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Fluid Mechanics:
Fluid mechanics deals with the study of all fluids under static
and dynamic situations. Fluid mechanics is a branch of continuous
mechanics which deals with a relationship between forces, motions,
and statically conditions in a continuous material. This study area
deals with many and diversified problems such as surface tension,
fluid statics, flow in enclose bodies, or flow round bodies (solid or
otherwise), flow stability, etc. In fact, almost any action a person is
doing involves some kind of a fluid mechanics problem. Furthermore,
the boundary between the solid mechanics and fluid mechanics is
some kind of gray shed and not a sharp distinction. For example, glass
appears as a solid material, but a closer look reveals that the glass is a
liquid with a large viscosity. A proof of the glass “liquidity” is the
change of the glass thickness in high windows in European Churches
after hundred years. The bottom part of the glass is thicker than the
top part. Materials like sand (some call it quick sand) and grains
should be treated as liquids. It is known that these materials have the
ability to drown people. Even material such as aluminum just below
the mushy zone also behaves as a liquid similarly to butter.
Furthermore, material particles that “behaves” as solid mixed with
liquid creates a mixture that behaves as a complex liquid. After it was
established that the boundaries of fluid mechanics aren‟t sharp, most
of the discussion in this book is limited to simple and (mostly)
Newtonian (sometimes power fluids) fluids which will be defined later.
2
Laminar flow
Solid Mechanics Fluid Mechanics
Constant
Fluid Statics Fluid Dynamics
Multi – Phase Boundaries Problems Stability
flow Problems
Internal flow
Figure. Diagram to explain part of relationships of fluid mechanics branches
Continuous Mechanics
Turbulent
flow
3
The fluid mechanics study involves many fields that have no
clear boundaries between them. Researchers distinguish between
orderly flow and chaotic flow as the laminar flow and the turbulent
flow. The fluid mechanics can also be distinguished between a single
phase flow and multiphase flow (flow made more than one phase or
single distinguishable material). The last boundary (as all the
boundaries in fluid mechanics)isn‟t sharp because fluid can go
through a phase change (condensation or evaporation)in the middle or
during the flow and switch from a single phase flow to a multiphase
flow. Moreover, flow with two phases (or materials) can be treated as a
single phase(for example, air with dust particle).After it was made clear
that the boundaries of fluid mechanics aren‟t sharp, the study must
make arbitrary boundaries between fields. Then the dimensional
analysis can be used explain why in certain cases one distinguish
area/principle is more relevant than the other and some effects can be
neglected. For example, engineers in Software Company analyzed a
flow of a complete still liquid assuming a complex turbulent flow
model. Such absurd analyses are common among engineers who do
not know which model can be applied. Thus, one of the main goals of
this book is to explain what model should be applied. Before dealing
with the boundaries, the simplified private cases must be explained.
There are two main approaches of presenting an introduction of fluid
mechanics materials. The first approach introduces the fluid kinematic
and then the basic governing equations, to be followed by stability,
turbulence, boundary layer and internal and external flow. The second
4
approach deals with the Integral Analysis to be followed with
Differential Analysis, and continue with Empirical Analysis.
Fluid mechanics one of the oldest braches of physics and the
foundation for the understanding of many other aspects of applied
sciences and engineering concerns itself with the investigation of the
motion and equilibrium of fluids. It is wide spread interest in all most
all fields of engineering as well as in Astro – Physics, biology, bio –
medicine, metrology, Physical Chemistry, Plasma Physics and Geo –
Physics. The frontier of fluid dynamic research has been extended into
the exotic regimes of hyper velocity flight and flow of electrically
conducting fluids. This has introduce new fields of interest such has
hypersonic flow and magneto fluid dynamics. In this connection it has
become necessary to combine knowledge of thermodynamic, mass
transfer, heat transfer, electromagnetic theory and fluid mechanics to
fully understand the physical phenomena involved. Anything that
occupies space is a Matter. Matter is divided into three state Solid,
Liquid and Gas. The matters have defined shape in given
thermodynamic condition and in the absence of external force are
called Solids. If the matter takes the shape of the container is called
Liquids. The liquid and gases collectively called Fluids. In a fluid the
volume does not change are called incompressible fluids (liquids), the
volume changes significantly are called compressible fluids (gases).
The properties of fluids are of general importance to study of
fluid mechanics. mass per unit volume is called Density, weight for
unit volume is called Specific Weight, volume per unit mass of a fluid
5
is called Specific Volume, the ratio of specific weight of the fluid to
that of standard fluid is called Specific Gravity, the property of fluid
that the flow of fluid is called the Viscosity, the capacity to do work is
called Energy, the energy of a particle possessed by virtue of its
position is called Kinematic Energy, sharing stress of fluid element is
directly proportional to the rate shearing strain is called Newton’s Law
of Viscosity, the amount of heat required to raise the temperature of
unit mass of the medium by one degree is called Specific Heat, an
incompressible fluid having no viscosity is called Ideal Fluid, the fluid
which possess viscosity is called Real Fluid, any fluid which obey
Newton‟s law of viscosity is called Newtonian Fluid, any fluid which
does not obey Newton‟s law of viscosity is called Non – Newtonian
Fluid.
Magnetohydrodynamics (MHD):
We can describe scientifically the interaction of electromagnetic
fields and fluids by the proper application of the principles of the
special theory of relativity. The practical applications of these
principles, in Physical Engineering, Astro – Physics, Geo – Physics
etc…, have become an important in recent years. The study of three
applications to continuum is known as Magnetohydrodynamics (MHD)
or Magneto fluid dynamics.
The study of Magnetohydrodynamics (MHD) plays an important
role in agriculture, engineering and petroleum industries. MHD has
won practical applications, for instance, it may be used to deal with
problems such as cooling of nuclear reactors by liquid sodium and
6
induction flow water which depends on the potential differencing the
fluid direction perpendicular to the motion and goes to the magnetic
field. The study of Magnetohydrodynamics (MHD) of viscous
conducting fluids playing a significant role, owing to its practical
interest and abundant applications, in Astro – physical and Geo –
physical phenomena. Astro – Physicsts and Geo – Physicsts realized
the importance of MHD in stellar and planetary processes. The main
impetus to the engineering approach to the electromagnetic fluid
interaction studies has come from the concept of the
magnetohydrodynamics, direct conversion generator, ion propulsion
study of flow problems of electrically conducting fluid, particularly of
ionized gasses is currently receiving considerable interest. Such
studies have made for years in connection with Astro – Physical and
Geo – Physical problems such as sun spot theory, motion of the
instellar gas etc… Recently, some engineering problems need the study
of the flow of an electrically conducting fluid, in ionized gas is called
plasma. Many names have been used in referring to the study of
plasma phenomena. Hartmann called it mercury dynamics, as he
worked with mercury. Astro – Physics called it comical electro
dynamics and some called it magnetohydrodynamics. Physics and
electrical engines commonly use the term plasma physics or plasma
dynamics. The aerodynamist has spoken of magnetohydrodynamics.
7
Applications of Magnetohydrodynamics (MHD):
MHD has applications in many areas. A few brief details are given
below.
1. The Earth:
The Earth The outer core of the Earth is composed primarily of molten
iron. It is here that it is believed that the Earth's magnetic field is
generated. Studying and solving the equations of MHD should permit
us to explain such phenomena as the gradual change of the field with
time and the infrequent and irregular reversals of the field. This is an
area of very active current research. MHD can also be used to describe
the ionosphere.
2. The Sun:
Much of the Sun is composed of ionized hydrogen. For MHD there are
two areas of interest. First there is the convection zone. In this, or just
below it, the solar magnetic field is generated. The basic mechanism
(interaction of a moving electrically conducting fluid with a magnetic
field) is similar to that operating in the Earth's core but results in a
rather different magnetic field. The Solar field reverses regularly on a
22 year cycle. Second, the Solar atmosphere (chromospheres and
corona) is much less dense than the convection zone. Here, features
such as flares and prominences can be observed and studied. One of
the major problems to be explained is the heating of the corona which
reaches temperatures of up to 106 K while the photosphere (the narrow
region separating the convection zone from the chromospheres) is only
at a few thousand degrees K.
8
3. Industry:
Industry Here there are many applications. For example
electromagnetic forces can be used to pump liquid metals (Example. in
cooling systems of nuclear power stations) without the need for any
moving parts. They can shape the flow of a molten metal and so aid
controlling its shape once solidified, and can even levitate and heat a
sample of metal to prevent any contact with (and consequent
contamination from) a container.
4. Fusion:
The goal of copying the Sun; releasing huge quantities of energy from
the fusing of hydrogen into helium, has so far eluded us. No material
can withstand the huge temperatures required. One promising way
around this problem is to contain the ionized hydrogen in a magnetic
container, so that there is no contact between the hydrogen and any
material container. Progress continues but so far the temperatures and
containment times achieved have fallen short of break – even where
the energy put in to the system equals the energy given out from
fusion.
Porous Medium:
A porous medium can be defined as a material consisting of solid
matrix with an interconnected void. In recent years, the investigation
of flow of fluids through porous media has become an important topic
due to the recovery of crude oil from the pores of reservoir rocks. Also
the flow through porous medium is of interest in Chemical Engineering
(absorption, filtration), Petroleum Engineering, Hydrology, Soil Physics,
9
Bio – Physics and Geo – Physics. With the growing importance of
Non – Newtonian fluids in modern technology and industries, the
thermal instability, thermal solution instability and Rayleigh – Taylor
instability, problems of Walters (model B) fluid and stress fluid are
desirable.
The definition of porosity of the porous medium can be given as
the ratio of pore volume to the total volume of a given sample of
material. A complete graduation exisists from large force easily,
accessible fluids to very small openings in minerals that are caused by
minor lattice imperfection. Moisture equivalent, effective porosity,
specific rententation, drainage coefficient of storage such as degree of
saturation, forces applied to the sample, length of test, degree of
interconnection of pores and fluid chemistry. Permeability of the
porous medium is a measure of ease with which fluids pass through a
porous material. The intrinsic permeability is an important property of
the solid material and it is independent of the density and viscosity of
the fluid.
The permeability K can be defined as
1
S
h
gA
QK
(1)
Where Q is the total discharge of the fluid, A is the crossectional area,
is the viscosity, is the density, g is the acceleration due to
gravity, S
h
is the hydraulic gradient in the direction of the flow. The
dimension of the permeability is 2L . The unit of permeability is named
as Darcy who is extensively used in petroleum industry. The value of
10
one Darcy is 2810987.0 mC . Permeability is very high with air and
other non polar fluids.
Boundary Conditions:
1. No – slip condition:
When the flow takes place over a rigid plate, the velocity
component vanishes at the boundaries. This is called no – slip
condition.
2. Free surface boundary condition:
Vertical component of velocity vanishes at a horizontal free
surface. Further, if there is no surface tension, the free surface will be
free form shear stress.
3. Beavers and Joseph slip condition:
When a fluid flows, an imperable surface, the no – slip condition
is valid on the boundary. But when a fluid flows over a permeable
surface, it is necessary to specify some condition on the tangential
component of the velocity of the free fluid at the boundary within the
permeable surface at the permeable interface. In this case, there will
be a migration of the fluid tangential drag due to the transfer of
forward momentum across the permeable interface. The velocity
inside the permeable bed will be different from the velocity of the fluid
past over the permeable bed. These two velocities are to be matched at
the normal boundary (surface) of the permeable bed.
The nominal boundary of a permeable bed is defined as a
smooth geometric surface with the assumption that the outermost
perimeters of all surface pores of the permeable material are in this
11
surface. Thus, if the surface is filled with solid material to the level of
their respective perimeters a smooth rigid boundary of the assumed
shape results. The Newtonian fluid flows past a permeable bed the no
– slip condition is not valid there. Firstly, Beavers and Joseph proved
that there exisists a slip on the velocity at the surface of the porous
bed.
Thermal Radiation:
The third mode of heat transmission due to electromagnetic
wave propagation, which can occur in a total vacuum as well as in a
medium. Thermal radiation is an important factor in the thermo
dynamic analysis of many high temperature systems like solar
connectors, boilers and furnaces. The simultaneous effects of heat and
mass transfer in the presence of thermal radiation play an important
role in manufacturing industries. For the design of fins, steel rolling,
nuclear power plants, cooling of towers, gas turbines and various
propulsion devices for aircraft, combustion and furnace design,
materials processing, energy utilization, temperature measurements,
remote sensing for astronomy and space exploration, food processing
and cryogenic engineering, as well as numerous agricultural, health
and military applications. Experimental evidence indicates that radiant
heat transfer is proportional to the fourth power of the absolute
temperature, where as conduction and convection are proportional to a
linear temperature difference. The fundamental Stefan – Boltzmann
law is
4'ATq (2)
12
When thermal radiation strikes a body, it can be absorbed by the body,
reflected from the body, or transmitted through the body. The fraction
of the incident radiation which is absorbed by the body is called
Absorptivity (symbol ). Other fractions of incident radiation which
are reflected and transmitted are called reflectivity (symbol 1 ) and
Transmissivity (symbol * )respectively. The sum of these fractions
should be unity i.e. 1* .
Free or Natural convection:
Free or Natural convection is a mechanism, or type of heat
transport, in which the fluid motion is not generated by any external
source (like a pump, fan, suction device, etc.) but only by density
differences in the fluid occurring due to temperature gradients. In
natural convection, fluid surrounding a heat source receives heat,
becomes less dense and rises. The surrounding, cooler fluid then
moves to replace it. This cooler fluid is then heated and the process
continues, forming convection current; this process transfers heat
energy from the bottom of the convection cell to top. The driving force
for natural convection is buoyancy, a result of differences in fluid
13
density. Because of this, the presence of a proper acceleration such as
arises from resistance to gravity, or an equivalent force (arising
from acceleration, centrifugal force or Coriolis effect), is essential for
natural convection. For example, natural convection essentially does
not operate in free fall (inertial) environments, such as that of the
orbiting International Space Station, where other heat transfer
mechanisms are required to prevent electronic components from
overheating. Natural convection has attracted a great deal of attention
from researchers because of its presence both in nature and
engineering applications.
In nature, convection cells formed from air raising above
sunlight-warmed land or water are a major feature of all weather
systems. Convection is also seen in the rising plume of hot air
from fire, oceanic currents, and sea-wind formation (where upward
convection is also modified by Coriolis forces). In engineering
applications, convection is commonly visualized in the formation of
microstructures during the cooling of molten metals, and fluid flows
around shrouded heat – dissipation fins, and solar ponds. A very
common industrial application of natural convection is free air cooling
without the aid of fans: this can happen on small scales (computer
chips) to large scale process equipment.
Couette flow:
In fluid dynamics, Couette flow refers to the free convection flow
of a viscous fluid in the space between two parallel plates, one of which
moving relative to the other. The flow is driven by virtue of viscous
14
drag force acting on the fluid and the applied pressure gradient
parallel to the plates. This type of flow is named in honor of Maurice
Marie Alfred Couette, a professor of physics at the French university of
Angers in the late 19thcentury. Couette flow is frequently used in
undergraduate physics and engineering courses to illustrate shear –
driven fluid motion.
Couette flows find widespread applications in geophysics,
planetary sciences and also many areas of industrial engineering. For
many decades engineers have studied such flows with and without
rotation and also for both the steady case and unsteady case.
Newtonian and non – Newtonian flows with for example magnetic field
effects and heat transfer have also been examined. Such studies have
entailed many configurations including the flow between rotating
plates, rotating concentric cylinders, etc. In rotating Couette flows a
viscous layer at the boundary is instantaneously set into motion.
15
Hall Effect:
The Hall Effect was discovered in 1879 by Edwin Herbert
Hall while he was working on his doctoral degree at Johns Hopkins
University in Baltimore, Maryland. His measurements of the tiny effect
produced in the apparatus he was used an experimental tour de force,
accomplished 18 years before the electron was discovered.
When the strength of applied magnetic field is very strong, one
cannot neglect the effect of hall currents. Due to the gyration and drift
of charged particles, the conductivity parallel to the electric field is
reduced and the current is induced in the direction normal to both
electric and magnetic fields. This phenomenon is known as the “Hall
Effect”. The Hall Effect is the production of a voltage
difference (the Hall voltage) across an electrical conductor, transverse
to an electric current in the conductor and a magnetic
16
field perpendicular to the current. The Hall Effect comes about due to
the nature of the current in a conductor. Current consists of the
movement of many small charge carriers, typically electrons, holes,
ions (see Electromigration) or all three. Moving charges experience a
force, called the Lorentz force, when a magnetic field is present that is
perpendicular to their motion.
When such a magnetic field is absent, the charges follow
approximately straight, 'line of sight' paths between collisions with
impurities, phonons, etc. However, when a perpendicular magnetic
field is applied, their paths between collisions are curved so that
moving charges accumulate on one face of the material. This leaves
equal and opposite charges exposed on the other face, where there is a
scarcity of mobile charges. The result is an asymmetric distribution of
charge density across the Hall element that is perpendicular to both
the 'line of sight' path and the applied magnetic field. The separation of
charge establishes an electric field that opposes the migration of
further charge, so a steady electrical potential builds up for as long as
the charge is flowing. It shall be noted that in the classical view, there
are only electrons moving in the same average direction both in the
case of electron or hole conductivity. This cannot explain the opposite
sign of the Hall Effect observed. The difference is that electrons in the
upper bound of the valence band have opposite group
velocity and wave vector direction when moving, which can be
effectively treated as if positively charged particles (holes) are moved in
opposite direction than the electrons do.
17
For a simple metal where there is only one type of charge
carrier (electrons) the Hall voltage HV is given by
ned
IBVH (3)
The Hall coefficient is defined as the ratio of the induced electric field
to the product of the current density and the applied magnetic field. It
is a characteristic of the material from which the conductor is made,
since its value depends on the type, number and properties of
the charge carriers that constitute the current.
The Hall coefficient is defined as
BJ
ER
x
y
H (4)
In SI units, this becomes
neIB
dV
BJ
ER H
x
y
H
1 (5)
18
As a result, the Hall Effect is very useful as a means to measure
either the carrier density or the magnetic field. One very important
feature of the Hall Effect is that it differentiates between positive
charges moving in one direction and negative charges moving in the
opposite. The Hall Effect offered the first real proof that electric
currents in metals are carried by moving electrons, not by protons. The
Hall Effect also showed that in some substances (especially p – type
semiconductors), it is more appropriate to think of the current as
positive "holes" moving rather than negative electrons.
A common source of confusion with the Hall Effect is that holes
moving to the left are really electrons moving to the right, so one
expects the same sign of the Hall coefficient for both electrons and
holes. This confusion, however, can only be resolved by modern
quantum mechanical theory of transport in solids. It must be noted
though that the sample in homogeneity might result in spurious sign
of the Hall Effect, even in ideal vander – Pauw configuration of
electrodes. For example, positive Hall Effect was observed in evidently
n – type semiconductors. The Hall Effect is a conduction phenomenon
which is different for different charge carriers. In most common
electrical applications, the conventional current is used partly because
it makes no difference whether you consider positive or negative charge
to be moving. But the Hall voltage has a different polarity for positive
and negative charge carriers and it has been used to study the details
of conduction in semiconductors and other materials which show a
combination of negative and positive charge carriers.
19
The Hall Effect can be used to measure the average drift velocity
of the charge carriers by mechanically moving the Hall probe at
different speeds until the Hall voltage disappears, showing that the
charge carriers are now not moving with respect to the magnetic field.
Other types of investigations of carrier behaviour are studied in the
quantum Hall Effect. The effect of hall currents on the fluid with
variable concentration has a lot of applications in MHD power
generators, several astrophysical and meteorological studies as well as
in flow of plasma through MHD power generators.
From the point of applications, this effect can be taken into
account within the range of magnetohydrodynamical approximation.
Hall probes are often used as magnetometers, i.e. to measure magnetic
fields, or inspect materials (such as tubing or pipelines) using the
principles of magnetic flux leakage. Hall Effect devices produce a very
low signal level and thus require amplification. While suitable for
laboratory instruments, the vacuum tube amplifiers available in the
first half of the 20th century were too expensive, power consuming and
unreliable for everyday applications. It was only with the development
of the low cost integrated circuit that the Hall Effect sensor became
suitable for mass application. Many devices now sold as Hall Effect
sensors in fact contain both the sensor as described above plus a high
gain integrated circuit ( IC ) amplifier in a single package. Recent
advances have further added into one package an analogue to digital
converter and CI 2 (Inter – integrated circuit communication protocol)
IC for direct connection to a microcontroller's OI / port.
20
Basic equations in vector form:
The basic equations in vector form of an unsteady
incompressible viscous, electrically conducting fluid are given as
follows.
1. Continuity Equation:
The continuity equation is 0)(.
q
t
(6)
Where is the fluid density, q is the fluid velocity vector.
t
is the rate of increase of the density in control volume.
)(. q is the rate of mass flux passing out of the control surface
(which surrounds control volume) per unit volume.
Where kwjviuq is the velocity of the fluid.
2. Momentum Equation:
The momentum equation is
BJq
Kqg
pqq
t
q
2).(
(7)
Where qqt
q).(
is the inertia forces,
p
is the pressure gradient,
BJ is the Lorentz force per unit volume,
B is the magnetic induction vector,
g is the acceleration due to gravity,
q2 is the viscous flow,
21
qK
is the porous media.
3. Energy Equation:
The energy equation is
TTQ
JT
C
kTq
t
T
p
22).( (8)
Where T is the temperature,
T is the temperature in the free stream,
pC is the specific heat at constant pressure ).1.( KKgJ ,
is the density,
is the viscous dissipation per unit volume,
2J
is the ohmic dissipation per unit volume,
J is the current conduction,
is the electrical conductivity,
Q is the heat source.
4. Species Diffusion Equation:
The species diffusion or species concentration equation is
CDCCKCDCqt
CT
22).(
(9)
Where TD is the chemical diffusivity,
C is the species concentration,
C is the species concentration in free stream,
D is the chemical molecular diffusivity ).( 12 Sm ,
K is chemical reaction.
22
5. Maxwell Equations:
The Maxwell‟s equations in RMKS are
lD . (Coulomb‟s law) (10)
D is the displacement.
0. B (Absence of free magnetic poles) (11)
B is the local magnetic field
t
B
(Faraday‟s law) (12)
t
DJH
(Ampere‟s law) (13)
qBqEJ (Ohm‟s law) (14)
The current conservation equation is 0.
t
lJ (15)
J is the conduction current.
Non – Dimensional Parameters:
Every physical problem involved some physical quantities, which
can be measured in different units. But the physical problem itself
should not depend on the unit used for measuring these quantities. In
dimensional analysis of any problem we write down the dimensions of
each physical quantity in term of fundamental units. Then by dividing
and rearranging the different units, we get some non – dimensional
numbers. Dimensional analysis of any problem provides information
on qualitative behaviors of the physical problem. The dimensionless
parameter helps us to understand the physical significance of a
23
particular phenomenon associated with the problem. There are usually
two general methods for obtains dimensionless parameters.
1. The inspection analysis
2. The dimensionless analysis
In this thesis the latter method has been used. In this method
the basic equations are made dimensionless using certain dependent
and independent characteristic values. In this processes certain
dimensionless numbers appears as the some of the dimensionless
parameters used in this thesis are explained below.
Grashof number for heat transfer )(Gr :
The Grashof number is usually occurring in free convection heat
transfer problems. This gives the relative importance of buoyancy force
to the viscous forces. This number is defined as:
Gr (Grashof number) =
3
o
w
V
TTg
Modified Grashof number for mass transfer )(Gc :
The Modified Grashof number is usually occurring in natural
convection mass transfer problems.
It is defined as Gc (Modified Grashof number) =
3
*
o
w
V
CCg
Prandtl number (Pr):
Prandtl number is the ratio of viscous forces to the thermal forces. It is
a measure of the relative importance of heat conduction and viscosity
of the fluid. The Prandtl number, like the viscosity and thermal
conductivity, is a material property and it thus varies from fluid to
24
fluid. Usually Prandtl number is large when thermal conductivity is
small and viscosity is large, and small when viscosity is small and
thermal conductivity is large.
It is defined as Pr (Prandtl number) =
p
p
c
c
Thus it gives the relative importance of viscous dissipation to the
thermal dissipation. Usually for gases Prandtl number is of the order of
unity and for the liquids the Prandtl number is large.
Schmidt number )(Sc : Schmidt number is a dimensionless number
defined as the ratio of momentum diffusivity (viscosity) and mass
diffusivity, and is used to characterize fluid flows in which there are
simultaneous momentum and mass diffusion convection processes. It
physically relates the relative thickness of the hydrodynamic layer and
mass transfer boundary layer.
It is defined as Sc (Schmidt number) D
Vo
Hartmann number (or) Magnetic parameter )(M : The dimensionless
quantity denoted by M is known as the Hartmann number. It was first
introduced by Hartmann in 1930, in the study of the plane Poiseuille
flow of an electrically conducting fluid in the presence of transverse
magnetic field, where the important forces are magnetic and viscous
force.
Therefore, M = Magnetic force/Viscous force and mathematically
defined as 2
2
o
o
v
vBM
25
Hall parameter )(m : The Hall parameter )(m in plasma is the ratio
between the electron gyro frequency )( e and the electron – heavy
particles collision frequency )( .
Mathematically, it is defined as e
e
m
Bem
Eckert number )(Ec : It is equal to the square of the fluid far from the
body divided by the product of the specific heat of the fluid at constant
temperature and the difference between the temperatures of the fluid
and the body .
It is denoted Ec by and mathematically defined as
TTC
vEc
wp
o
2
Reynold’s number (Re) : In fluid mechanics, the Reynold‟s number
(Re) concept was introduced by George Gabriel Stokes in 1851,but the
Reynolds number is named after Osborne Reynolds (1842 – 1912), who
popularized its use in 1883. The Reynold‟s number (Re) is a
dimensionless number that gives a measure of the ratio of inertial
forces to viscous forces and consequently quantifies the relative
importance of these two types of forces for given flow conditions (or)
The ratio between total momentum transfer and molecular momentum
transfer is Reynolds number. It is defined as
xU oRe
Darcy number )( : In fluid mechanics, Darcy number )( is a
non – dimensional number used in the study of the flow of fluids in
26
porous media, equal to the fluid velocity times the flow path divided by
the permeability of the medium. It is defined as 2
0
22
wk
Skin – friction )( : The dimensionless shearing stress on the surface
of a body, due to a fluid motion, is known as skin – friction and is
defined by the Newton‟s law of viscosity is given by y
ux
.
We can calculate the shearing stress component in dimensionless form
as 0
2
yo
xx
y
u
V
Rate of heat transfer (or) Nusselt number )(Nu : The heat transfer
co – efficient is generally known as Nusselt number )(Nu is the ratio of
the heat flow by convection process under a unit temperature gradient
to the heat flow by conduction under a unit temperature gradient
through a stationary thickness of meter.
We can calculate the dimensionless coefficient of heat transfer as
follows
TT
y
T
xNuw
y 0
0
1Re
y
xy
Nu
Rate of mass transfer (or) Sherwood number )(Sh : The mass transfer
coefficient is generally known as Sherwood number )(Sh is a diffusion
rate constant that relates the mass transfer rate, mass transfer area
and concentration gradient as driving force.
It is defined as
CC
y
C
xShw
y 0
0
1Re
y
xy
CSh
27
The objective of chapter – 1 is to find the numerical solution of
unsteady magnetohydrodynamic free convective Couette flow of
viscous incompressible fluid confined between two vertical permeable
parallel plates in the presence of thermal radiation is performed. A
uniform magnetic field which acts in a direction orthogonal to the
permeable plates, and uniform suction and injection through the
plates are applied. The magnetic field lines are assumed to be fixed
relative to the moving plate. The momentum equation considers
buoyancy forces while the energy equation incorporates the effects of
thermal radiation. The fluid is considered to be a gray absorbing –
emitting but non – scattering medium in the optically thick limit. The
Rosseland and approximation is used to describe the radiative heat
flux in the energy equation. The two plates are kept at two constant
but different temperatures and the viscous and Joule dissipations are
considered in the energy equation. The non – linear coupled pair of
partial differential equations are solved by an efficient Crank Nicholson
method. With the help of graphs, the effects of the various important
flow parameters entering into the problem on the velocity, temperature
and concentration fields within the boundary layer are discussed. Also
the effects of these flow parameters on skin friction coefficient and
rates of heat and mass transfer in terms of the Nusselt and Sherwood
numbers are presented numerically in tabular form.
Jha and Apere [7] extended the work of Jha [4] by considering
the unsteady MHD free convection Couette flow between two vertical
parallel porous plates with uniform suction and injection. The cases
28
where the magnetic field is considered fixed relative to the fluid and
fixed relative to the moving plate were considered. The velocity and
temperature distributions were obtained using the Laplace transform
technique. The results revealed that both temperature and velocity
decrease with increasing Prandtl number and with increasing
suction/injection parameter. The effect of magnetic field strength on
the velocity is consistent with the results obtained in [3] and [5]. The
velocity has also been found to increase with increasing Grashof
number. An early study of unsteady Couette flow was reported by
Vidyanidhi and Nigam [8] who studied the viscous flow between
rotating parallel plates under constant pressure gradient. Verma and
Sehgal [9] used the micropolar flow model to obtain analytical
solutions for the Couette flow of fluids which can support couple
stresses and distributed body couples. Liu and Chen [10] investigated
computationally the transient rotating Couette flow problem. Jana and
Datta [11] studied the steady Couette flow of a viscous incompressible
fluid between two infinite parallel plates, one stationary and the other
moving with uniform velocity in a rotating frame of reference. Heat
transfer rates were shown to decrease with an increase in rotation
parameter. Mandal and Mandal [13] obtained analytical solutions for
the effects of magnetic field and Hall currents on rotating parallel plate
Couette flow. They are also studied the cases, where the plates have
arbitrary conductivity and thickness. The transient dusty suspension
Couette flow problem was studied by Kythe and Puri [14]. Singh et al.
[15] obtained closed form solutions for velocity and skin friction for
29
rotating hydromagnetic Couette flow, showing that the Ekman number
decreases primary velocities but boosts the secondary velocity values.
The converse effect was reported for the magnetic parameter
(Hartmann number). Other studies of rotating Couette flows include
those by Ghosh [18] who considered magnetic field effects, Hayat et al.
[19] who studied non – Newtonian visco – elastic hydromagnetic fluids,
Choi et al. [20] who reported on free convection effects who considered
the transient Couette flow in a rotating infinitely long parallel plate
system. These studies were all confined to purely fluid regimes.
Chauhan and Rastogi [30] analyzed the effects of thermal
radiation, porosity and suction on unsteady convective hydromagnetic
vertical rotating channel. Ibrahim and Makinde [31] investigated
radiation effect on chemically reaction MHD boundary layer flow of
heat and mass transfer past a porous vertical flat plate. Pal and
Mondal [32] studied the effects of thermal radiation on MHD Darcy –
Forchheimer convective flow pasta stretching sheet in a porous
medium. Palani and Kim [33] analyzed the effect of thermal radiation
on convection flow past a vertical cone with surface heat flux. Recently,
Mahmoud and Waheed [34] examined thermal radiation on flow over
an infinite flat plate with slip velocity. The effects of thermal radiation
and heat source/sink on the natural convection in unsteady
hydromagnetic Couette flow of a viscous incompressible electrically
conducting fluid confined between two vertical parallel plates with
constant heat flux at one boundary are analyzed by Rajput and Sahu
[35]. The magnetic lines of force are assumed to be fixed relative to the
30
moving plate. In deriving the governing equations, a temperature
dependent heat source/sink term is employed and the Rosseland
approximation for the thermal radiation term is assumed to be valid.
The non – dimensional governing equations involved in the present
analysis are solved analytically, to the best possible extent.
Main purpose of this chapter is to find the numerical solution of
unsteady magnetohydrodynamic free convective Couette flow of
viscous incompressible fluid confined between two vertical permeable
parallel plates in the presence of thermal radiation is performed. A
uniform magnetic field which acts in a direction orthogonal to the
permeable plates and uniform suction and injection through the plates
are applied. The magnetic field lines are assumed to be fixed relative to
the moving plate. The momentum equation considers buoyancy forces
while the energy equation incorporates the effects of thermal radiation.
The fluid is considered to be a gray absorbing – emitting but non –
scattering medium in the optically thick limit. The Roseland‟s
approximation is used to describe the radiative heat flux in the energy
equation. The non – linear coupled pair of partial differential equations
are solved by an efficient Crank Nicholson method which is more
economical from computational point of view. The resulting system of
equations are solved to obtain the velocity and temperature
distributions. These solutions are useful to gain a deeper knowledge of
the underlying physical processes and it provides the possibility to get
a benchmark for numerical solvers with reference to basic flow
configurations.
31
Chapter – 2 investigates the effect of thermal radiation on an
unsteady magnetohydrodynamic free convective oscillatory Couette
flow of an optically, viscous thin fluid bounded by two horizontal
porous parallel walls under the influence of an external imposed
transverse magnetic field embedded in a porous medium. The fluid is
considered to be a gray, absorbing – emitting but non – scattering
medium and the Rosseland approximation is used to describe the
radiative heat flux in the energy equation. The non – dimensional
governing coupled equations involved in the present analysis are
solved by an efficient, accurate, and extensively validated and
unconditionally stable finite difference scheme of the Crank Nicholson
method and the expressions for velocity, temperature, Skin friction and
rate of heat transfer has been obtained. Numerical results for velocity
and temperature are presented graphically and the numerical values of
Skin friction and Nusselt number have been tabulated. The effect of
different parameters like thermal Grashof number, Magnetic field
(Hartmann number), Prandtl number, Porosity parameter and Thermal
radiation parameter on the velocity, temperature, Skin friction and
Nusselt number are discussed.
Sharma and Pareek [40] explained the behaviour of steady free
convective MHD flow past a vertical porous moving surface. Singh and
his co – workers [41] have analyzed the effect of heat and mass
transfer in MHD flow of a viscous fluid past a vertical plate under
oscillatory suction velocity. Makinde et al. [42] discussed the unsteady
free convective flow with suction on an accelerating porous plate.
32
Sarangi and Jose [43] studied the unsteady free convective MHD flow
and mass transfer past a vertical porous plate with variable
temperature. Das and his associates [44] estimated the mass transfer
effects on unsteady flow past an accelerated vertical porous plate with
suction employing finite difference analysis. Das et al. [45] investigated
numerically the unsteady free convective MHD flow past an accelerated
vertical plate with suction and heat flux. Das and Mitra [46] discussed
the unsteady mixed convective MHD flow and mass transfer past an
accelerated infinite vertical plate with suction. Bestman and Adjepong
[52] studied the magnetohydrodynamic free convection flow, with
radiative heat transfer, past an infinite moving plate in rotating
incompressible, viscous and optically transparent medium. Das et al.
[53] have analyzed radiation effects on flow past an impulsively started
infinite isothermal vertical plate. Raptis and Perdikis [54] considered
the effects of thermal radiation and free convection flow past a moving
vertical plate. The governing equations were solved analytically. Ghaly
and Elbarbary [58] have investigated the radiation effect on MHD free
convection flow of a gas at a stretching surface with a uniform free
stream. In all the above studies, only steady state flows over a semi –
infinite vertical plate have been considered. The unsteady free
convection flows over a vertical plate has been studied by Gokhale [59]
and Muthucumaraswamy and Ganesan [60]. Bejan and Khair [64]
have investigated the vertical free convective boundary layer flow
embedded in a porous medium resulting from the combined heat and
mass transfer. Lin and Wu [65] were analyzed the problem of
33
simultaneous heat and mass transfer with the entire range of
buoyancy ratio for most practical and chemical species in dilute and
aqueous solutions. Rushi Kumar and Nagarajan [66] studied the mass
transfer effects of MHD free convection flow of an incompressible
viscous dissipative fluid past an infinite vertical plate. Mass transfer
effects on free convection flow of an incompressible viscous dissipative
fluid have been studied by Manohar and Nagarajan [67]. Choi et al.
[68] studied the buoyancy effects in plane Couette flow heated
uniformly from below with constant heat flux. Attia and Sayed –
Ahmed [69] investigated the problem of the effect Hall currents on
unsteady MHD Couette flow and heat transfer of a Bingham fluid with
suction and injection. The effectiveness of variation in the physical
variables on the generalized Couette flow with heat transfer in
presence of porous medium studied by Attia [70]. Makinde and Osalusi
[71] considered the problem of MHD steady flow in a channel filled
with porous material with slip at the boundaries, while, Narahari [72]
studied the effects of thermal radiation and free convection currents on
the unsteady Couette flow between two vertical parallel plates with
constant heat flux at one boundary. Israel – Cookey et al. [73]
discussed oscillatory magnetohydrodynamic Couette flow of a radiating
viscous fluid in a porous medium with periodic wall temperature.
The object of this chapter is to analyze the effect of thermal
radiation on an unsteady magnetohydrodynamic free convective
oscillatory Couette flow of an optically, viscous thin fluid bounded by
two horizontal porous parallel walls under the influence of an external
34
imposed transverse magnetic field embedded in a porous medium. The
fluid is considered to be a gray, absorbing – emitting but
non – scattering medium and the Rosseland approximation is used to
describe the radiative heat flux in the energy equation. The
non – dimensional governing coupled equations involved in the present
analysis are solved by an efficient, accurate, and extensively validated
and unconditionally stable finite difference scheme of the Crank
Nicholson method which is more economical from computational view
point. The effects of various governing parameters on the velocity,
temperature, skin friction coefficient and Nusselt number are shown in
figures and tables and discussed in detail. From computational point
of view it is identified and proved beyond all doubts that the Crank
Nicholson method is more economical in arriving at the solution and
the results obtained are good agreement with the results of
Israel – Cookey et al. [73] in some special cases.
Chapter – 3 is an investigation on the non – linear problem of
the effect of Hall current on the unsteady magnetohydrodynamic free
convective Couette flow of incompressible, electrically conducting fluid
between two permeable plates is carried out, when a uniform magnetic
field is applied transverse to the plate, while the thermal radiation,
viscous and Joule‟s dissipations are taken into account. The fluid is
considered to be a gray, absorbing – emitting but non – scattering
medium and the Rosseland approximation is used to describe the
radiative heat flux in the energy equation. The dimensionless governing
coupled, non – linear boundary layer partial differential equations are
35
solved by an efficient, accurate, and extensively validated and
unconditionally stable finite difference scheme of the Crank Nicholson
method. The effects of thermal radiation and Hall current on primary
and secondary velocity, skin friction and rate of heat transfer are
analyzed in detail for heating and cooling of the plate by convection
currents. Physical interpretations and justifications are rendered for
various results obtained.
Hellums and Churchill [76], using an explicit finite difference
method. Because the explicit finite difference scheme has its own
deficiencies, a more efficient implicit finite difference scheme has been
used by Soundalgekar and Ganesan [77]. A numerical solution of
transient free convection flow with mass transfer on a vertical plate by
employing an implicit method was obtained by Soundalgekar and
Ganesan [78]. Hossain et al. [88] analyzed the influence of thermal
radiation on convective flows over a porous vertical plate. Seddeek [89]
explained the importance of thermal radiation and variable viscosity on
unsteady forced convection with an align magnetic field.
