Chapter 1: Introduction to Fluid Mechanics...Chapter 1: Introduction to Fluid Mechanics Page | 3 need to solve waste (sewage) and some basic understanding was created. At some point,

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CHAPTER 1

Introduction to

Fluid Mechanics

1.1 About Fluid Mechanics

Fluid mechanics is the study of fluids and the forces acting on them. (Fluids

include liquids, gases, and plasmas.) Fluid mechanics can be divided into fluid statics,

the study of fluids at rest; fluid kinematics, the study of fluids in motion; and fluid

dynamics, the study of the effect of forces on fluid motion. It is a branch of continuum

mechanics, a subject that models matter without using the information that it is made

out of atoms, that is, it models matter from a macroscopic viewpoint rather than from

a microscopic viewpoint.

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. For example, various aircrafts and rocket engines

involve fluid flow fields, reaction forces. In the food and chemical process industries,

fluid mechanics is required for the design of transport system and in process design.

In fact, almost any action of a person is doing involves some kind of a fluid

mechanics problem.

Fluid mechanics, especially fluid dynamics, is an active field of research with many

unsolved or partly solved problems. Fluid mechanics can be mathematically complex.

Sometimes it can best be solved by numerical methods, typically using computers. A

modern discipline, called computational fluid dynamics (CFD), is devoted to this

approach to solving fluid mechanics problems. Most of the discussion is limited to

simple and (mostly) Newtonian fluid.

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 multi-phase flow (flow made more than one phase or

single distinguishable material).

http://en.wikipedia.org/wiki/Liquid

http://en.wikipedia.org/wiki/Gas

http://en.wikipedia.org/wiki/Plasma_(physics)

Chapter 1: Introduction to Fluid Mechanics

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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 approach deals with the Mathematical

Analysis to be followed with Differential Analysis, and continue with Empirical

Analysis. This thesis attempts to find a hybrid approach in which the kinematic is

presented first follow by similarity analysis and continued by Differential analysis.

The ideal flow (frictionless flow) should be expanded compared to the regular

treatment. This thesis is unique in providing chapters with the latest developments.

Figure 1. 1: Fluid mechanics is a sub discipline of continuum mechanics

1.2 Brief History

* The need to have some understanding of fluid mechanics started with the need

to obtain water supply. For example, people realized that wells have to be dug and

crude pumping devices need to be constructed. Later, a large population created a

* http://en.wikipedia.org/wiki/History_of_fluid_mechanics

Continuum mechanics

(The study of the physics of continuous

materials)

Fluid Mechanics


material, which take the shape of their

container)

Solid Mechanics


materials with a defined rest shape)

Plasticity

Elasticity Newtonian

fluids

Non-Newtonian

fluids

Rheology

(The study of materials with both solid and

fluid characteristics)


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need to solve waste (sewage) and some basic understanding was created. At some

point, people realized that water could be use to move things and provide power.

When cities increased to a larger size, aqueducts were constructed. These aqueducts

reached their greatest size and grandeur in those of the City of Rome and China.

Yet, almost all knowledge of the ancients can be summarized as application of

instincts, with the exception Archimedes (250 B.C.) on the principles of buoyancy.

For example, larger tunnels built for a larger water supply, etc. There were no

calculations even with the great need for water supply and transportation. The first

progress in fluid mechanics was made by Leonardo Da Vinci (1452-1519) who built

the first chambered canal lock near Milan. He also made several attempts to study the

flight (birds) and developed some concepts on the origin of the forces. After his initial

work, the knowledge of fluid mechanics (hydraulic) increasingly gained speed by the

contributions of Galileo, Torricelli, Euler, Newton, Bernoulli family, and D‘Alembert.

At that stage theory and experiments had some discrepancy. This fact acknowledged

by D‘Alembert who stated that, ―The theory of fluids must necessarily be based upon

experiment.‖ For example, the concept of ideal liquid that leads to motion with no

resistance, conflicts with the reality.

This discrepancy between theory and practice is called the ―D‘Alembert

paradox‖ and serves to demonstrate the limitations of theory alone in solving fluid

problems. As in thermodynamics, two different of school of thoughts had created: the

first believed that the solution will come from theoretical aspect alone, and the second

believed that solution is the pure practical (experimental) aspect of fluid mechanics.

On the theoretical side, Euler, La Grange, Helmholtz, Kirchhoff, Rayleigh, Rankine,

and Kelvin made considerable contribution. On the ―experimental‖ side, mainly in

pipes and open channels area, were Brahms, Bossut, Chezy, Dubuat, Fabre, Coulomb,

Dupuit, d‘Aubisson, Hagen, and Poisseuille.

