1 i tV V Ae
is
fluctuating with respect to time is considered. A co-ordinate system is employed
with wall lying vertically in x y plane. The x axis is taken in vertically upward
direction along the vertical porous plate and y axis is taken normal to the plate.
Since the plate is considered infinite in the x direction, hence all physical quantities
will be independent of x . Under these assumption, the physical variables are purely
the functions of y and t only. In the fairness of the realistic situation by neglecting
viscous dissipation and then assuming variation of density in the body force term
(Boussinesq's approximation) the problem can be governed by the following set of
equations:
22
* 0 0
0 21 i tu u uV Ae g T T g C C u u
t y y K
(1)
2
*
0 21 i t
PT T TC V Ae kt y y
(2)
2
*
0 21 i tC C CV Ae D
t y y
(3)
The boundary conditions of the problem are:
*
*
, , at 0
0, , as
i t i t i tw w w wu B e T T T T e C C C C e y
u T T C C y
(4)
74 G.Dharmaiah, Uday Kumar.Y. and N.Vedavathi
We now introduce the following non-dimensional quantities into Eqs. (1) to (4)
2
0 0
2
0 0
3 3
0 0
2 2
0 0
2 2
0 0
4, , , , ,
4
. , ,
, , , , .
w
w w
w
p p
y V t V T Tuy t uV V T T
g T T g C CC CC Gr GcC C V V
C C K V BPr Sc M K Bk k D V V
The subscript denotes the free stream condition. Then equations (1) to (3) reduce
to the following non-dimensional form:
2
2
11
4
i tu u u uAe Gr GcC Mut y Ky
(5)
2
2
1 11
4 Pr
i tAet y y
(6)
2
2
1 11
4
i tC C CAet y Sc y
(7)
The boundary conditions to the problem in the dimensionless form are:
, 1 , 1 at 0
0, 0, 0 as
i t i t i tu Be e C e yu C y
(8)
3. SOLUTION OF THE PROBLEM
Assuming the small amplitude oscillations (ε << 1), we can represent the velocity u,
temperature θ and concentration C near the plate as follows:
0 1, i tu y t u y u y e (9)
0 1, i ty t y y e (10)
0 1, i tC y t C y C y e (11)
Substituting (9) to (11) in (5) to (7), equating the coefficients of harmonic and non
harmonic terms, neglecting the coefficients of 2 , the solutions are given by
Pr
0
yy e (12)
0
ScyC y e (13)
Magneto Hydro Dynamics Convective Flow Past a Vertical Porous Surface … 75
2 Pr
0 1 3 4
m y y Scyu y m e m e m e (14)
6 Pr
1 5 7
m y yy m e m e (15)
9
1 8 10
m y ScyC y m e m e (16)
6 912 2 Pr
1 11 13 14 15 16 17
m y m ym y m y y Scyu y m e m e m e m e m e m e (17)
Using equations (12)-(17), the velocity, temperature and concentration can be
obtained as follows:
6 912
2
2
11 13 14Pr
1 3 4 Pr
15 16 17
,
m y m ym ym y y Scy i t
m y y Scy
m e m e m eu y t m e m e m e e
m e m e m e
6Pr Pr
5 7,m yy y i ty t e m e m e e
9
8 10,m yScy Scy i tC y t e m e m e e
4. RESULTS AND CONCLUSION The effect of Grashof number on velocity profiles is illustrated in Fig. 1. It is noticed
that increase in Grashof number contributes to the increase in velocity of the fluid.
Further, it is noticed in the boundary layer region the velocity increases and thereafter,
it decreases. Also, far away from the plate, not much of significant effect of Grashof
number is noticed.
Fig.1. Effect of Gr on velocity profiles
The variation of velocity with respect to Gc is noticed in Fig. 2. It is observed that as
Gc increases, in general the velocity also increases. As was seen in earlier case, the
76 G.Dharmaiah, Uday Kumar.Y. and N.Vedavathi
rise in velocity is noticed in the boundary layer region and thereafter, it decreases. The
contribution of Schmidt number on the velocity profiles is noticed in Fig. 3. It is seen
that as Schmidt number increases, the velocity decreases. It is seen that the
contribution by Sc is not that significant at the boundary but as we move far away
from the plate, the dispersion due to Sc is found to be more distinct and the effect of
Sc could be noticed. The contribution of the magnetic intensity over the velocity field
is observed in Fig. 4. It is observed that, as the magnetic intensity is increased, the
fluid velocity decreases. Such an observation is in tune with the realistic situation that,
as the magnetic intensity suppresses the fluid velocity. The influence of the porosity
of the fluid bed on the velocity profiles is shown in Fig. 5. It is noted that, as the pore
size of the fluid bed increases, the velocity decreases. This is in agreement with the
real life situation. As the pore size of the fluid bed increases, the fluid over the bed
gets trapped into the pores resulting in the decrease of the fluid velocity. Variation in
the velocity of the fluid medium with respect to suction parameter is illustrated in Fig.
6. It is noticed that as the suction parameter is increased, the velocity of the fluid
medium is found to be decreasing. At the boundary, the suction parameter does not
show any influence. However, as we move far away from the plate the influence
appears to be more predominant. The influence of the amplitude on the velocity
profiles is shown in Fig. 7. It is noticed that, as the amplitude increases, the fluid
velocity decreases and also at times a backward flow is noticed. Due to the
percolation of the fluid into the boundary such a backward flow is noticed. However,
as we move far away from the bounding surface, the effect of such backward flow is
found to be negligible and the influence of amplitude is not seen.
Fig. 8 illustrates the influence of Prandtl number on the temperature field. It is
observed that the prandtl number has significant contribution over the temperature
field. From the illustrations it is seen that as the Prandtl No increases, the temperature
decreases. Further, not much of significant contribution by the Prandtl number is
noticed at the boundary and also as we move far away from the plate. The influence
of suction parameter on the temperature field is illustrated in Fig. 9. It is seen that as
the suction parameter increases, the temperature field decreases. It is observed that the
increase in the suction parameter contributes to the parabolic nature of the
temperature profiles. Further, far away from the plate, it is noticed that the suction
parameter tends to loose its significance. Fig. 10 illustrates the effect of Sc on
concentration profiles. It is seen that as Sc increases, the concentration is found to be
decreasing. Further, increase in Sc contributes to the parabolic nature of the
concentration profiles. Also, the dispersion in concentration is found to be more
distinctive as we move far away from the bounding surface.
Magneto Hydro Dynamics Convective Flow Past a Vertical Porous Surface … 77
Fig. 2. Effect of Gc on velocity profiles
Fig. 3. Effect of Sc on velocity profiles
Fig. 4. Effect of Magnetic field on velocity profiles.
78 G.Dharmaiah, Uday Kumar.Y. and N.Vedavathi
Fig. 5. Effect of Permeability parameter on velocity
Fig. 6. Effect of Suction parameter on velocity
Fig. 7. Effect of B on velocity profiles.
Magneto Hydro Dynamics Convective Flow Past a Vertical Porous Surface … 79
Fig. 8. Effect of Pr on temperature profiles.
Fig. 9. Effect of Suction parameter on temperature.
Fig.10. Effect of Sc on concentration
80 G.Dharmaiah, Uday Kumar.Y. and N.Vedavathi
ACKNOWLEDGMENTS
The authors are very much grateful to the reviewers for their constructive and
valuable suggestions for further improvement in this paper.
REFERENCES:
[1] D .Nield, A. Bejan “Convection in porous media”,Newyork, Springer, 1999.
[2] P.Cheng, “Heat transfer in geothermal system”, Adv Heat Transfer 978,14:1-
105.
[3] N. Rudraiah, “Flow through and past porous media”,Encyclopedia of Fluid
mechanics, Gulf Publ,1986,5:567-647.
[4] Y.J. Kim, “Unsteady MHD convective heat transfer past a semi-infinite
vertical porous moving plate with variable suction”, Int J Eng Sci, 2000,
38:833–45.
[5] M. A. Hossain, “Viscous and Joule heating effects on MHD free convection
flow with variable plate temperature”, Int. J. Heat Mass Transfer, 35, 3485,
1992.
[6] R. Moreau, Magnetohydrodynamics, Kluwar Academic, Dordrecht, The
Netherlands, 1990.
[7] Gebhart, B. and L. Pera, "The nature of vertical natural convection flow from
the combined buoyancy effects on thermal and mass diffusion", Int. J. Heat
Mass Transfer, 14, 2024-2050 (1971).
[8] Pera, L. and B. Gebhart, "Natural convection flows adjacent to horizontal
surface resulting from the combined buoyancy effects of thermal and mass
diffusion", Int. J. Heat Mass Transfer, 15, 269-278 (1972).
[9] Soundalgekar, V.M., "Viscous dissipation effects on unsteady free convective
flow past an infinite vertical porous plate with constant suction", Int. J. Heat
Mass Transfer, 15, 1253-1261 (1972).
[10] Chen, T.S., C.F. Yuh and A. Moutsoglou, "Combined heat and mass transfer
in mixed convection along a vertical and inclined plates", Int. J. Heat Mass
Transfer, 23, 527-537 (1980).
[11] Soundalgekar, V.M. and P.D. Wavre, "Unsteady free convection flow past an
infinite vertical plate with constant suction and mass transfer", Int. J. Heat
Mass Transfer, 20, 1363-1373, (1977a).
[12] Sharma, P.K. and R.C. Chaudhary, "Effect of variable suction on transient free
convection viscous incompressible flow past a vertical plate with periodic
temperature variations in slip-flow regime", Emirates Journal for Engineering
Research, 8, 33-38 (2003).
Magneto Hydro Dynamics Convective Flow Past a Vertical Porous Surface … 81
APPENDIX:
1
1,M M
K
1 3 4 ,m m m
1
2
1 1 4,
2
Mm
3 2
1
,Pr Pr
GrmM
4 2
1
,Gcm
Sc Sc M
5
4 Pr1 ,
iAm
2
6
Pr Pr Pr,
2
im
7
4 Pr,
iAm
8
41 ,
iAScm
2
9 ,2
Sc Sc i Scm
10
4,
iAScm
11 13 14 15 16 17 ,
i tBm e m m m m m
1
12
1 1 4,
2
M im
5
13 2
6 6 1
,/ 4
m Grm
m m M i
8
14 2
9 9 1
,/ 4
m Gcm
m m M i
1 2
15 2
2 2 1
,/ 4
m m Am
m m M i
3 7
16 2
1
Pr,
Pr Pr / 4
m A m Grm
M i
4 10
17 2
1
./ 4
m ASc m Gcm
Sc Sc M i
82 G.Dharmaiah, Uday Kumar.Y. and N.Vedavathi