Muthucumaraswamy and Senthil [90] studied the effects of thermal
radiation on heat and mass transfer over a moving vertical plate. Pal
[91] investigated convective heat and mass transfer in stagnation –
point flow towards a stretching sheet with thermal radiation. Aydin
and Kaya [92] justified the effects of thermal radiation on mixed
convection flow over a permeable vertical plate with magnetic field.
Mohamed [93] studied unsteady MHD flow over a vertical moving
porous plate with heat generation and Soret effect. Chauhan and
36
Rastogi [94] analyzed the effects of thermal radiation, porosity, and
suction on unsteady convective hydromagnetic vertical rotating
channel. Ibrahim and Makinde [95] investigated radiation effect on
chemically reaction MHD boundary layer flow of heat and mass
transfer past a porous vertical flat plate. Pal and Mondal [96] studied
the effects of thermal radiation on MHD Darcy Forchheimer convective
flow pasta stretching sheet in a porous medium. Gebhart [99] has
shown the importance of viscous dissipative heat in free convection
flow in the case of isothermal and constant heat flux at the plate.
Gebhart and Mollendorf [100] have considered the effects of viscous
dissipation for external natural convection flow over a surface.
Soundalgekar [101] has analyzed viscous dissipative heat on the two –
dimensional unsteady free convective flow past an infinite vertical
porous plate when the temperature oscillates in time and there is
constant suction at the plate. Maharajan and Gebhart [102] have
reported the influence of viscous dissipation effects in natural
convective flows, showing that the heat transfer rates are reduced by
an increase in the dissipation parameter. Israel Cookey et al. [103]
have investigated the influence of viscous dissipation and radiation on
an unsteady MHD free convection flow past an infinite heated vertical
plate in a porous medium with time dependent suction. Suneetha et al.
[104] have analyzed the effects of viscous dissipation and thermal
radiation on hydromagnetic free convective flow past an impulsively
started vertical plate.
37
Katagiri [112] has studied the effect of Hall currents on the
magnetohydrodynamic boundary layer flow past a semi – infinite flat
plate. Hall effects on hydromagnetic free convection flow along a
porous flat plate with mass transfer have been analyzed by Hossain
and Rashid [113]. Hossain and Mohammad [114] have discussed the
effect of Hall currents on hydromagnetic free convection flow near an
accelerated porous plate. Pop and Watanabe [115] have studied the
Hall effects on the magnetohydrodynamic free convection about a
semi – infinite vertical flat plate. Hall effects on magnetohydrodynamic
boundary layer flow over a continuous moving flat plate have been
investigated by Pop and Watanabe [116]. Sharma et al. [117] have
analyzed the Hall effects on an MHD mixed convective flow of a viscous
incompressible fluid past a vertical porous plate immersed in a porous
medium with heat source/sink. Effects of Hall current and heat
transfer on the flow in a porous medium with slip condition have been
described by Hayat and Abbas [118]. Guria et al. [119] have
investigated the combined effects of Hall current and slip condition on
unsteady flow of a viscous fluid due to non – coaxial rotation of a
porous disk and a fluid at infinity. Hall currents in MHD Couette flow
and heat transfer effects have been investigated in parallel plate
channels with or without ion – slip effects by Soundalgekar et al. [129],
Soundalgekar and Uplekar [130] and Attia [131]. Hall effects on MHD
Couette flow between arbitrarily conducting parallel plates have been
investigated in a rotating system by Mandal and Mandal [132]. The
same problem of MHD Couette flow rotating flow in a rotating system
38
with Hall current was examined by Ghosh [133] in the presence of an
arbitrary magnetic field. The study of hydromagnetic Couette flow in a
porous channel has become important in the applications of fluid
engineering and geophysics. Krishna et al. [134] investigated
convection flow in a rotating porous medium channel. Beg et al. [135]
investigated unsteady magnetohydrodynamic Couette flow in a porous
medium channel with Hall current and heat transfer.
Motivated the above research work, we have proposed in the
present chapter to investigate the effect of Hall current on the
unsteady magnetohydrodynamic free convective Couette flow of
incompressible, electrically conducting fluid between two permeable
plates is carried out, when a uniform magnetic field is applied
transverse to the plate, while the thermal radiation, viscous and
Joule‟s dissipations are taken into account. The fluid is considered to
be a gray, absorbing – emitting but non – scattering medium and the
Rosseland approximation is used to describe the radiative heat flux in
the energy equation. The dimensionless governing coupled,
non – linear boundary layer partial differential equations are solved by
an efficient, accurate, and extensively validated and unconditionally
stable finite difference scheme of the Crank Nicholson method which is
more economical from computational view point. The behaviours of the
velocity, temperature, skin friction coefficient and Nusselt number
have been discussed in detail for variations in the important physical
parameters.
39
The objective of Chapter – 4 is to find the numerical solution of
unsteady magnetohydrodynamic flow of an electrically conducting
viscous incompressible non – Newtonian Bingham fluid bounded by
two parallel non – conducting porous plates is studied with thermal
radiation considering the Hall Effect. An external uniform magnetic
field is applied perpendicular to the plates and the fluid motion is
subjected to a uniform suction and injection. The lower plate is
stationary and the upper plate moves with a constant velocity and the
two plates are kept at different but constant temperatures. The fluid is
considered to be a gray, absorbing emitting but non – scattering
medium and the Rosseland approximation is used to describe the
radiative heat flux in the energy equation. Numerical solutions are
obtained for the governing momentum and energy equations taking the
Joule and viscous dissipations into consideration. The dimensionless
governing coupled, non – linear boundary layer partial differential
equations are solved by an efficient, accurate, and extensively
validated and unconditionally stable finite difference scheme of the
Crank Nicholson method. The effects of the Hall term, the parameter
describing the non – Newtonian behaviour, thermal radiation
parameter and the velocity of suction and injection on both the velocity
and temperature distributions are studied through graphs and tabular
form.
Singh [171] studied the effect of free convection in Couette
motion. He has considered the unsteady free convective flow of a
viscous incompressible fluid between two vertical parallel plates at
40
constant but different temperatures and one of which is impulsively
started in its own plane and the other is kept stationary. This problem
was further extended for magnetohydrodynamic case by Jha [172].
