1

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.

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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: