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Chapter – 1

INTRODUCTION

The science which explains the behavior of fluid under the application of a system of

forces is called Fluid Mechanics which involves the study of kinematics, dynamics and

statics of fluids. Fluid can be defined as a substance which deforms continuously due

to shear stress and can be classified into two parts: liquids and gases. This branch of

Mathematics involves the problems related with the flow of fluid in different physical

conditions in different sectors of science and technology e.g. aerodynamics, rocket

propulsion, ship motion on water, power generation by nuclear reactors, petroleum

industry and hydroelectric power generation etc. The applications of fluid flow are

very wide as it plays a very important role in the industries of steel, plastic, electric

wire, glasses, flow of oil, gases, and molten metals etc. It also helps in maintaining

the temperature of computer chips, vehicle engines and high power machines. In our

daily life, lubrication reduces the friction in different parts of moving objects and saves

energy.

Also in the field of agriculture, the fluid flow has got a very important role in carrying

dissolved minerals and nutrients from soil to the different parts of the plants. The

blood carries oxygen and nutrients to the different organs through arties and veins. In

medical science, the applications of fluid flow require a deep knowledge for the

problems of cardiology and respiratory track. Under different conditions, the

knowledge of fluid motion helps us to solve many types of problem in dairy-plants,

solar science, metrology, cosmic fluid dynamics, flood control and water purification

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system. In observing the operation of cyclones, swirl-burners and scrubbers, the

knowledge of aerosols in curvilinear fluid flow is very necessary. In most of the cases,

the fluids occur in more than one phase therefore multiphase flow gets a significant

role. Also such flows are highly affected by the magnetic field, so we have considered

a two phase flow (i.e. dusty) of incompressible fluids, flowing through different types

of inclined channels placed in a transversely applied magnetic field. Also most of the

fluids occurring in nature are non Newtonian, therefore, seeing their wide range of

applications, it becomes very necessary to thrust on new researches of this area for

its better applications as above described, so we have concentrated our study on non

Newtonian incompressible fluid flowing downward through parallel plates, circular

cylinder, co axial cylinder and hexagonal channel when all are placed in inclined

position with the horizontal direction under transversely applied magnetic field.

Here we have mentioned some important definitions and equations which are to be

used in this work.

Types of fluids:

1. Ideal Fluid: The fluid for which 0 if o is called an ideal fluid. The ideal fluid

is clean and free of solid particles.

2. Newtonian Fluid: The fluids which follow the Newton‟s law of viscosity,

.du

i edy

(1.1)

are called Newtonian fluids. Glycerin, light-hydrocarbon oils, silicone oils, air and gases

are Newtonian fluids.

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Where is called dynamic coefficient of viscosity, u is the velocity of fluid layer

at a height y above the base and is shearing stress.

3. Non Newtonian Fluid: The fluid which has no yield stress and nonlinear

relation between and dy

du are called non Newtonian.

4. Viscous fluid: Every fluid particle experiences stress on it exerted by surrounding

particles. The stress at each part of the fluid surface is resolved into two components

known as pressure and shear stress. First component occurs in moving or rest fluid

while second component occurs only in moving fluid. A fluid is said to be viscous if both

components i.e. pressure and shearing stress exist. By the experiments, it has been

found that in viscous fluid the shear stress is directly proportional to the rate of change

of velocity (u) with respect to height (y).

5. Inviscid fluid: In this fluid, the stress is zero and hence no slip condition can be

satisfied.

6. Dusty viscous fluid: The viscous fluid in which the dust particles are present and its

aerodynamics resistance is less than that of a clean gas is called dusty viscous fluid.

7. Multiphase fluid: A fluid with several different immiscible fluids (oil, water or gas) is

called multi-phase fluid. When the fluid medium is gas, the particulate phase may

consist of solid particles, gas bubbles or liquid droplets immiscible to the fluid phase.

Nitrified fluid is a very useful multi-phase fluid.

Some important types of flows

1. Uniform Flow: A flow in which the velocities of fluid particles are equal at each section

of the channel is called uniform flow.

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2. Non-uniform Flow: A flow in which the velocities of fluid particles are different at each

section of the channel is called non- uniform flow.

3. Compressible flow: A flow is considered to be a compressible flow if the

density of the fluid changes with respect to pressure.

4. Incompressible flow: A flow is considered to be a incompressible flow if the density

of the fluid cannot change with respect to pressure.

5. Rotational Flow: The flow in which the fluid particles rotate about their own axis is

called rotational.

6. Irrotational Flow: The flow in which the fluid particle does not rotate about their own

axis is called Irrotational.

7. Couette flow: It refers to the laminar flow of a viscous fluid in the space

between two parallel plates, one of which is moving relative to the other. The

flow is driven by virtue of viscous drag force acting on the fluid and the applied

pressure gradient parallel to the plates.

8. Inviscid flow: The flow of a fluid that is assumed to have no viscosity is called inviscid

flow.

9. Steady flow: A flow in which properties and conditions of fluid motions do not change

with change of time, is called steady i.e. 0

t

K where K may be velocity, temperature,

pressure, density etc.

10. Unsteady flow: A flow in which properties and conditions of fluid motions depend

upon time, it is called unsteady flow.

11. Laminar flow: A flow in which every fluid particle traces out a definite curve and the

curves traced out by any two different particles never intersect is called laminar flow.

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12. Turbulent flow: A flow in which every fluid particle does not trace out a definite curve

and the curves traced out by the particles intersect, is called turbulent flow.

13. (Multi phase) Two-phase flow: It occurs in a system containing gas and

liquid with a meniscus separating the two phases.

Basic Theory of Multiphase Flow

We have two methods of approach to study the multiphase fluid dynamics:

1. First approach is that the dynamics of a single particle and then tries to apply it to the

multiple particle system containing that analogue.

2. Second approach that by modifying single-phase fluid in such a way as to account for the

presence of particles.

Other related subjects of multiphase fluid dynamics are electrodynamics, electron

state conductivity, electric charges phenomenon and properties of solid. The shape of

particulate phase of matter encountered in multiphase system are in general non –

spherical but while studying the dynamics of multiphase system , we assume that the

particulate phase consists of spherical particles to avoid the unnecessary

complications. When gravity is significant, they attain shapes having small surface

resistance.

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Importance of Dusty Fluid Motion

In recent years, many problem in applied -sciences related with flow of non –

Newtonian fluids with more than one phase have come into picture as most of the

liquids or gases are impure and contain a distribution of solid particles e.g fluidization

process, the process of inhaling oxygen in respiration, formation of rain drops by the

coalescence of small droplets, the movement of dust laden air, using dust in gas-

cooling system to enhance heat transfer process. Scientists and technologists have

taken interest in the study of the problems of gas – solid particle flow occurring in

industries. Dusty fluid phenomena are important in sedimentation, pipe-flow, gas

purification, and transport-process. The gas particle flow is important in fall-out of

pollutant in air and water. It has an important role in exhausting the gas through the

nozzle of rockets with added metal powders. In physiological science, motion of blood

cells in the liquids plasma through arteries can give vital information for

cardiovascular problem. The power-generation by MHD generator, as an alternative

source of energy, can also be the dusty fluid phenomenon. The problem of two

components fluids under the influence of temperature difference is useful in soil

science and geo-physics. The amount of solid particle present in such systems is

variable but definitely effective. Therefore, a consideration for non –Newtonian flows

having dust - factor is to be included to describe the motion in a more precise way,

which is to be emphasized in this work.

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Assumptions in the study

The presence of the dust particles in a homogeneous fluid makes the study of the

dynamics of flow complicated. However, these problems are investigated under

various simplifying assumptions. In order to formulate the fundamental equations of

two phase fluid flow in a reasonably simple form to highlight the essential features,

certain-assumptions are made as given below:

1. The particles of dusty – fluid are assumed spherical in shape having uniform radius.

2. The fluid is incompressible and non-deformable.

3. Reynolds- number of the relative motion between dust and fluid is small compared to

unity.

4. Chemical- reactions, mass-transfer, radiations and interactions have been ignored.

5. The velocity of sedimentation is negligible in comparison to the characteristics velocity

for sufficiently small dust particle.

6. The dust-particle show negligible distortion effect in the flow around them.

7. The magnetic field is constant with respect to the channel.

8. The included magnetic field is neglected, as the magnetic Reynolds‟s number is small.

9. The Hall- effect is ignored.

10. The density of the material of the dust particles (ρ1) is very high as compared to that of

the fluid- density (ρ) so that mass-concentration f is treated as a significant fraction of

unity while the bulk- concentration (f. ρ/ ρ1) is small.

11. The buoyancy force on the particles is neglected as (ρ/ ρ1) is very small as the entire

pressure is being exerted by fluid.

12. The concentration by volume i.e. bulk- concentration of dust is very small so that the

net-effect of the dust on the fluid particles is equivalent to an extra force K N (vp-v) per

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unit volume, N is the number density of the dust particles (i.e. the number of dust

particles per unit volume ) & K (= 6 𝜋 µ a) is called Strokes- drag constant, a is the

radius of spherical particles,µ is the viscosity of clean fluid, vp and v are the velocities of

dust particles and fluid respectively.

In general, the dust is described by two parameters: (i) the mass concentration of dust and

(ii) Its relaxation time 𝜏 = (m / k) which give the rate at which the velocity of a dust particles

adjust to changes in the gas velocity and depends on the size of the individual particle. For

fine dust 𝜏 is less than of a coarse dust.

Two- Dimensional Boundary Layer Equations

The equations governing the two dimensional flow of a dusty fluid are:

0

y

v

x

u (1.2)

2 21( )

2 2

pu u u u u KNu v u upt x y x x y

(1.3)

vpvKN

y

v

x

v

y

p

y

vv

x

vu

t

v

2

2

2

21 (1.4)

0)()(

pNv

ypNuxt

N

(1.5)

puum

K

y

pu

pvx

pu

put

pu

(1.6)

pvvm

K

y

pv

pvx

pv

put

pv

(1.7)

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Where u, v are the components of velocity of fluid phase and up, vp are the

components of velocity of particle phase along the axes.

Since the fluid is incompressible all dissipation terms can be neglected. Also k and

can be assumed to be constants and then for steady flow the boundary layer

equations are:

0

y

v

x

u

(1.8)

upuKN

y

u

x

p

y

uv

x

uu

2

21 (1.9)

0)(

pvN

ypNux

(1.10)

puum

K

y

pu

pvx

pu

pu

(1.11)

For uniformly distributed dust particles i.e., where N is constant throughout the flow

field, there are only four unknowns viz., u, v, up, vp, which can be found by solving

the four equations (1.8) to (1.11)

Basic Equations for the Dusty Fluid

1. Equation of continuity for the fluid phase:

0Div u (1.12)

2. Equation of conservation of momentum of the fluid phase:

2( . ) ( )u

u u p u KN v ut

(1.13)

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3. The continuity equation for the particle phase:

0)(

vNdiv

t

N

(1.14)

4. The conservation of momentum equation for the particle phase:

)().( vum

Kvv

t

v

(1.15)

Boundary Conditions

The fundamental equations stated in the previous section are to be solved under

appropriate boundary conditions to determine the flow fields of the fluid and the dust

particles. In general, the boundary conditions are given below:

1. There will be no mass transfer at a solid boundary.

2. The fluid velocity vanishes at a solid boundary.

3. The dust particles do not slip at the boundary and the boundary conditions are to

be taken from suitable conditions.

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Electromagnetohydrodynamics (EMHD)

The study of the interaction between electromagnetic field and hydrodynamics is

called “Electro-Magneto-Hydrodynamics”. There are many problems in which the

energy in electric field is much smaller than that in magnetic field. In these cases,

electric-field can be ignored and electromagnetic quantities can be expressed in terms

of magnetic field. So in these cases we shall study only the interaction between

magnetic field and hydrodynamics which is known as Magnetohydrodynamics. The

idea of MHD is that the magnetic field can induce currents in a moving conductive

fluid which create forces on the fluid and also change the magnetic field itself. The set

of equations which describe MHD is a combination of the Navier-Stokes equations of

fluid dynamics and Maxwell's equations of electromagnetism. These differential

equations have to be solved simultaneously either analytically or numerically.

Ideal MHD

The MHD in which it is assumed that the fluid has so little resistivity that it can be

treated as a perfect conductor is called an ideal MHD. In ideal MHD, Lenz's law

dictates that the fluid is in a sense tied to the magnetic field lines. The connection

between magnetic field lines and fluid in ideal MHD fixes the topology of the magnetic

field in the fluid.

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Applications of MHD

The application of MHD has a very wide range. MHD applies quite well to astrophysics

as over 99% of baryonic matter content of the Universe is present in plasma, stars,

interplanetary medium (space between the planets), interstellar medium (space

between the stars), nebulae and jets. Many astrophysical systems are not in local

thermal equilibrium and therefore require an additional kinematics treatment to

describe all the phenomena within the system. Sunspots are caused by the Sun‟s

magnetic fields. The solar wind is also governed by MHD. The differential solar

rotation may be long term effect of magnetic drag at the poles of the Sun. The

problems of MHD are related to engineering such as plasma confinement, liquid metal

cooling of nuclear reactors and electromagnetic casting. MHD power generations

fueled by potassium seeded coal combustion gas have showed better performance.

MHD is being extensively made in the study of the hemodynamic flow of blood as well

as in the field of biomedical engineering. During recent years Bio-magnetic Fluid

Dynamics (BFD) has been emerging as a new area for fluid dynamical behavior of

biological fluids in the presence magnetic fields.

Maxwell’s Equations

Maxwell‟s equations for electromagnetism used in MHD are:

JeBCurl 0

(1.18)

00BDiv

(1.19)

0Div J (1.20)

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0B

Curl Et

(1.21)

Ohm’s law

0J E q B (1.22)

Equations of MHD Flow of Dusty Viscous Fluid

In case of uniform transverse magnetic field, it can be assumed that the dust particles

are non-conducting and dust concentration is as small as unable to disturb the

continuity of electromagnetic effects. The equation to represent the motion of a dusty

viscous incompressible fluid in a straight channel under the influence of applied

external transverse uniform magnetic field are given by

)(.2 HJeuv

KNu

puu

t

u

(1.23)

.. vuKvt

vm

(1.24)

0.. vu (1.25)

0.

vN

t

N

(1.26)

If the displacement current is neglected, the Maxwell‟s electromagnetic equations

become

t

HeE

(1.27)

JH 4 (1.28)

J E u He

(1.29)

0. H (1.30)

Standard Differential Equations

Bessel’s Differential equations

The differential equation

02

21

1

2

2

y

x

n

dx

dy

xdx

yd

is called Bessel‟s Differential equations of order n and its solution is given by

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)()( xnJQxnJPy (1.31)

Where

knx

k knk

kxnJ

2

20 1!

1)(

And

knx

k knk

kxnJ

2

20 1!

1)(

are the Bessel‟s Function of the first kind of order n and –n respectively .P and Q are

constants and the value of n is not zero.

If n=o, then the solutions of Bessel‟s equations is

).(0

)(0

xYQxJPy (1.32)

Where

kx

k

kxJ

k

2

21

1)(

0 !

and

kx

kk k

kxJxY

2

2

1.......

3

1

2

11

1 2!

11

log0

)(0

is the Bessel‟s Function of the second kind of order zero.

For the Bessel‟s function of the second kind of order n i.e. for

x

nJx

dxxnJxnY

2

the solution of Bessel‟s equations is

).()( xnYQxnJPy (1.33)

Laplace Transform

If the kernel P(s, t) is defined as ( , )0 0

0P s t

for t

st for te

,

Then ( ) ( ) .f s ste F t dto

(1.34)

is called Laplace transform of the function F(t).

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Inverse Laplace Transform

If f(s) is the Laplace transform of the function F (t), then F (t) is called inverse

Laplace transform of the function f(s).

Convolution Theorem

Let F (t) and G (t) are two functions and their Laplace transform are f (s) and g (s)

respectively, then t

odxxtGxFsgsfL )()()()(1

(1.35)

where 1L is inverse Laplace transform.

Finite Fourier Sine Transform

Let f x is a sectionally continuous function over a finite interval (0, r), then Finite

Fourier sine transform of f x on this interval is defined as

,sin)()( dxr

or

xpxfpsf

where p is an integer. (1.36)

The inversion formula for Finite Fourier Sine Transform is given by

,sin)(1

2)(sin)(

1

2)( pxp

psfxfor

r

xpp

psfr

xf

if interval is ),0( (1.37)

Finite Fourier Cosine Transforms

Let f x is a sectionally continuous function over a finite interval (0, r), then finite

Fourier Cosine transform of f x on this interval is defined as

,cos)()( dxr

or

xpxfpcf

where p is an integer. (1.38)

The inversion formula for Finite Fourier cosine Transform is given by;

,cos)(1

2)0(

1)(cos)(

1

2)0(

1)( pxp

pcfc

fxforr

xpp

pcfrc

fr

xf

if interval is (0, ) (1.39)

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Symbols : Physical quantities

m : Mass of the dust particle

N : Number of the particles

p : Pressure applied in the motion

t : Time

K : Strokes resistance coefficient

: Relaxation – Parameter equal to (m/k)

: Electrical – conductivity of the dust particles

: Coefficient of viscosity of the dust particles

: Kinetic coefficient of viscosity

: Frequency – Parameter of the applied pressure

: Density of the dust particles

u : Fluid velocity

v : Dust velocity

g : gravity

B0 : Magnetic induction

E : Electric field

θ : Inclination of the channel

𝜇𝑒 : Permeability of the fluid

All the physical quantities under consideration are in cgs unit and vector signs are

ignored.

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Non dimensional parameters:

These are the numbers which are used to characterize a particular type of dynamical

similarity to be used in the governing equations of motion including boundary

conditions in this work. These are evolved using two methods:

1. Inspectional analysis: Here we reduce the fundamental equations to non

dimensional form and then the non dimensional parameters are obtained from the

resulting equations.

2. Dimensional analysis: Here the non dimensional parameters are created from the

physical quantities available in the problem.

Thus it is evident that the two phase flow of fluid through different shaped channels

placed under magnetic field and influenced by gravity plays a vital role in various

fields so it needs a major thrust and attention for its development. Therefore we have

stressed our study on this topic with the following objectives.

Objectives:

1. To study the flow of non Newtonian dusty fluids in the presence of magnetic field through

the following channels placed in inclined position to the horizontal direction.

i. Parallel plates

ii. Circular cylinder

iii. Co axial circular cylinder

iv. Hexagonal channels

2. To evolve a new mathematical model governing these flows and justify them with the

physical nature of the problems under assumed parameters.