Fully developed laminar free convection Couette flow between two
vertical parallel plates with transverse sinusoidal injection of the fluid
at the stationary plate and its corresponding removal by constant
suction through the plate in uniform motion has been analyzed by
Jain and Gupta [173]. The physical effect of external shear in the form
of Couette flow of a Bingham fluid in a vertical parallel plane channel
with constant temperature differential across the walls was
investigated analytically by Barletta and Magyari [174]. Steady fully
developed combined forced and free convection Couette flow with
viscous dissipation in a vertical channel has been investigated
analytically by Barletta et al. [175]. In their study, the moving wall is
thermally insulated and the wall at rest is kept at a uniform
temperature. Viskanta and Grosh [183] were one of the initial
investigators to study the effects of thermal radiation on temperature
distribution and heat transfer in an absorbing and emitting media
flowing over a wedge. They used Rosseland approximation for the
radiative flux vector to simplify the energy equation. Cess [184] studied
laminar free convection along a vertical isothermal plate with thermal
radiation. The text books by Sparrow and Cess [185] and Howell et al.
[186] present the most essential features and state of the art
applications of radiative heat transfer. Takhar et al. [187] analyzed the
effects of radiation on MHD free convection flow of a gas past a semi –
41
infinite vertical plate. Raptis and Massalas [188] studied oscillatory
magnetohydrodynamic flow of a gray, absorbing emitting fluid with
non-scattering medium past a flat plate in the presence of radiation
assuming the Rosseland flux model. Chamkha [189] discussed thermal
radiation and buoyancy effects on hydromagnetic flow over an
accelerating permeable surface with heat source or sink. Cookey et al.
[190] considered the influence of viscous dissipation and radiation on
unsteady MHD free convection flow past an infinite heated vertical
plate in a porous medium with time dependent suction. Satya
Narayana et al. [197] studied the effects of Hall current and radiative
absorption on MHD natural convection heat and mass transfer flow of
a micropolar fluid in a rotating frame of reference. Seth et al. [198]
investigated effects of Hall current and rotation on unsteady
hydromagnetic natural convection flow of a viscous, incompressible,
electrically conducting and heat absorbing fluid past an impulsively
moving vertical plate with ramped temperature in a porous medium
taking into account the effects of thermal diffusion.
The aim of the present chapter is to find numerical solutions of
unsteady magnetohydrodynamic the numerical solution of unsteady
magnetohydrodynamic flow of an electrically conducting viscous
incompressible non – Newtonian Bingham fluid bounded by two
parallel non – conducting porous plates is studied with thermal
radiation considering the Hall Effect. The dimensionless governing
coupled, non – linear boundary layer partial differential equations are
solved by an efficient, accurate, and extensively validated and
42
unconditionally stable finite difference scheme of the Crank Nicholson
method, which is more economical from a computational point of view.
These solutions are useful to gain a deeper knowledge of the
underlying physical processes and it provides the possibility to get a
benchmark for numerical solvers with reference to basic flow
configurations. The behaviour of the velocity, temperature, skin friction
coefficient and Nusselt number has been discussed in detail for
variations in the physical parameters.
The objective of Chapter – 5 is to find the numerical solution of
natural convection in unsteady hydromagnetic Couette flow of a
viscous incompressible electrically conducting fluid between two
vertical parallel plates in the presence of thermal radiation is obtained
here. The fluid is considered to be a gray, absorbing – emitting but non
– scattering medium and the Rosseland approximation is used to
describe the radiative heat flux in the energy equation. The
dimensionless governing coupled, non – linear boundary layer partial
differential equations are solved by an efficient, accurate, and
extensively validated and unconditionally stable finite difference
scheme of the Crank Nicholson method. Computations are performed
for a wide range of the governing flow parameters, viz., the thermal
Grashof number, Solutal Grashof number, Magnetic field parameter
(Hartmann number), Prandtl number, Thermal radiation parameter
and Schmidt number. The effects of these flow parameters on the
velocity and temperature are shown graphically. Finally, the effects of
various parameters on the on the skin – friction coefficient and Rate of
43
heat and mass transfer at the wall are prepared with various values of
the parameters. These findings are in quantitative agreement with
earlier reported studies.
Bejan and Khair [219] investigated the vertical free convection
boundary layer flow in porous media owing to combined heat and
mass transfer. The suction and blowing effects on free convection
coupled heat and mass transfer over a vertical plate in a saturated
porous medium was studied by Raptis et al. [220] and Lai and Kulacki
[221] respectively. Hydromagnetic flows and heat transfer have become
more important in recent years because of its varied applications in
agriculture, engineering and petroleum industries. Raptis [222] studied
mathematically the case of time varying two – dimensional natural
convective flow of an incompressible, electrically conducting fluid along
an infinite vertical porous plate embedded in a porous medium.
Soundalgekar [223] obtained approximate solutions for
two – dimensional flow of an incompressible viscous flow past an
infinite porous plate with constant suction velocity, the difference
between the temperature of the plate and the free stream is moderately
large causing free convection currents. Soundalgekar et al. [225]
analyzed the problem of free convection effects on Stokes problem for a
vertical plate under the action of transversely applied magnetic field
with mass transfer. Elbashbeshy [226] studied heat and mass transfer
along a vertical plate under the combined buoyancy effects of thermal
and species diffusion, in the presence of magnetic field. Alagoa et al.
[227] studied radiative and free convection effects on MHD flow
44
through porous medium between infinite parallel plates with time –
dependent suction. Bestman and Adjepong [228] analyzed unsteady
hydromagnetic free convection flow with radiative heat transfer in a
rotating fluid. Promise Mebine and Emmanuel Munakurogha Adigio
[229] investigates the effects of thermal radiation on transient MHD
free convection flow over a vertical surface embedded in a porous
medium with periodic temperature. Mebine [235] studied the effect of
thermal radiation on MHD Couette flow with heat transfer between two
parallel plates. The natural convection in unsteady Couette flow of a
viscous incompressible fluid confined between two vertical parallel
plates in the presence of thermal radiation has been studied by
Narahari [236]. Seth et al. [237] have studied unsteady MHD Couette
flow of a viscous incompressible electrically conducting fluid, in the
presence of a transverse magnetic field, between two parallel porous
plates. Deka and Bhattacharya [238] obtained an exact solution of
unsteady free convective Couette flow of a viscous incompressible heat
generating or absorbing fluid confined between two vertical plates in a
porous medium.
The objective of the present chapter is to study the radiation,
heat and mass transfer effects on an unsteady two – dimensional
natural convective Couette flow of a viscous, incompressible,
electrically conducting fluid between two parallel plates with suction,
embedded in a porous medium, under the influence of a uniform
transverse magnetic field. The problem is described by a system of
coupled nonlinear partial differential equations, whose exact solutions
45
are difficult to obtain, whenever possible. Thus, the finite difference
method is adopted for the solution, which is more economical from a
computational point of view. The behaviour of the velocity,
temperature, concentration, skin friction coefficient, Nusselt number
and Sherwood number has been discussed in detail for variations in
the physical parameters.
Future research work to this present research
work: