Transient free convection MHD flow between two vertical
walls with one wall moving in the presence of induced magnetic field and heat
sink
P. Suriyakumar and S.P. Anjali Devi
Department of Applied
Mathematics, Bharathiar University, Coimbatore-46, Tamil Nadu, India
*Corresponding Author: suriyakumar_08@yahoo.co.in,
anjalidevi_s_p@yahoo.co.in
ABSTRACT:
Free convection in real fluids
has been studied extensively due to its widespread applications in industry and
geophysics. Most of these studies have focused on the effects of different fluid dynamical processes and flow
geometry on the ensuing flow. The objective of
this paper is to study the transient free convection flow of an electrically
conducting, viscous, incompressible fluid between two vertical walls, with one
wall moving, in the presence of induced magnetic field and heat sink is
considered. A uniform external magnetic field is applied normal to the walls.
The governing non-dimensional equations are solved numerically using Matlab. Effects of non-dimensional parameters such as
Hartmann number, Prandtl number, Magnetic Prandtl number, Buoyancy distribution parameter, heat sink
parameter and the wall parameter over the velocity, induced magnetic field and
temperature distribution have been discussed with the help of graphs.
KEY WORDS: Transient free
convection, MHD, heat sink, vertical wall, buoyancy distribution parameter.
INTRODUCTION:
Buoyancy driven flows
are of common occurrence in technological, atmospheric and oceanic phenomena,
the buoyancy stratification being achieved often by the temperature field.
Investigations on these types of buoyancy driven natural convection flows are
very frequently encountered in engineering devices and natural environment. Investigation of the natural
convection transport processes due to the coupling of the fluid flow and heat
transfer is a challenging as well as interesting phenomenon. It has been
extensively studied between vertical walls because of its importance in many
engineering applications.
Unsteady natural convection flows, where devices are either heated or
cooled have its wide applications in technological processes, such as the
cooling of the core of a nuclear reactor in the case of power or pump failures
and the warming and cooling of electronic components. Transient natural convection is of fundamental interest in many
industrial and environmental situations such as air conditioning systems, human
comfort in buildings, atmospheric flows, motors, thermal regulation process,
cooling of electronic devices and security of energy systems. The governing
equations of such coupled flows are much more complex than the conventional
fluid flow equations for the non-magnetic case.
Hartmann [1937]
investigated experimentally as well as theoretically the hydromagnetic
flow between two infinite parallel plates.
This work provided
fundamental knowledge for development of several MHD devices such as MHD pumps,
generators, brakes and flow meters. Ostrach [1952]
examined steady laminar convection in a viscous incompressible fluid between
two vertical walls. An analysis of the fully-developed MHD free convective flow
between two vertical, electrically conducting plates has been carried out by Soundalgekar and Haldavnekar
[1973]. Mishra and Mohapatra
[1975] have presented the unsteady free convection flow
from a vertical plate in the presence of a uniform magnetic field.
Joshi [1988] presented a numerical study of the transient natural convection
flow between two vertical parallel plates. Gourla et
al. [1991] studied unsteady free convection MHD flow between heated vertical
plates in the presence of uniform magnetic field. Takhar
et al. [1993&1999] studied induced magnetic field
effects in transient laminar MHD boundary layer convection along an
impulsively-started semi-infinite flat plate with an aligned magnetic field, showing numerically that a reduction in
magnetic Prandtl number increases the surface shear
stress, surface component of the induced magnetic field
and also the surface heat transfer.
Paul et al. [1996] have
studied the flow behavior of a transient free convective flow of a viscous
incompressible fluid in a vertical channel when one of the channel walls is
moving impulsively and the walls are heated asymmetrically. Singh et al. [1996]
have presented the transient free convection flow of a viscous incompressible
fluid in a vertical parallel plate channel, when the walls are heated
asymmetrically. Borkakati and Chakrabarty
[1999] investigated the unsteady free convection MHD flow between two heated
vertical parallel plates in induced magnetic field.
Higuera and Ryazantsev
[2002] have presented the laminar natural convection flow due to a localized
heat source on the center line of a long vertical channel or pipe whose walls
are kept at a constant temperature. Mansour et al.
[2003] investigated the analytical solutions for hydromagnetic
natural convection flow of a particulate suspension through a channel with heat
generation or absorption effects. Jha et al. [2003] have presented an
analytical solution for the transient free convection flow in a vertical
channel as a result of symmetric heating.
Singh and Paul [2006] has investigated Transient free convection flow of
a viscous and incompressible fluid between two vertical walls as a result of
asymmetric heating or cooling of the walls. Ajibade
and Jha [2009] analyzed the transient natural
convection flow between vertical parallel
plates with temperature-dependent heat sources/sinks. Analytical solution to the
problem of MHD free convective flow of an
electrically conducting fluid between two
heated parallel plates in the presence of an induced magnetic field was analyzed by Singha
[2009]. Recently, Ghosh
et al. [2010] has investigated the hydromagnetic
natural convection boundary layer flow past an infinite vertical flat
plate under the influence of a transverse
magnetic field with the inclusion of magnetic
induction effects. Singh et al. [2010] studied hydromagnetic
free convection in the presence of induced magnetic field.
In the proposed study,
we analyze the transient MHD free convection of an electrically conducting
fluid flowing between two vertical walls, with one wall moving, in the presence
of induced magnetic field and heat sink. The governing partial differential
equations are reduced to non-dimensional form involving the non-dimensional
parameters Hartmann number M, Prandtl number Pm, the
wall parameter
3. Numerical Solution
of the Problem
The system of linear parabolic partial differential equations (6) to (8)
subject to the initial and boundary conditions have been solved numerically
using MATLAB. The increment step along t and y directions is chosen as 0.02 in
the numerical computations. The mesh for the variables y and t in the interval
[0, 1] is accordingly selected.
It is worth mentioning here that the present convection problem is
described by a multi-parameter system involving the seven parameters Gr, M, Pr, Pm, R, S and Uw. A complete study of the system thus
necessitates the consideration of the effects of all these parameters and the
study of their effects on the free convection.
As the physical problem involves mixing of too many parameters, the
value of Gr is chosen as unity to avoid complexity.
Regarding key parameter R which arises in the temperature boundary condition,
the values of R are chosen such that -1 < R < 1.
Three different cases are considered.
We have fixed the values of
Gr = 1 and time t = 0.5 throughout our results. We have discussed all the results for the case R = -0.7
when -1 < R < 0. Figures are drawn for numerical results for different
values of the non-dimensional parameters. Two dimensional sketches have been
presented.
4. RESULTS AND DISCUSSION:
Numerical results for the velocity , induced magnetic field and
temperature distribution are obtained by
fixing various values of the non-dimensional parameters M, Pr, Pm ,S and Uw
for the case of R = -0.7
when -1 < R < 0 and the obtained results are shown graphically.
In order to compare
our results for verification purpose, Fig.1 is drawn by fixing the values of
non-dimensional parameters as that of Singh [2010] and Fig.1 demonstrates the
velocity profiles in the absence of S and Uw.
The results are identical to those of Singh [2010] in the absence of S and Uw which are justified
through Fig. 1.
Figures (2) to (13)
represent the two-dimensional plots for illustrating the velocity, induced
magnetic field and temperature.
Fig.2 elucidates the
effect of the Hartmann number over velocity, when the wall at y* = L
remains stationary. The presence of a magnetic field in an electrically
conducting fluid introduces a force called Lorentz force which acts against the
flow if the magnetic field is applied in the normal direction as considered.
The effect of Hartmann number is to accelerate the velocity in minimum part of
the channel and decelerate the remaining part. It is interesting to note both
the increasing and decreasing effect over the velocity. However the influence
is dominant to decelerate the velocity in major portion of the channel when
time t = 0.5.
The influence of the
Hartmann number M on the induced magnetic field through the boundary layer is
depicted in Fig.3. As Hartmann number M increases, the induced magnetic field
decreases and it is parabolic in nature. The dimensionless velocity profiles
for various values of Pr are shown in Fig. 4. From this figure we observe that
velocity decreases with the increasing values of Prandtl
number. Fig. 5 displays the effect of Prandtl number
Pr over the induced magnetic field distribution. The effect of Prandtl number Pr is to decrease the induced magnetic field
upto certain part of the channel from the wall at
Fig.7 depicts the
dimensionless velocity for various values of Pm. For increasing values of Pm,
it is noted that the velocity increases upto certain
region and the reverse effect takes place in the remaining part of the channel.
The effect of Pm over the dimensionless velocity is less significant in the
case of time t = 0.5. The effect of Magnetic Prandtl
number Pm over the induced magnetic field distribution can be visualized through Fig.8. For
increasing values of Magnetic Prandtl number Pm, the
induced magnetic field also increases and it is parabolic in nature.
From Fig.9, it can be observed that the influence of heat sink parameter
S is to decelerate the velocity of the fluid. The induced magnetic field
distribution for various values of heat Sink parameter S is represented in
Fig.10. It is very interesting to note that when the strength of Sink
increases, the induced magnetic field decreases from
Fig.11 illustrates the temperature distribution for different values of
heat sink parameter. The effect of heat sink parameter is having both the
decreasing and increasing effect over the temperature. However the influence is
dominant to reduce the temperature in major portion of the channel and to
increase it in the remaining part of the channel. Fig.12 portrays the effect of
Uw over the
dimensionless velocity. The effect of the speed of the moving wall increases
the velocity of the fluid for its increasing values. Fig.13 demonstrates the
speed of the moving wall over the induced magnetic field. The trend of the
induced magnetic field distribution is to increase for increasing values of Uw throughout the region of the channel.
5. CONCLUSION:
The problem of transient free convection flow of an electrically
conducting, viscous, incompressible fluid between two vertical walls, with one
wall moving, in the presence of induced magnetic field and heat sink has been
analyzed. Numerical solutions of the problem have been obtained for certain
values of the non-dimensional parameters and the results are displayed
graphically.
From the obtained numerical results the following conclusions are drawn
Ĝ The effect of Hartmann
number M over the velocity has both increasing and decreasing trend when the
wall is moving with a constant speed and the Hartmann number M is to reduce the
induced magnetic field.
Ĝ The effect of Prandtl number is to reduce both the velocity and
temperature while it has both decreasing and increasing trend over the induced
magnetic field for its increasing values of Pr.
Ĝ When the wall at y*
= L is moving with a constant speed, magnetic Prandtl
number has both the increasing and decreasing effect over the velocity while it
has increasing effect alone over the induced magnetic field.
Ĝ Velocity gets
decelerated for increasing values of heat sink parameter. The effect of heat
sink parameter has both decreasing and increasing trend over the induced
magnetic field and similar trend is also observed temperature distribution for
its increasing values.
Ĝ Both the velocity and
induced magnetic field increase due to increasing values of Uw,
where Uw is the speed of the moving wall
at y* = L.
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Received on 11.01.2013 Accepted
on 10.02.2013
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