In the middle of the nineteen century, first Navier in the molecular level and

later Stokes from continuous point of view succeeded in creating governing equations

for real fluid motion. Thus, creating a matching between the two schools of thoughts:

experimental and theoretical. However, as in thermodynamics, people cannot


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relinquish control. As results, it created today ―strange‖ names Hydrodynamics,

Hydraulics, Gas Dynamics, and Aeronautics.

The Navier-Stokes equations describe the flow (or even Euler equations),

considered unsolvable during the mid nineteen century because of the high

complexity. This problem led to two consequences. Theoreticians tried to simplify the

equations and arrive at approximated solutions representing specific cases. Examples

of such work are Hermann von Helmholtz‘s concept of vortexes (1858), Lanchester‘s

concept of circulatory flow (1894), and the Kutta-Joukowski circulation theory of lift

(1906). The experimentalists, at the same time proposed many correlations to many

fluid mechanics problems, for example, resistance by Darcy, Weisbach, Fanning,

Ganguillet, and Manning. The obvious happened without theoretical guidance, the

empirical formulas generated by fitting curves to experimental data (even sometime

merely presenting the results in tabular form) resulting in formulas that the

relationship between the physics and properties made very little sense.

At the end of the twentieth century, the demand for vigorous scientific

knowledge that can be applied to various liquids as opposed to formula for every fluid

was created by the expansion of many industries. This demand coupled with new

several novel concepts like the theoretical and experimental researches of Reynolds,

the development of dimensional analysis by Rayleigh, and Froude‘s idea of the use of

models change the science of the fluid mechanics. Perhaps the most radical concept

that affects the fluid mechanics is of Prandtl‘s idea of boundary layer, which is a

combination of the modeling, and dimensional analysis that leads to modern fluid

mechanics. Therefore, many call Prandtl as the father of modern fluid mechanics.

This concept leads to mathematical basis for many approximations. Thus, Prandtl and

his students Blasius, von Karman, Meyer, and Blasius and several other individuals as

Nikuradse, Rose, Taylor, Buckingham, Stanton, and many others, transformed the

fluid mechanics to today modern science.

While the understanding of the fundamentals did not change much, after

World War 2, the way in which it was calculated changed. The introduction of the

computers during the 60s and much more powerful personal computer has changed

the field. Many open source programs can analyze many fluid mechanics situations.


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Today many problems can be analyzed by using the numerical tools and provide

reasonable results. These programs in many cases can capture all the appropriate

parameters and adequately provide a reasonable description of the physics. However,

there are many other cases that numerical analysis cannot provide any meaningful

result (trends). For example, no weather prediction program can produce good

engineering quality results (where the snow will fall within 50 kilometers accuracy.

Building a car with this accuracy is a disaster). In the best scenario, these programs

are as good as the input provided. Thus, assuming turbulent flow for still flow simply

provides erroneous results (see for example, http://ekkinc.com/).

1.3 Some Basic Definitions

In this section, we defined some basic definitions, which are quite useful in the

subsequent chapters. Some of these are taken from Fox and McDonald (1985). Others

are referred to in the text.

1.3.1 Fluid

A fluid is a substance that continually deforms (flows) under an applied shear

stress, no matter how small. Fluids are a subset of the phases of matter and include

liquids, gases, plasmas and, to some an extent, plastic solids. In common usage,

‗fluid‘ is often used as a synonym for ‗liquid‘, with no implication that gas could also

be present.

1.3.2 Density or Mass Density

Density or Mass density of a fluid is defined as the ratio of mass and volume.

Thus mass per unit volume of a fluid is called density and is denoted by . The SI

unit of density is 3/kg m .

The density of liquids may be considered as constant while that of gases changes with

the variation of pressure and temperature.

Mathematically, mass density if written as

Mass of fluid

Volume of fluid (1.1)


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1.3.3 Viscosity and Shear Stress

Viscosity is defined as the property of a fluid, which offers resistance to the

movement of one layer of fluid over another adjacent layer of the fluid. When two

layers of a fluid apart a distance dy , move one over another at different velocity say

and u u du as show in Fig. (1.2), the viscosity together with relative velocity

causes the force acing between the fluid layers, is defined as ‗shear stress‘ and is

denoted by the Greek symbol τ or τyx (for two-dimension).

This shear stress is proportional to the rate of change of velocity with respect to y

that is the shear stress between layers is proportional to the velocity gradient in the

direction perpendicular to the layers. Hence

yx

u

y

(1.2)

Figure 1. 2: Schematics of viscosity variation and the shear stress

Where is proportionality constant and is known as co-efficient of dynamic

viscosity or simply viscosity. Further, u

y

represent the rate of shear strain, a

normalized measure of deformation representing the displacement between particles

in the body relative to a reference length. Thus viscosity is also defined as the shear

stress required producing unit rate of shear strain and its unit is Ns/m2.

Equation (1.2) also describes the ‗Newton‘s law of viscosity.‘

u du

u dy du

Y- Dimension

Velocity profile

Boundary plate (2D, Stationary)

Boundary plate (2D, Moving)


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1.3.4 Kinematic Velocity

The kinematic velocity is defined as the ration of viscosity and density. It is denoted

by Greek symbol and its unit is m2/s. Thus mathematically

Viscosity

Density

(1.3)

1.4 Types of Flow

1.4.1 Compressible and Incompressible flows

An incompressible flow is a flow in which the variation of the density within

the flow is considered constant. In general, all liquids are treated as the

incompressible fluids. On the contrary, flows that are characterized by a varying

density are said to be compressible. Gases are normally used as the compressible

fluids. However, all fluids in reality are compressible because any change in

temperature or pressure result in changes in density. In many situations, though, the

changes in temperature and pressure are so small that the resulting changes in density

are negligible.

The mathematical equation that describes the incompressibility property of the fluid is

given by

0D

Dt

(1.4)

Where D

Dt is the material derivative defined by

DV

Dt t

(1.5)

in which V represents velocity of flow and is the vector differential operator.

1.4.2 Steady and Unsteady Flows

A steady flow is one for which the velocity does not depend on time. When

the velocity varies with respect to time then the flow is called unsteady.


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1.4.3 Laminar and Turbulent Flows

Figure 1. 3: Schematics of laminar and turbulent flow

Laminar flow (Streamlines) is one in which each fluid particle has a definite

path. In such flow, the paths of fluid particulars do not intersect each other. In

turbulent flow, the paths of fluid particles may intersect each other.

Consider water flowing through a pipe at low speeds, there is a nice smooth

condition, which is call laminar flow. The mixing of warm and cold air in the

atmosphere by wind, which causes clear-air turbulence experienced during airplane

flight, as well as poor astronomical seeing.

1.5 Types of Fluids

The fluids are classified in to the following five types:

1. Ideal fluid 2. Real fluid

3. Newtonian fluid 4. Non-Newtonian fluid

5. Ideal plastic fluid

1 Ideal fluid: A fluid, which is incompressible and is having no viscosity, is known

as an ideal fluid. Ideal fluid is only an imaginary fluid as all the fluid, which

exists, has some viscosity.

2 Real fluid: A fluid, which possesses viscosity, is known as real fluid. All the

fluids in actual practice are real fluids.

3 Newtonian fluid: A real fluid, in which the shear stress is directly, proportional to

the rate of shear strain (or velocity gradient), is known as Newtonian fluid.

Laminar Turbulent


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Alternately, a fluid, which obeys the Newton‘s law of viscosity, given by equation

(1.2), is known as Newtonian fluid.

4 Non-Newtonian fluid: A real fluid, in which the shear stress is not proportional to

the rate of shear strain (or velocity gradient), is known as Non-Newtonian fluid.

In case of such a fluid, the relationship between the shear stress and the rate of

strain is an arbitrary functional relation, either implicit or explicit, given by

, 0yx

u

y

(1.6)

5 Ideal plastic fluid: A real fluid, in which the shear stress is more that the yield

value and the shear stress is proportional to the rate of shear strain (or velocity

gradient), is known as ideal plastic fluid.

Figure 1. 4: Types of fluids

Velocity gradient u

y

Shea

r st

ress

Ideal fluid

Newtonian fluid

Non-Newtonian fluid

Ideal Plastic fluid Ideal solid

O


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1.6 Newtonian and Non-Newtonian Fluids

1.6.1 Newtonian Fluids

Fluids can be classified into two broad categories: Newtonian and Non-

Newtonian. A Newtonian fluid is a fluid that has a constant viscosity independent of

the magnitude of the shear stress. For a parallel flow of a Newtonian fluid, as

discussed in section (1.3), Newton‘s law of viscosity holds [Eq. (1.2)]. Also it is

observed that when shear stress yx is plotted against the rate of strain u

y

, the

result for a Newtonian fluid is a straight line going through the origin of the

coordinates as shown in Figure (1.3), the graphs yx plotting versus u

y

are called

rheograms. Physically, u

y

is the velocity gradient or the rate of angular

deformation of the fluid, also known as rate of strain.

The slope of the straight line in any rheogram of a Newtonian fluid represents

the viscosity (or more specifically, the dynamic viscosity) of the fluid. The higher

viscosity of a fluid becomes the steeper the slope in the rheogram. Fluids such as air,

water, oil, glycerin, and honey are described as Newtonian fluids. Figure (1.5) gives

the rheogram of several Newtonian fluids of quite different values of viscosity.

1.6.2 Classification of Non-Newtonian Fluids

For Non-Newtonian fluids, the line in the rheogram either is curved or does

not pass through the origin, or both. Figure (1.6) shows the rheograms of various

types of Non-Newtonian fluids. Note that while the rheogram of a pseudoplastic

fluid curves downward (i.e., having decreasing slope with increased shear), for a

dilatant fluid the rheogram curves upward (i.e. having increasing slope with increased

shear). Otherwise, the two are similar: they both pass through the origin of the

rheogram, as is the case with Newtonian fluids.

As shown in Fig. (1.6), a Bingham plastic (or simply Bingham) fluid is

represented by a straight line in rheograms, but the line does not pass through the

origin. It takes a certain minimum shear stress, called the yield stress, y to cause a

Bingham fluid to behave like a fluid. For yx less than y a Bingham plastic fluid


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Figure 1. 5: Rheogram of various Newtonian fluids

Figure 1. 6: Rheogram of various Non-Newtonian fluids

u

y

O

yx

Dilatan

t

Newtonian

Pseudoplastic

Yield dilatant

Yield-pseudoplastic

Bingham plastic

u

y

Air

O

Water

Oil

Glycerin

yx 1


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behaves like a solid rather than a fluid. When yx becomes greater than y , behaves

like a Newtonian fluid. Finally, a yield-pseudo plastic fluid is similar to a

pseudoplastic fluid except that it requires a minimum shear (yield stress) to behave

like a fluid, and a yield-dilatant fluid is similar to a dilatant fluid but requires a yield

stress.

*Non-Newtonian fluids are solutions or suspensions of particulates (i.e., large

molecules or fine solid particles suspended in a pure fluid). Whether a Non-

Newtonian fluid is pseudoplastic, dilatant, or another type depends on not only the

kind but also the concentration of the suspended particles. In some cases, at low

concentration of particulates the fluid is pseudoplastic. It changes to Bingham plastic

when the concentration is moderate, and then changes to dilatant when the

concentration is high. At very low concentration of solids, all fluids behave like a

Newtonian fluid, with increasing viscosity as the concentration of solids increases.

The reason that a pseudoplastic fluid has a decreasing viscosity when the shear

increases is believed to be a reversible breakdown of loosely bonded aggregates by

the shearing action of the flow.

Examples of pseudoplastic fluids include aqueous suspension of limestone, aqueous

and non-aqueous suspension of certain polymers, hydrocarbon greases etc.

The reason that a dilatant fluid has an increasing viscosity when shear

increases is believed to be due to the shift, under shear, of a closely packed particulate

system to a more open arrangement, which entraps some of the liquid. Examples of

such fluids are aqueous suspensions of magnetite, galena, and ferrosilicon.

Examples of Bingham plastic fluids include water suspensions of clay, fly ash,

sewage sludge, paint, and fine minerals such as coal slurry. The yield stress y for a

Bingham fluid may be very small (less than 0.1 dynes/cm2 for some sewage sludge),

or very large (more than 1010 dyne/cm2 for some asphalts and bitumen). Finally,

some clay-water suspensions at intermediate level of concentration exhibit yield-

pseudoplastic properties.

http://en.wikipedia.org/wiki/Non-Newtonian_fluid


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The aforementioned Non-Newtonian fluids are time-independent. This means their

viscosity under shear does not change with time. Some other Non-Newtonian fluids,

however, have time-dependent rheological properties. For instance, a thixotropic fluid

is a pseudoplastic fluid whose viscosity under constant shear decreases with time.

This is due to particle agglomeration. Water suspension of bentonitic clay, the drilling

fluid used by the petroleum industry, is a thixotropic fluid. Crude oil at low

temperature, such as the oil from the Pembina Field in Canada, is another example of

thixotropic fluid. Another type of time-dependent Non-Newtonian fluid is the

rheopectic fluid. It exhibits negative thixotropic behavior (i.e., the viscosity of the

fluid under shear increases with time).

The development of modern chemical engineering, lubrication technology,

biophysics, biomechanics, biorheology, soil mechanics and other branches of science

and technology, which deal with high polymers, suspensions, pastes, oils, lubricants

and physiological fluids, have made the study of Non-Newtonian fluids important.

The frequent occurrences of these fluids in industries and in day-to-day life have

provided a great impetus to the detailed study of their flow behavior.

1.7 Rheological Aspects of Non-Newtonian Fluids

Non-Newtonian fluids form a part of the wider field called ‗rheology‘, which

is the science of deformation, and flow matter. The main aim of rheology is to predict

the force system necessary to cause a given deformation, flow, or vice versa. The

basis of these predictions is the constitutive equations, which are the relationships

between the stress and the rate of strain tensors. The theoretical study of Non-

Newtonian systems is based on non-linear constitutive equations in contrast to

Newtonian systems that have linear constitutive equations.

For Non-Newtonian fluids, the shearing stress bears a non-linear relation to

the rate of strain. Stress, at a given temperature and pressure, is not a linear function

of the spatial variation of velocity. These types of fluids are classified into three broad

types:

1. The fluid for which the rate of strain at any point is a function of instantaneous shearing

stress

2. Fluids for which the relation between the rate of strain and shearing stress depends on the

time, the fluid is sheared


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3. Fluids, which exhibit both elastic and viscous properties

1.8 Development of Models

The steady-state behavior of Newtonian fluids is expressed by the rheological

equation (1.2) while for Non-Newtonian fluids, the steady-state relation between

shearing stress and the rate of strain shows non-linearity and it is given by any arbitrary functional

relationship, explicit or implicit, given by rheological equation (1.6).

(Bird et. al., 1960; Wilkinsion, 1960; Skelland, 1967)

1.8.1 Power-Law Fluids

The relationship between the shear stress yx and the velocity gradient u

y

for certain Non-Newtonian fluids can expressed satisfactorily with the following

power-laws:

n

yx

uK

y

(1.7)

From which

1nu

Ky

(1.8)

Equation (1.8) is applicable to pseudoplastic fluids when 1n , dilatant fluids when

1n , and Newtonian fluids when 1n . From Equation (1.8), the two rheological

properties of pseudoplastic and dilatant fluids that can be represented by the equation

are the coefficient K and the power n. The constant K is usually referred as the

consistency index or power-law coefficient, whereas the constant n is referred to as

the flow-behavior index, or power-law exponent. The constant μ, in Equation (1.8) is

the apparent viscosity, which reduces to the dynamic viscosity when the fluid is

Newtonian 1n .


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*

Viscoelastic

Kelvin material

"Parallel" linearstic

combination of

elastic and viscous

effects

Anelastic

Material returns to a

well-defined "rest

shape"

Time-dependent

viscosity

Rheopectic

Apparent viscosity

increases with

duration of stress

Some lubricants,

whipped cream

Thixotropic

Apparent viscosity

decreases with

duration of stress

Some clays, some

drilling mud, many

paints, synovial fluid

Time-independent

viscosity

Shear thickening

(dilatant)

Apparent viscosity

increases with

increased stress

Suspensions of corn

starch or sand in

water

Shear thinning

(pseudoplastic)

Apparent viscosity

decreases with

increased stress[

Paper pulp in water,

latex paint, ice,

blood, syrup,

molasses

Generalized Non-

Newtonian fluids

Viscosity is constant

Stress depends on

normal and shear

strain rates and also

the pressure applied

on it

Blood plasma,

custard

Table 1. 1: Classification of Non-Newtonian fluids

http://en.wikipedia.org/wiki/Non-Newtonian_fluid

http://en.wikipedia.org/wiki/Non-Newtonian_fluid#cite_note-3


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1.8.2 Bingham Fluids

For any Bingham plastic fluid (or Bingham fluids, for short), the following law holds:

yx y

u

y

(1.9)

Where y is the yield stress, and is the coefficient of rigidity, or simply the rigidity

of the fluid.

1.8.3 Yield Fluids

For yield-pseudoplastic fluids and yield-dilatant fluids, the following law can be used:

n

yx y

uK

y

(1.10)

This is a combination of equations (1.8) and (1.9). The exponent n in Equation (1.10)

is greater than one for yield-dilatant fluids, and less than one for yield-pseudoplastic

fluids. When 1n , equation (1.10) reduces to equation (1.9), which is for Bingham

fluids. Apart from above, many empirical models are proposed to express this

relationship (Bird et. al., 1960; Wilkinsion, 1960; Skelland, 1967). Some of these

models are listed in Table 1.2.

1.9 Assumptions and Basic Equations

Like any mathematical model of the real world, fluid mechanics makes some

basic assumptions about the materials being studied. These assumptions are turned

into equations that must be satisfied if the assumptions are to be held true. For

example, consider an incompressible fluid in three-dimensions. The assumption that

mass is conserved means that for any fixed closed surface (such as a sphere) the rate

of mass passing from outside to inside the surface must be the same as rate of mass

passing the other way. (Alternatively, the mass inside remains constant, as does the

mass outside). This can be turned into an integral equation over the surface.

Fluid mechanics assumes that every fluid obeys the following basic rules:

(Currie, 1974; Massey and Ward-Smith, 2005; Falkovich and Gregory, 2011)


Page | 17

1.9.1 Transport Equation

The generic scalar transport equation is a general partial differential equation

that describes transport phenomena such as heat transfer, mass transfer, fluid

dynamics (momentum transfer), etc. A general form of the equation is

, , , , ,f t x g t xt

(1.11)

where f is called the flux, and g is called the source.

All the transfer processes express a certain conservation principle. In this

respect, any differential equation addresses a certain quantity as its dependent variable

and thus expresses the balance between the phenomena affecting the evolution of this

quantity. For example, the temperature of a fluid in a heated pipe is affected by

convection due to the solid-fluid interface, and due to the fluid-fluid interaction.

Furthermore, temperature is also diffused inside the fluid. For a steady-state problem,

with the absence of sources, a differential equation governing the temperature will

express a balance between convection and diffusion.

In general, it can be inferred that all the dependent variables seem to obey a

generalized conservation principle (A conservation law states that a particular

measurable property of an isolated physical system does not change as the system

evolves.). If the dependent variable (scalar or vector) is denoted by , the generic

differential equation is

Source TermDiffusion TermConvection Term

Transient Term

V St

(1.12)

Here,

is the diffusion coefficient or diffusivity and is the density, V is velocity .

The transient term,

t

, accounts for the accumulation of in the concerned

control volume.

The diffusion term, , accounts for the transport of due to its gradients.


Page | 18

The source term, S , accounts for any sources or sinks that either create or destroy

. Any extra terms that cannot be cast into the convection or diffusion terms are

considered as source terms.

1.9.2 Mass Conservation Law (Equation of Continuity)

Let W be a fixed sub-region (controlled volume) of D (W does not change with

time) and W denote the boundary of W. The rate of change of mass in W is

, ,W W

d dm W t W t dV dV

dt dt t

(1.13)

Let W denote the boundary of W, assumed to be smooth; let n denote the unit

outward normal defined at points of W and let dA denotes the area element on W .

The volume flow rate across W per unit area is V n and the mass flow rate per unit

area is V n .

The principle of conservation of mass can be more precisely stated as follows:

The rate of increase of mass in W equals the rate at which mass is crossing ∂W in the

inward direction; that is

W W

dW V n dA

dt

(1.14)

This is the integral form of the law of conservation of mass. By the divergence

theorem, this statement is equivalent to

0W

V Wt

(1.15)

Because this is to hold for all W, it is equivalent to

0Vt

(1.16)

The last equation is the differential form of the law of conservation of mass, also

known as the continuity equation.

In the case of an incompressible fluid, the density does not vary and the

continuity equation (1.16) is equivalent to the following equation

0V (1.17)


Page | 19

1.9.3 Balance of Momentum (Equation of Motion)

The motion of a fluid is generally govern by the equation of continuity derived

above along with the conservation of linear momentum. The conservation law of

momentum states that the rate of change of linear momentum over a control volume

W bounded by W must equal what is created by external forces acting on the

control volume, minus what is lost by the fluid moving out of the boundary. Thus

W W W

dV W f dW nd W

dt

(1.18)

Where V is velocity of the fluid, is the density, f is the body force per unit mass

and is the linear momentum current density given by

V V (1.19)

in which is the Cauchy stress tensor and ⊗ denotes the tensor product.

Unless the fluid is made up of spinning degrees of freedom like vortices, is a

symmetric tensor. In general, (in three dimensions) has the form:

xx xy xz

yx yy yz

zx zy zz

Where ' s are normal stresses and ' s are tangential stresses. (Shear stresses)

Using the divergence theorem, the last part of the right hand side of equation (1.18)

can be written as

W W

n d W dW

(1.20)

Substituting equation (1.20) into equation (1.18), we have

W W W

dVdW f dW dW

dt (1.21)

Since the control volume W is invariant in time, we can take the derivative under the

integral and equation (1.21) becomes

W W

V dW f dWt

(1.22)

For an arbitrary volume W we can drop the integral and we have

V ft

(1.23)


Page | 20

Hence, from equation (1.19) and (1.23), the general form of the equation of motion is

written as

V

V V ft

(1.24)

Equation (1.24) is also known as general form of Navier–Stokes equations for the

conservation of momentum. In this equation, the force f is the body force per unit

volume, for example, gravity and electromagnetic forces. In this thesis, flows of Non-

Newtonian fluids with applied magnetic field are considered. Such a fluid is known as

Magnetohydrodynamics (MHD) and the equation of motion is given by

V

V V J Bt

P (1.25)

Here J is the current density and 0B B b is total magnetic field, 0 and B b are

applied and induced magnetic fields, respectively.

1.9.4 Conservation of Energy (Equation of Energy)

So far, we developed the continuity equation and equation of motion in earlier

sections. In general, for three-dimensional studies, these vector forms of the equations

together provide the system of three equations in the four scalar quantities like

components of the fluid velocity V , the density and the pressure p . Thus, one

might suspect that to specify the fluid motion completely, one more equation is

needed. This is in fact true, and conservation of energy will supply the necessary

equation in fluid mechanics. This situation is more complicated for general continua,

and issues of general thermodynamics would need to be discussed for a complete

treatment. At present, we shall confine ourselves to two special cases of fluid. (There

may be other cases) For fluid moving in a domain W D , with velocity field V , the

kinetic energy

2kinetic

1

2 WE V dW (1.26)

Where 2 2 2 2V u v w is the square length of the vector functionV . We

assume that the total energy of the fluid can be written as

total kinetic internalE E E (1.27)


Page | 21

where internalE is the internal energy, which is energy we cannot ―see‖ on a

macroscopic scale, and derives from sources such as intermolecular potentials and

internal molecular vibrations. If energy is pumped into the fluid or if we allow the

fluid to do work totalE will change. The rate of change of kinetic energy of a moving

portion tW of fluid is calculated using the transport theorem as follows:

2 2kinetic

1 1

2 2t tW W

d d DE V dW V dW

dt dt Dt

(1.28)

Where D

Dt is the material derivative, given by equation (1.5), equation (1.28)

becomes

kinetictW

d VE V V V dW

dt t

(1.29)

A general discussion of energy conservation requires more thermodynamics than we

shall need. At present, we limit ourselves here for two examples of energy

conservation.

1.9.4.1 Incompressible Fluid

For such a fluid, we assume all the energy is kinetic and that the rate of change

of kinetic energy in a portion of fluid equals the rate at which the pressure and body

forces do work:

kinetic

t tW W

dE pV n dA V f dW

dt

(1.30)

By divergence theorem and for incompressible fluid 0V , we get from (1.29) and

(1.30) as

t tW W

VV V V dW V p V f dW

t

(1.31)

The preceding equation is also a consequence of balance of momentum. This

argument, in addition, shows that if we assume kineticE E , then the fluid must be

incompressible (unless 0p ). In summary, in this incompressible case, the Euler

equations are:


Page | 22

0

0

DVp f

Dt

D

Dt

V

(1.32)

With the boundary conditions

0 on V n D (1.33)

1.9.4.2 Isentropic Fluid

A compressible flow will be called isentropic if there is a function w, called

the enthalpy, such that

1w p

(1.34)

This terminology comes from thermodynamics. We shall not need a detailed

discussion of thermodynamics concepts in this thesis, and so it is omitted, for the sake

of convenience, we just make a few general comments. In thermodynamics, one has

the following basic quantities,

p : Pressure

ρ : Density

T : Temperature

s : Entropy

w : Enthalpy

w p : Internal energy per unit mass

These quantities are related by the First Law of Thermodynamics, which we accept as

a basic principle:

1dw Tds d

(1.35)

The first law is a statement of conservation of energy; a statement equivalent to (1.35)

is, as it readily verified,


Page | 23

2

pd Tds d

(1.36)

Hence, according to the law of conservation of energy we have

dq r

dt

(1.37)

Where is the specific internal heat, q is the heat flux vector, r is the radiant heat

vector, is the Cauchy stress tensor, V is the velocity gradient and as

denotes the tensor product.

The velocity gradient for 1 1 2 2 3 3V v e v e v e , is defined as

31 2

1 1 1

31 2

2 2 2

31 2

3 3 3

=j

i j j i ji i

vv v

x x x

v vv vV e v e e e

x x x x x

vv v

x x x

In the absence of radiant heating, the equation (1.37) takes the following form:

2p

dTC k T

dt (1.38)

Where pT C , q k T , pC is the specific pressure heat, k is the thermal

conductivity and T is the temperature.


Page | 24

1.10 Some Interesting Examples Related to Fluid Mechanics

1.10.1 Hydrodynamic Lubrication

We know about the use of oil to lubricate moving machine parts. How does

oil "lubricate"? One of the objectives of lubrication is to prevent direct contact

between metallic surfaces subjected to relative motion, and thus eliminate solid-to-

solid friction. This is achieved by a phenomenon known as hydrodynamic lubrication.

Consider two flat surfaces inclined at a small angle to each other and moving

relative to each other as shown in figures (a) and (b). The gap between the two

surfaces is very small, and this gap becomes the characteristic length dimension L for

the flow. Consequently, this is a very low Re (Reynolds Number) flow. The liquid is

drawn between the two surfaces by viscous action and the no-slip boundary condition

will give the velocity profiles shown [Fig. (a)]. These velocity profiles are untenable

from the point of view of the continuity equation and continuity is satisfied by the

generation of pressure in the space between the surfaces. This pressure tends to drive

the liquid outwards at both ends, thereby altering the velocity profiles so that they

satisfy the continuity condition [Fig. (b)]. The pressure is maximum at some interior

location and falls off to the ambient value at the two ends. This pressure supports the

load and prevents the surfaces from touching.

Hydrodynamic lubrication is responsible when we slip on a wet pavement, although a

second hydrodynamic mechanism, known as the squeeze film effect, also comes into

play. The squeeze film effect has to do with the relative velocity normal to the

surfaces. This normal relative motion tends to drive out the liquid between the

surfaces. The inertial resistance of the liquid film to rapid acceleration causes the

build-up of pressure in the fluid film. Viscosity enhances the effect.

Pressur

e

(a) (b)


Page | 25

1.10.2 The Swing of a Cricket Ball

The picture illustrates the situation in the so-called "out swing". The plane of

the seam is inclined towards first slip. The air stream divides at the leading end of the

ball. One branch flows over the polished side, stays laminar and experiences early

boundary layer separation. The other branch trips over the seam, becomes turbulent

and flows over the "rough" side of the ball. Because of turbulent mixing, the

separation on this side is delayed and the boundary layer stays attached to the surface

of the ball most of the way (the seam acts as a vorticity generator). In the separated

region of the flow on the onside, the pressure is atmospheric, whereas the pressure

over most of the other side of the ball (the off side) is below atmospheric. This

pressure difference moves the ball towards the offside, away from a right-handed

batsman. Another way to look at it is to recognize that the asymmetrical boundary

layer separation on the two sides effectively deflects the wake to the onside. The

deflection must be due to a force, whose reaction on the ball is what causes the ball to

deviate from its original path.

For the swing to be sustained, the seam must retain its orientation. That is achieved by

giving the ball a slight backspin at the instant of delivery. The backspin stabilizes the

seam inclination by virtue of the gyroscopic effect.

Polished side

Rough side ON SIDE

1st

Slip

OFF SIDE


Page | 26

1.10.3 A Spoon Against the Flow from a Tap (Coanda Effect)

Here is an experiment we can carry out easily. Dangle a

spoon lightly from your fingers and bring its convex side in

contact with the stream of water issuing from a tap. Just let the

surface of the spoon touch the stream gently. We will find that

the spoon is pulled towards the jet quite strongly. Why does the

jet draw the spoon towards itself and not push it away? This is

due to Coanda effect, (named after Romanian aerodynamics

pioneer Henri Coandă) that has tendency of a fluid jet to be

attracted to a nearby surface.

* The Coanda effect has important applications in various high-lift devices on

aircraft, where air moving over the wing can be "bent down" towards the ground

using flaps and a jet sheet blowing over the curved surface of the top of the wing. The

bending of the flow results in aerodynamic lift. The flow from a high-speed jet engine

mounted in a pod over the wing produces enhanced lift by dramatically increasing the

velocity gradient in the shear flowin the boundary layer. In this velocity, gradient

particles are blown away from the surface, thus lowering the pressure there.

1.10.4 The Wind at the Base of a Tall Building

Figure 1. 7: Front and rear stagnation point (S) in the flow past an object

We may know from experience that the foot of a tall building is a windy place! The

reason for that is as speed of the ground height, from the ground. When this

follows, the airflow over the increases with zero exactly at surface (no slip!), flow an

obstacle like a tall building, it stagnates against the surface of the building. A

stagnation point is a point of zero velocity and it occurs where a streamline and a solid

* http://en.wikipedia.org/wiki/Coand%C4%83_effect

S S


Page | 27

surface meet at right angles to each other (the impermeable wall and the no-slip

boundary condition). By Bernoulli's equation, all the kinetic energy is converted to

pressure energy at the stagnation point, the pressure going as the square of the

velocity. Now if we look at the picture of the building (c). The stagnation points near

the top of the building, where the faster layers of air are brought to stagnation, are at

higher pressure than those near the foot of the building. A pressure gradient exists

down the face of the building and this pressure gradient drives a downward flow of

air, which fans out into the "wind" as it approaches the ground.

(c)

Wind

Stagnation pressure


Page | 28

Fluid Model Stress Vs Rate of Strain

1. Prandtl

1 1

siny x

uA

C y

2. Powell Eyring 11 1

sinhy x

u u

y B C y

3. Williamson yx

Au

uB y

y

4. Power Law

1n

yx

u uK

y y

5. Eyring 1 1

siny x y x

uC

B y A

6. Ellis

1n

y x y x

uA B

y

7. Prandtl-Eyring 1 1

sinhy x

uA

B y

8. Sisko

n

y x

u uA B

y y

9. Reiner-Philippoff

02

0

1yx yx

u

y

10. Sutterby

11

0 sinh

n

y x

uu uB B

yy y

Table 1. 2: Different models for Non-Newtonian fluids

Here A, B, C, n 0 and are material constants, that characterize the fluid.

Chapter 1: Introduction to Fluid Mechanics...Chapter 1: Introduction to Fluid Mechanics Page | 3 need to solve waste (sewage) and some basic understanding was created. At some point,

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