Fluids Lecturenotes

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Fluid Mechanics – MECH 2403

Y. A. Gengel, R. H. Turner, Thermal-fluid sciences, McGraw Hill, 2001. ; F. M. White, Fluid mechanics,

McGraw Hill, 1999.; P.M. Gerhart, R. J. Gross, J. I. Hochstein, Fundamentals of fluid mechanics,2rd edition,

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School of Mechanical Engineering

University of Western Australia

Fluid Mechanics MECH 2403

2007

CLASS INSTRUCTIONS

Lecturer: Dr. Xiaolin Wang

School of Mechanical Engineering

ME building, Rm 2.84

[email protected]

Aims

Fluid mechanics aims to introduce the basic concepts of how fluid behaves under varying conditions. At the end of this course, you shall have a working knowledge of

fluid properties, hydrostatics, and mass & energy conservation of fluids. In particular,

you should have an understanding of how to find and solve the typical engineering problems that involve fluids. For example, you will learn how to use the pressure

manometer to check the energy loss along the piping system, to use the dimensional

analyses to analyze the experimental results.

Most of these engineering problems will require understanding of many parts of this

course. You should try to continually integrate each new concept you learn with work

you have already covered.

Tutors

Tianran Lin Room 2.68

Laboratory demonstrators

(all are in the Centre for Water Research Building, unless otherwise noted)

Assessment

Assessment for the course will be by:

1) Examination 70%

2) Laboratory experiment 15%

3) Tutorial assignments 15%






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1) ExaminationThis will consist of: i. one set of answer questions on concepts covered by the unit ii. four to five problems similar to those you would have dealt with in

your tutorial and assignements.

There will be no choice of questions.

2) Laboratory experiment

You will do two laboratory experiments and complete in the laboratoryfollowing the instruction by laboratory demonstrators. The assessment will be

given by the laboratory demonstrator according to the lab quiz which will begiven during lab session. After you have completed the experiment, the

demonstrator will mark them during the session and return them to you.

If you do not understand any concepts covered in the lab, ask your

demonstrator before you do the quiz!

Note: If there is no quiz mark recorded (i.e. the student did not attend the

laboratory session), then the student will automatically score a ZERO for the

laboratory experiment.

3) Tutorial assignments

Two to three assignments: 1 is due every other week during semester.

Go to tutorials for hints on how to answer the problems.

Assignments will be marked by tutors and returned to you.

Solutions will be put on website a week after they were due to be handed in. This

does mean that late assignments will not be accepted.

Recommended Reading

1 Y. A. Gengel, R. H. Turner, Thermal-fluid sciences, McGraw Hill, 2001.2 F. M. White, Fluid mechanics, McGraw Hill, 1999.

3 P.M. Gerhart, R. J. Gross, J. I. Hochstein, Fundamentals of fluid

mechanics,2rd edition, 1992.

4 R. W. Fox, A. T. McDonald, Introduction to fluid mechanics, 5th edition, 1998.






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Faculty of Engineering and Mathematical Sciences

Policy on Plagiarism

“Plagiarise: take an another person’s thoughts, writings or inventions as one’s own”

The Concise Oxford Dictionary, Sixth Edition.

Whilst co-operation is normally encouraged, and sometimes required, plagiarism is

totally unacceptable.

Where plagiarism is discovered in a student’s work, it is the normal policy of the

faculty to apply the following penalties,

1, When the work has a mark allocated to it, the mark given will be negative with a

magnitude up to the total allocation for the particular work.

2,When the work has no mark allocation, but is required to be preformed, a penalty of up to 20 marks will be deducted from the student’s score for the unit.

3,Where two or more students are involved, the penalty will be applied to all of

them except that no penalty will be applied to an innocent party who did not permit

the copying of his/her work.

4,All cases of plagiarism will be reported to the Dean for recording on the

Faculty’s records.






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LECTURE OUTLINE

LECTURE 1 INTRODUCTION

1.1 Fluid mechanics in engineering1.2 The concept of a fluid and continuum hypothesis1.3 Thermal properties

LECTURE 2FUNDAMENTAL PROPERTIES OF FLUIDS AND

DEFINITION OF TERMS

2.1 Viscosity

2.2 Density

2.3 Laminar/transition/turbulent flow

2.4 Steady/unsteady flow

2.5 Viscous/inviscid flow2.6 Compressible/incompressible flow

LECTURE 3 FLUID STATICS

3.1 Pressure at point3.2 Hydrostatic pressure distributions


4.1 Pressure measurement - application to Manometer 4.2 Hydrostatic forces on plane surface


5.1 Hydrostatic forces on curved surface

5.2 Buoyancy and stability

LECTURE 6-9 FLUID DYNAMICS

LECTURE 10 DIMENSIONAL ANALYSES

10.1 The principle of dimensional homogeneity

10.2 The Pi theorem

10.3 Nondimensionalization of the basic equations

10.4 Similitude - model studies

LECTURE 11 PIPE FLOW

11.1 Reynolds-Number regimes

11.2 Laminar flow and turbulent flow

11.3 Fully developed laminar flow in pipes

LECTURE 12 PIPE FLOW






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12.1 Fully developed turbulent flow in pipes12.2 minor losses

12.3 Piping networks and pumps selection






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Lecture notes






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LECTURE 1

Fluid Mechanics application in Engineering

Fluid mechanics is the branch of engineering science that deals with the

behaviors of fluids at rest and in motion, and the interaction between fluid and fluid,fluid and solid. The study of fluid mechanics involves applying the fundamental

principles of mechanics and thermodynamics to develop physical understanding and

analytic tools that engineers can use to design and evaluate equipment and processes

involving fluids.

Fluid mechanics involves a wide variety of fluid flow problems encountered in

practice. Its principles and methods find many technological applications in fields

such as:

• Fluid transport

• Energy generation• Environmental control

• Transportation

Fluid transport discusses how to move a fluid from one place to another so

that the fluid may be used or processed. Examples include tap water supply system,oil-and-gas transportation pipe, chemical plant piping. The fluid transport system may

include pumps, compressors, pipes, valves and a host of components. In order to

design the new systems, engineers may evaluate existing systems to meet new

demands or they may maintain or upgrade existing systems.

Energy generation always involves fluid movement. Typical energy

conversion devices such as steam turbines, reciprocating engines, gas turbines,hydroelectric plants, and windmills involve many complicated flow processes.

Environmental control involves fluid motion. Building ventilation, heating andair-conditioning systems use a fluid to transport energy from a source to required

environment. For example, in the heating system, the fluid brings the energy from a

combustion process or other heat source to the heated place. In the air-conditioning

system, the circulating air is cooled by a flowing refrigerant and then is distributed to

the cooled place.

With the exception of space travel, all transportation takes place within a fluid

medium. The application of fluid mechanics to vehicle design can minimize thedesired force which is generated by the relative motion between the fluid and vehicle

and hence opposes the desired force. The fluid often contributes in a positive way

such as by floating a ship or generating lift on airplane wings. In addition, these

transportation devices derive propulsive forces that also interact with the surrounding

fluid.

These examples are by no means exhaustive. There are plenty of other examples such as design of harbors, bridges, canals and dams. In the environmental

engineering, engineers must deal with naturally occurring flow processes in the

atmosphere and lakes, rives and oceans. The phenomena of fluid motion are central to

the field of meteorology and weather forecasting.






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The fundamentals of fluid mechanics include a knowledge of the nature of

fluids and the properties used to describe them, the physical laws that govern fluid

behavior, the ways in which these laws may be cast into mathematical form, and the

various methodologies that may be used to solve the engineering problem.

The Fluids concept

What is a fluid? From the point of fluid mechanics, all matter consists of onlytwo states, fluid and solid. A solid can resist a shear stress by a static deformation but

a fluid cannot. In order to develop a formal definition of a fluid, consider imaginarychunks of both a fluid and a solid as shown in figure 1.1. The chunks are fixed along

the bottom and a shear force is applied along the top surface. A short time after

application of the force, the difference between fluid and solid is obvious. The solid

assumes a deformed shape which can be measured by the angle θ. This angle will not

change with the time. That means the deformation of solid is exactly the same at

different time and it will return to the original undeformed shape if the force isremoved. On the contrary, the angle for fluid will become greater when the

application of the force maintain for a short time. In fact, the fluid continue to deform

as long as the force is applied. If the force is removed, the fluid will not return to its

original shape but it will retain whatever shape it had. Now we can define a fluid:

Figure 1.1 Response of samples of solid and fluid to applied shear force:

Condition (a) corresponds to the instant of application of the force; condition (b)

to a short time after the application of the force; condition (c) to a later time and

(d) removal of the force.

A fluid is a substance that deforms continuously under the action of an applied

shear force or stress.

F F Fθ1

θ2

t1 t2θ1 = θ2

F F Fθ1 θ2

t1 t2θ1 < θ2

θ2

(a) (b) (c) (d)






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The process of continuous deformation is called flowing. A fluid is a substance that isable to flow. Because of flowing, it is not possible to analyze or discuss fluid behavior

in terms of stress and deformation as is done in solid mechanics. It is necessary to

consider the relation to between stress and the time rate of deformation. A fluid

deforms at a rate related to the applied stress. The fluid attains a state of “dynamic

equilibrium” in which the applied stress is balanced by the resisting stress. Thusalternative definition of a fluid is:

A fluid is a substance that can resist shear only when moving.

The continuum hypothesis

All substances are composed of an extremely large number of discrete particles called

molecules. In a pure substance such as water, all molecules are identical; other

substances such as air are mechanical mixtures of different types of molecules.

Molecules interact with each other via collisions and intermolecular forces. The phase

of a sample of matter-solid, liquid and gas- is a consequence of the molecular spacingand intermolecular forces. The molecules of a solid are relatively close (spacing on

the order of a molecular diameter) and exert large intermolecular forces; themolecules of a gas are far apart (spacing of an order of magnitude larger than

molecular diameter) and exert relatively weak intermolecular forces. Thus a gas caneasily change its volume and shape, however, the solid maintain both its volume and

shape due to stronger intermolecular forces. In a liquid, the intermolecular forces are

sufficiently strong to maintain volume but not shape.

Because of the large number of molecules involved in the fluid, it is impossible to

describe a sample of fluid in terms of the dynamics of its individual molecules. For

most cases of practical interest, it is possible to ignore the molecular nature of matter

and to assume that matter is continuum. This assumption is called the “continuummodel” and may be applied to both solid and fluids. This model assumes that

molecular structure is so small relative to the dimensions involved in problems of practical interest that we may ignore it. Because of the continuum model, the fluid can

be described in terms of its properties, which represent average characteristics of its

molecular structure. As an example, we use the mass per unit or density rather than

the number of molecules and the molecular mass. Furthermore, because the fluid

properties and velocity are continuous functions, we can use calculus to analyze a

continuum rather than applying discrete mathematics to each molecule.

Fluid properties

1. Density and specific volume

The “density” of a fluid is its mass per unit volume. It has a value at each point in

a continuum and may vary from one point to another. Assume an arbitrary volume _ V

in a fluid and the mass is _m. The average density ρ is

V

m

Δ

Δ=ρ

This average density depends on the size and location of the chosen volume.

Therefore an exact definition of the density must involve a limit. With the volume






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reducing, our nature of inclination is to take the limit of ρ as _ V approaches zero. The

local density is defined by

V

m

V V Δ

Δ=

→→Δ 0

limδ

ρ

The small volume V δ represents the size of a typical “point” in the continuum. For

fluids near atmospheric pressure and temperature, such small volume is of the order

of 10-9

mm3. We interpret all fluid properties as representing an average of the fluid

molecular structure over this small volume. The unit of density in SI units is kg/m 3.

a. Specific volume

The “specific volume” of a fluid is its volume per unit mass which is directly

related to density. The “specific volume” is defined by

ρ ν

1= ,

Is of considerable use in thermodynamics but is seldom used in fluid mechanics.

b. Specific weight

The “specific weight” is the weight of the fluid per unit volume; thus

g ρ γ = ,

Where g is the local acceleration of gravity.

c. Specific gravity

The specific gravity of a fluid is the ration of the density of the fluid to the density

of a reference fluid. The defining equation is

f

S Re

ρ

ρ =

For liquids, the reference fluid is pure water at 4oC and 5

100133.1 × N/m2,

1000Re

= f ρ kg/m3.

Note: all the above properties are directly related to each other, once density is

constant so that all the others are constant.

2. Pressure

Pressure is a fluid property of utmost importance. Most fluid mechanics problem

involve prediction of fluid pressure or with the integrated effects of pressure over

some surface or surface in contact with the fluid. The definition of pressure is

Pressure is the normal compressive force per unit area acting on a real or

imaginary surface in the fluid.

Consider a small surface area AΔ within a fluid as shown in figure 1.2, A forces

n F Δ acts normal to the surface. Tangential forces may also be present but are not

relevant to the definition of pressure. The pressure is defined by






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A

F p n

A A Δ

Δ=

→→Δ 0

limδ

The limiting value of Aδ represents the lower bound of the continuum assumption.

The pressure can vary from point to point in a fluid. It presents the molecular

momentum and intermolecular force within the fluid microscopically and it is positive

for compression.

The unit of pressure is force per unit area, N/m 2. Gage pressure refers to the

pressure measured relative to local atmospheric pressure. Pressure measured relativeto zero pressure is called absolute pressure. Gage and absolute pressures are related

as:

Absolute pressure = Gage pressure + Atmospheric pressure in vicinity of gage.

Pressure below local atmospheric pressure is sometimes called vacuum pressures. It isnormally expressed as a positive number,

Vacuum pressure = Atmospheric pressure – Absolute pressure = -Gage pressure.

In many situations that arise in fluid mechanics, we are more concerned with

differences of pressure than with levels of pressure. Normally, pressure differences

are the same whether the pressure are considered as absolute or gage, so long as all

pressures are based on a common datum.

3. Temperature

Temperature is a property that is familiar to everyone but rather difficult to definewith the same exactness as density or pressure. We normally associate temperaturewith the degree of “hotness” or “coldness”, but this is hardly a precise definition. For

introduction purposes, we define the temperature of a fluid as a measure of the energy

contained in the molecular motions of the fluid.

The most common temperature scales are relative scales such as “Fahrenheit

scale”, “Celsius Scale” and absolute scales such as “Kelvin scale” and “Rankine

scale”.

T (Rankine) = T (Fahrenheit) + 459.67

T (Kelvin) = T (Celsius) + 273.15

n F Δ

Surface area AΔ






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T (Rankine) = 1.8 T (Kelvin).

Internal energy (U) is the energy contained in random molecular motions and

intermolecular forces. The specific internal energy is the internal energy per unit

mass. For a single liquid, the internal energy is primarily a function of temperature.

T cuv

=

Where Cv is specific heat at a constant volume.






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LECTURE 2

2.1 Thermal properties (continue)

1. Viscosity

As you have learned that a fluid is a substance that undergoes continuous

deformation when subjected to a shear stress and that the shear stress is a function of the rate of deformation. For many common fluids, this shear stress is proportional to

the rate of deformation. This constant of proportionality, called Viscosity which is

important fluid property. To develop this equation for viscosity, we must first obtain

an expression for the rate of deformation of a fluid particle. Consider the behavior of

a fluid element between the two infinite plates shown in figure 2.1. The upper plate

moves at constant velocity,uδ

, under the influence of a constant applied force, x

F δ .

The shear stress, τ , applied to the fluid element is given by

dA

dF

A

F x x

A

==

→ δ

δ τ

δ 0

lim

Where Aδ is the area of contact of a fluid element with the plate, and x

F δ is the force

exerted by the plate on that element. During the time interval t δ , the fluid element is

deformed from position “BCMN” to position “BDON”. The rate of deformation of

the fluid is given by

deformation rate =dt

d

t t

α

δ

δα

δ

=

→0

lim

Figure 2.1 Deformation of a fluid element

x

y

yδ Fluid element

at time, t

Fluid element

at time, dt t +

xδ

C D

N

M OForce,

x F δ

Velocity, uδ

l δ

δα

B






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To calculate the shear stress τ , it is desirable to express dt d α in terms of

readily measurable quantities. The distance, l δ , between the points C and D is given

by

tδ δ δ ul =

For the small angle,

δα δ δ yl =

By equating the above two equations, we obtain

y

u

t δ

δ

δ

δα =

Taking the limits of both sides of the quality, we obtain

dy

du

dt

d =

α

Thus, the fluid element of figure 2.1, when subjected to shear stress,τ ,

experiences a rate of deformation (shear rate) given by dydu . According to the

definition of a fluid, the shear stress is directly proportional to the rate of deformation,

dy

duµ τ =

The coefficient µ is the viscosity, we called absolute viscosity and its unit is

N-s/m2. In fluid mechanics, the ratio of absolute viscosity to density is very useful and

this ratio is given the name “kinetic viscosity” and its unit is m2/s.

2.2 Newtonian Fluid & non-Newtonian Fluid

Fluids in which the shear stress is directly proportional to rate of deformation

are Newtonian fluids. Thus in terms of the coordinates of the figure 2.1, Newton’s law

of viscosity is given for one-dimensional flow by

dy

duµ τ =

Most common fluids such as water, air and gasoline are Newtonian under

normal conditions.

Non-Newtonian Fluid is used to classify all fluids in which shear stress is not

directly proportional to shear rate. Blood and plastics are examples of non-newtonian

fluids. In this text we only consider Newtonian fluids.

For liquids, both the dynamic and kinematic viscosities are practically

independent of pressure and any small variation with pressure is usually disregarded,except at extremely high pressures. For gases, this is also the case for dynamic

viscosity (at low to moderate pressures), but not for kinematic viscosity since the

density of a gas is proportional to its pressure. For example,

Air at 20oC and 1 bar: s/m0.000015 s;kg/m000018.0 2

=⋅= υ µ

Air at 20oC and 3 bars: s/m0.00005 s;kg/m000018.0 2=⋅= υ µ

The viscosity of a fluid is a measure of its “stickiness” or “resistance to

deformation”. This is due to the internal frictional force that develops between layers

as they are forced to move relative to each other. Viscosity is caused by cohesive

forces between molecules in liquids and by molecular collisions in gases, and it variesgreatly with temperature. The viscosity of liquids decreases with temperature,






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whereas the viscosity of gases increases with temperature as shown in figure 2.2. Thisis because, in a liquid, the molecules possess more energy at higher temperatures, and

they can oppose the large cohesive intermolecular forces more strongly. As a result,

the energized liquid molecules can move freely.

In a gas, on the other hand, the intermolecular forces are negligible, and the

gas molecules at high temperatures move randomly at higher velocities, this results inmore molecular collisions per unit volume per unit time, and therefore in greater

resistance to flow. The viscosity of a fluid is directly related to power needed to

transport a fluid in a pipe.

Figure 2.2 The viscosity of liquids decreases and the viscosity of gases increases with

temperature.

The kinetic theory of gases predicts the viscosity of gases to be proportional to thesquare root of temperature,

T ∝µ

This has been confirmed by practical observations, but deviations for different gasesneed to be accounted for by incorporating some correction factors. According to

Sutherland correlation, it is as

T b

aT

/1

2/1

+

=µ

Where T is absolute temperature and a & b are experimentally determined constant,for air a=1.458 x 10-6Pa-s/ K 1/2; b=110.4K at atmospheric conditions.

For a liquid: )/(10 cT ba

−

=µ

For water, a=2.414 x 10-5 N-s/m2, b=247.8K and c=140K results in less than 2.5%

error in the temperature range of 0oC to 370oC.

2.3 Viscous versus inviscid flow

As we explained early, it is quite clear of the viscous flow in which the effects of viscosity are significant. The effects of viscosity are very small in some flows, and

Temperature

Viscosity

Liquids

Gases






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neglecting those effects greatly simplifies the analysis without much loss in accuracy.Such idealized flows of zero-viscosity fluids are called inviscid flows.

2.4 Internal flow and external flow

A fluid flow is classified as being either internal or external, which depends onwhether the fluid is forced to flow in a confined channel or over a surface. The flow

of an unbounded fluid over a surface such as a plate, a wire, or a pipe is external flow.

The flow completely bounded by a solid surface such as in a pipe and duct are calledinternal flow.

Figure 2.3 Internal flow in a pipe and the external flow of air over the same pipe(basic concept for gas-liquid heat exchanger.

2.5 Compressible and Incompressible flow

Flows in which variation of density are negligible are termed incompressible; when

the density variations within a flow are not negligible, the flow is called compressible.

Gases are normally considered as compressible flow and liquids are usually classified

as incompressible.

For many liquids, density is only a weak function of temperature. At modest pressure,

liquids may be considered incompressible. However, at high pressures,

compressibility effects in liquids can be important. Pressure and density changes in

liquids are related by the bulk compressibility modulus, or modulus of elasticity,

ρ ρ d

dp E v =

If the bulk modulus is independent of temperature, then density is only a function of

pressure (the fluid is barotropic).

Internal flow

Water External

flow

Air or gas






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Water hammer and cavitation are examples of the important of compressibility effectsin liquid flow. Water is caused by acoustic waves propagating and reflecting in a

confined liquid, for example, when a valve is closed abruptly. The resulting noise can

be similar to “hammering” on the pipes hence the term.

Here, we would like to know one more thermal property, vapor pressure of a liquidwhich is the partial pressure of the vapor in contact with the saturated liquid at a given

temperature. When pressure in a liquid is reduced to less than the vapor pressure, the

liquid may change phase suddenly and “flash” to vapor.

2.6 Laminar flow and Turbulent flow

Some flows are smooth and orderly while others are rather chaotic. The highly

ordered fluid motion characterized by smooth streamline is called laminar flow. The

flow of high-viscosity fluids such as oil at low velocities is typically laminar. The

highly dis-ordered fluid motion that typically occurs at high velocities is characterized

by velocity fluctuations and is called turbulent flow. Air at high velocities is typicallyturbulent. This flow regime greatly influences the heat transfer rates and required

power for pumping.

The nature of the Laminar flow and Turbulent flow is determined by the value of adimensionless parameter, called Reynolds number (Re). For a piping flow,

Re= µ ρ /VD . Generally, Re<2300, it is laminar flow.

Note: in order to understand more about the nature of laminar flow and turbulent flow

and the behavior of the transition from the laminar flow to turbulent flow, one

laboratory experiment is prepared for students.

2.7 Steady and unsteady (transient) flow

The term steady and uniform are used commonly in engineering. Steady flow means

flow does not change with time. The opposite of steady is unsteady flow or called

transient. The term uniform implies no change with location over a specified region.

Many devices such as turbines, compressors and nozzles operate for long period of

time under the same conditions and they are classified as steady flow devices. During

the steady flow, the fluid properties can change from point to point within a device

but at any fixed point they remain constant.

2.8 Natural flow and forced flow

A fluid flow is said to be natural or forced, depending on how fluid motion is

initiated. In forced flow, a fluid is forced to flow over a surface or in a pipe by

external means such as a pump or a fan. In natural flows, any fluid motion is due to

natural means such as the buoyancy effect, which manifests itself as the rise of

warmer fluid and the fall of cooler fluid. Figure 2.4 shows a thermo-siphoning system

which is typical natural flow system.






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Figure 2.4 Natural circulation of water in a solar water heater by thermo-siphoning.

2.9 Surface tension and Capillary effect

Figure 2.5 Some consequences of surface tensionFrom: Y. A. Gengel, R. H. Turner, Thermal-fluid Sciences, 2001

It is often observed that a drop of blood forms a hump on a horizontal glass, a drop of

mercury forms a near-perfect sphere and can be rolled just like a steel ball over asmooth surface, water droplets from rain or dew hang from branches or leaves of

trees, a liquid fuel injected into an engine forms a mist of spherical droplets, water

dripping from a leaky faucet falls as spherical droplets, and a soap bubble released

into air forms a spherical shape. All these observances demonstrate a pulling force

that causes a tension acts parallel to the surface and is due to the attractive forces

between the molecules of the liquid. The magnitude of this force per unit length iscalled surface tension.

To understand the surface tension effect better, consider a liquid film such as the film

of soap bubble suspended on a U-shape frame with a movable side as shown in figure

2.6. Normally, the liquid film will tend to pull the movable side inward in order tominimize the surface area. A force F needs to be applied on the movable frame in the

Solar

collector

Solar

radiation

Hot water

storage (top

part)

Cold

water

Hot

water






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opposite direction to balancing this pulling effect. The thin film in the device has twosurfaces exposed to air, and thus the length along which the tension acts in this case is

2l . Then a force balance on the movable frame gives

sl F σ 2=

Thus the surface tension can be expressed as

l

F s

2=σ

The unit of surface tension is N/m. An apparatus of this kind with sufficient precision

can be used to measure the surface tension of various fluids.

Figure 2.6 Stretching a liquid with U-shaped frame and the force acting on themovable frame

In the U-shape frame, the force F remains constant as the movable frame is pulled to

stretch the film and increase its surface area. When the movable frame is pulled a

distance xΔ , the surface area increases by xl A Δ=Δ 2 and the work done W during

this stretching process is

A xl x F W s s

Δ⋅=Δ⋅⋅=Δ⋅= σ σ 2

Since the force remains constant in this case. This result also can be interpreted as the

surface energy of the film is increased by an amount of A sΔσ during this stretching

process, which is consistent with the alternative interpretation of s

σ as surface energy.

In the case of liquid film, the work is used to move liquid molecules from the interior

parts to the surface against the attraction forces of other molecules. Therefore, surface

tension also can be defined as the work done per unit increase in the surface area of the liquid.

There are a few factors which affect the surface tension. Firstly, the surface tension

varies greatly from substance to substance, and with temperature for a given

F

Movableframe

Ri id frame

l

x

F

sσ

sσ

Liquid film






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substance. For example, at 20oC, the surface tension is 0.073 N/m for water and 0.440

N/m for mercury surrounded by atmospheric air. In general, the surface tension of

liquid decreases with temperature and becomes zero at the critical point. The effect of

the pressure on tension is usually negligible. The table 2.1 shows the surface tension

for some substance at different temperature.

Fluids Surface tension s

σ , N/m

Water

0 oC 0.076

20oC 0.073

100oC 0.059

300 oC 0.014

Glycerin 0.063

SAE 30 oil 0.035

Mercury 0.440

Ethyl alcohol 0.023Blood, 37oC 0.058

Gasoline 0.022

Ammonia 0.021

Soap solution 0.025

Kerosene 0.028

In addition, the surface tension can also be affected considerably by impurities.Therefore, certain chemicals, called surfactants can be added to a liquid to decrease its

surface tension. For example, soaps and detergents lower the surface tension of water

and enable it to penetrate through the small openings between fibers for more

effective washing. But this also means that devices whose operation depends on

surface tension (such as heat pipes) can be destroyed by the presence of impurities

due to poor workmanship.

Capillary effect

Capillary effect is one of most useful and interesting consequence of surface tension,which is the rise or fall of a liquid in a small-diameter tube inserted into the liquid.

Such small or narrow tubes or confined flow channels are called capillaries. The rise

of kerosene through the cotton wick inserted into the reservoir of a kerosene lamp is

due to this effect. The capillary effect is also partially responsible for the rise of water to the top of tall trees.

It is commonly observed that water in a glass container curves up slightly at the edges

where it touches the glass surface; but the opposite occurs for mercury: it curvesdown at edges. This effect is usually expressed by saying that water wets the glass by

sticking to it while mercury does not. This strength of the capillary effect is quantified by the contact angle φ which the tangent to the liquid surface makes with the solid

surface at the point of contact. When this angle is bigger than 90o, the liquid does not

wet the surface; when it is smaller than 90o, it wets the surface. This contact angle is

different in different environments such as another gas or liquid in place of air. For

example, in atmospheric air, the contact angle of water (most other organic liquids)with glass is nearly 0

o, thus the surface tension force acts upwards on water in a glass






1992.

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21

tube along the circumference, tending to pull the water up. As a result, water rises inthe tube until the weight of the liquid in the tube above the liquid level of the reservoir

balances the surface tension force. However, the contact angle is 130o for mercury-

glass so that it does not wet the glass.

The magnitude of the capillary rise in a circular tube can be determined from a force balance on the cylindrical liquid column of height h in the tube. The bottom of the

liquid column is at the same level as the free surface of the reservoir, and thus the pressure there must be atmospheric pressure. This will balance the atmospheric

pressure acting at the top surface, and thus these two effects will cancel each other.

The weight of the liquid column is

)( 2h R g Vg mg W π ρ ρ ===

Equating the vertical component of the surface tension force to the weight gives

φ σ π π ρ cos22

s surface Rh R g F W =⇒=

Solving for h gives the capillary rise to be

Capillary rise: φ ρ

σ cos

2

gRh s=

This is also valid for nonwetting liquids as shown in the figure 2.9.

Water

φ

Mercury

φ

a. Wetting fluid b. Non-wetting

fluid

Figure 2.7 Contact angle for wetting and

non-wetting fluids

φ

h

s Rσ π 2

W

2R

Figure 2.8 the forces acting

on a liquid column due to

capillary effect






1992.

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It is noted that the capillary rise is inversely proportional to the radius of the tube.Therefore, the thinner the tube is, the greater the rise of the liquid will be in the tube.

In practice, the capillary tube is usually negligible in tubes whose diameter is greater then 9mm. when pressure measurement are made using manometers and barometers

(we will talk about this in the next few lectures), it is important to use sufficiently

large tubes to minimize the capillary effect. Also, it can be found that the capillary

rise is also inversely proportional to the density of the liquid.

θ hΔ

Tube

hΔ

Tube

θ

a. Capillary rise ( o

90<ϕ ) b. Capillary depression ( o

90>φ )

Figure 2.9






1992.

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LECTURE 3 FLUID STATICS ONE

This lecture and following two lectures will discuss the mechanics of fluids that are

not flowing; that is, the particles of fluid are not experiencing any deformation.

1.1 Pressure at a point: Pascal’s law

In this discuss, we neglect the surface tension and assume that no electromagneticforces act on the fluid. If there is no deformation, there are no shear stresses acting on

the fluid; the forces on any fluid particle are the result of gravity and pressure only.We are mainly concerned here with calculating pressure distribution and evaluating

the resultant pressure forces at interfaces between a fluid and as solid.

Pascal’s law gives a clear statement for the pressure: the pressure at any point in a

nonflowing fluid has a single value, independent of direction.

In order to prove Pascal’s law, we consider the equilibrium of forces for the small

fluid wedge shown in figure 3.1 which is assumed static.

Figure 3.1 wedge of fluid at rest

The forces on the fluid are the result of gravity and pressure and we ignore x directionforce. From the force balance, the wedge is at rest so

==

=−−=

=−=

θ θ

γ θ

θ

dssindz ;cos

0)2/(cos)()(

0sin)()(

dsdy

dydxdz dsdx pdxdy p F

dsdx pdxdz p F

s z z

s y y

Therefore,

2 ;

dz

p p p p s z s y γ +==

Ps

Pzg

x

y

z

θ

θ

dx

dy

dz

ds






1992.

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24

To evaluate this relation at a point, we take the limit as dx,dy,dz approach zero, whichresults in

s z s yp p p p == ;

As θ is arbitrary, these equations are valid for any angle. Note that x and y axes are

not unique; rotating the wedge 90o about the z axis would interchange the x and y

axes, and we would conclude that s x

p p = . Therefore, pressure is independent of the

direction.

1.2 Pressure variation in a static fluid

Even through the pressure at a point is the same in all directions in a static fluid, the

pressure may vary from point to point. To evaluate the pressure variation, lets

examine a cubical fluid element at rest as shown in figure 3.2.

Figure 3.2 Cubic fluid element in a static fluid

The pressure at the center of the element is p. For simplicity; we assume that the

pressure does not vary in the x direction. As the fluid is continuous, we express the pressure at the faces of the element from the center by Taylor series as shown in the

figure 3.2. where the H.O.T. means “higher order terms” involving dy and dz. The

forces balance for the rest fluid element is as

0 ;0 == z y F F

By combining and canceling the terms where possible, consider the higher order

terms vanish as we take the limit, it can get

0 ;0 =+

∂

∂=

∂

∂γ

z

p

y

p

These two equations show: (i) the pressure does not vary in a horizontal plane (y

direction); (ii) the pressure increases if we go “down” and decreases if we go “up”.

...)2

(21)

2( 2

2

2

T O H dz z pdz

z p +

∂

∂+

∂

∂+ρ

...)2

(2

1)

2( 2

2

2

T O H dz

z

pdz

z

p+

∂

∂+

∂

∂−ρ

(x,y,z)

G dx

dy

dz

...)2

(2

1)

2( 2

2

2

T O H dy

y

pdy

y

p+

∂

∂+

∂

∂+ρ

...)2

(2

1)

2( 2

2

2

T O H dy

y

pdy

y

p+

∂

∂+

∂

∂−ρ

dx

dy

dz






1992.

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25

Because pressure changes in only one direction, we can replace the partial derivativewith an ordinary derivative:

γ −=

dz

dp

This equation is the basic equation of fluid statics.

3.2.1 Application of Pressure variation

In a constant-density fluid

If the specific weight of fluid is constant, the above equation can be easily integrate as

o p z z p +−= γ )(

Whereo

p is the pressure at reference point, that is z=0.

If we consider the pressure in the liquid which has a free surface as shown in thefigure 3.3, the free surface is at constant pressure and it is horizontal. We introduce

the depth of the fluid, h, measured downward from the free surface. As z=-h, we canhave

h ph p o )( γ +=

This pressure distribution is called a hydrostatic pressure distribution. The last term in

the above equation is called hydrostatic pressure.

Based on the above, we can state the hydrostatic condition:

Pressure in a continuously distributed uniform static fluid varies only with

vertical distance and is independent of the shape of the container. The pressure is the

same at all points on a given horizontal plane in the fluid. The pressure increases with

depth in the fluid.

An illustration of this is shown figure 3.4. the free surface of the container is

atmospheric and forms a horizontal plane. Points a, b, c, and d are at equal depth in a

horizontal plane and are interconnected by the same fluid, water. Therefore all points

have the same pressure. The same is true of points A,B,C on the bottom, which all

have the same higher pressure than at a, b, c, and d. However, point D although at the

h

Symbol for

free surfaceFree surface

x

yGas ( po)

Figure 3.3 Body of liquid with a free surface.






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same depth as A, B and C has a different pressure because it lies beneath a differentfluid, mercury.

In a variable-density fluid and the standard atmosphere

In the above section, we discussed the constant specific weight. For some cases, the

specific weight is variable so that we must relate the specific weight to the pressure

and evaluation before we can integrate the equation. Let’s take the ideal gas as anexample,

RT

pg g

dz

dp−=−=−= ρ γ

Solving for pressure, we can get:

∫ ∫ +−== C dz RT

g p

p

dp)ln(

Now the problem becomes the temperature variation with the z. A useful application

of this is calculation of the variation of pressure with altitude in the earth’s

atmosphere. The temperature profile of the U.S. standard atmosphere is shown infigure 3.5a and the pressure variation with altitude could be calculated according to

the temperature profile as shown in figure 3.5b.

Free surface

Atmospheric pressure

Depth 1

Depth 2

a bcd

ABCD

Mercury Figure 3.4 Hydrostatic pressure distributions






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(a) (b)

Figure 3.5 Temperature and pressure distribution in the U.S. Standard.

3.2.2 Manometry

1 Manometers:

Recall the equation for pressure in a constant density:h p p o γ =−

A simple and effective way to measure the pressure is to measure the height of a

column of liquid supported by the pressure. Such device based on this principle is

called a manometer. One of most common types of manometer is the U-tube, as

shown in Fig. 3.6. Since the pressure is the same at the point A’, we can have

1122gh p gh P oa ρ ρ +=+

)( 22110 gh gh p pa ρ ρ −=−

2211gh gh p gage ρ ρ −=

If the fluid 2 is air inside the tube, then the gage pressure is just the height of theliquid, fluid 1.

60

0

50

40

30

20

10

-60 -40 -20 0 20

20.1km

11.0km

Temperature, oC

Altitude z, km

60

0

50

40

30

20

10

40 80 120

Altitude z, km

Pressure, kPa

1.20kPa

101.33kPa

Ionosphere

Stratosphere

Tropospher






1992.

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Figure 3.6 U-tube manometer

A

h1

h2

B

Fluid

density2

ρ

Fluid

density1

ρ

B’

Open to

atmosphere

A’






1992.

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29

LECTURE 4

4.1 (Con) Manometry and pressure measurement

In this section, we would like to illustrate the principles by which the equation for any

manometer can be developed. The essence of the method is the application of thehydrostatic pressure equation and the balance of pressure at a point.

Figure 4.1 shows a manometer with several different fluids. We can find anexpression for the pressure difference by dividing the manometer at the high point

“D” and the low point “C”.

Figure 4.1 multi-liquid manometer

2211gh gh P P D A ρ ρ ++=

4332gh gh P P DC ρ ρ ++=

6453gh gh P P C B ρ ρ −−=

Therefore, we can obtain

6454323211 )()( ghhh g hh g gh P P B A ρ ρ ρ ρ +−−−−

=−

This shows that any type of manometer can be developed. We can avoid the slight

complexity of solving simultaneous equations by following a three-step manometer

rule:

1) write the pressure at one end of the manometer;

2) proceed through the manometer, adding the hydrostatic pressure if you are

going down and subtracting if going up.

3) At any point, the algebraic sum of pressure is equal to the pressure at that point.

The sensitivity of the pressure depends on the h per unit change in pressure and the

maximum readable pressure depends on the change of liquid.

PA PB

C

D

1

2

3

4

h1

h2

h3

h4

h5

h6






1992.

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30

For the small pressure change, we can use the inclined manometer as shown in thefigure 4.2 to increase the sensitivity.

Figure 4.2 Inclined manometer

θ ρ ρ

sinl g gh P ==

By reading the length, l , which is much longer than h, the sensitivity increases which

depends on the angle θ .

4.2 pressure measurement

Example 1

Figure 4.3 Pressure drop measurement

Application for measuring the pressure drop across the fittings, valves, flowmeters,

orifices and so on.

gh P ab )( 12 ρ ρ −=Δ

Example 2

Application for measuring the pressure inside the pipe flow.

l

h

θ

P

h

1ρ

2ρ

Flow

devices

a b






1992.

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31

Figure 4.4 Pressure inside the water flow pipe

1324 )( gH gH H H g P P wom B A ρ ρ ρ −+−+=

Example 3

This figure shows how to measure the absolute pressure at point 1 and 2 by using

different type of manometers. However, if the pressure difference is desired, this

method is not perfects since the significant error is incurred by the subtracting twoindependent measurements. It is far better to connect both ends of one instrument to

the two static holes 1 and 2 so that one manometer reads the difference directly. The

figure c shows an elastic-deformation pressure measurement device which is a

popular, inexpensive and reliable device, called bourdon tube. The deflection can be

measured by a linkage attached to a calibrated dial pointer. This idea can use the

diaphragm or membrane to replace the bourdon tube. As shown in the other

measurement devices.

A

Water flow

C

Mercury

SAE 30 Oil

Gage B

H1

H2

H3

H4






1992.

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Schematics of (b)

Figure 4.5 From F. M. White, Fluid Mechanics, 1999

Some other pressure measurement devices

Except the above-mentioned pressure sensors, the commonly used pressure devices

include the following types, Capacitive sensors, Strain gages and Frequency sensor.






1992.

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33

Figure 4.6a Capacitive pressure sensor From F. M. White, Fluid Mechanics, 1999

Figure 4.6b Strain gage From F. M. White, Fluid Mechanics, 1999






1992.

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34

Figure 4.6 c Frequency pressure sensor From F. M. White, Fluid Mechanics, 1999

4.2 Pressure force on surface

When a surface is in contact with a fluid, fluid pressure exerts a force on the surface.

This force is distributed over the surface; however, it is often helpful in engineering

calculations to replace the distributed force by a single resultant. To completely

specify the resultant force, it is necessary to determine its magnitude, direction, and point of application which is called center of pressure. According to the definition:

dA

dF p n=

The resultant force can be obtained by integration:

∫ = pdA F n

There are two difficulties in performing this integration:

1) The pressure and area must be expressed in terms of a common variable to

permit integration.

2) If the surface is curved, the normal direction varies from point to point on the

surface, and n F is meaningless because no single “normal” directioncharacterizes the entire surface. In this case, the force at each point on the

surface must resolved into components and then each component integrated.

4.2.1 Pressure force on plane surface

The simplest pressure on a place surface is uniform pressure acting on a surface such

as a container as shown in figure 4.7 below:

The pressure force can be easily integrated as:

A gh p A p pdA F l ao )( ρ +=== ∫






1992.

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Figure 4.7 Uniform pressures in the liquid tank

From this, it is obvious that the force on a plane surface caused by a uniform pressure

is equal to the product of the pressure and the area. The force is normal to the area.

On an arbitrary surface

Consider the top surface of a flat plate of arbitrary shape completely submerged in a

liquid as shown in figure 4.8 together with its top view. The plane of this surface

(normal to the paper) intersects the horizontal free surface with an angle θ , and we

take the line of intersection to be the x axis.

Figure 4.8 Hydrostatic force on an inclined plate surface completely submerged in a

liquid

The distance from the free surface to the plate is

h

g

F

o Liquid

Px

Free surface

θ

h(x,y)

dA=dxdy

(dA)

CP

C

F=PCAPc

C: CentroidCP Center of pressure

hc

0






1992.

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36

θ sin yh =

The absolute pressure on the plate is

θ ρ sin gy P P o +=

The resultant hydrostatic force F R acting on the surface is determined by integrating

the force PdA acting on a differential area dA over the entire surface area,

( ) ∫ ∫ ∫ +=+== A A

oo A

R ydA g A P dA gy P PdA F θ ρ θ ρ sinsin

The first moment of area ∫ A ydA is related to the y coordinate of the centroid (or

center) of the surface by

∫ =

AC ydA

A y

1

Substituting,

A P A P A gy P F aveC C o R ==+= )sin( θ ρ

Where C c gh P ρ = is the pressure at the centroid of the surface which is equivalent to

the average pressure on the surface, and hc is the vertical distance fo the centroidfrom the free surface of the liquid.

Note: the magnitude of the resultant force acting on a plane surface of a completely

submerged plate in a homogeneous (constant density) fluid is equal to the product of

the pressure Pc at the centroid of the surface and the area A of the surface.

Now we need to know solve the second problem how to determine the line of action

of the resultant force. Two parallel force systems are equivalent if they have the samemagnitude and the same moment about any point. The line of action of the resultant

hydrostatic force, in general, does not pass through the centroid of the surface, it

normally lies underneath where the pressure is higher. This point is called the “center of pressure”. This vertical location of the line of action is determined by equating the

moment of the resultant force to the moment of the distributed pressure force about

the x axis. It gives

θ ρ θ ρ sinsin ,

2 g I A y P dA y g ydA P yPdA F y o xxC o A A A

o R p +=+== ∫ ∫ ∫ Where y p is the distance of the center of pressure from the x axis, I xx,o is the second

moment of area (also called the area moment of inertia) about x axis. This second

moment of area is commonly used in the engineering handbooks, but they are usually

given about the axes passing through the centroid of the area. However, the second

moments of area about the two parallel axes are related to each other by the parallel

axis theorem, which in this case is expressed as A y I I C C xxo xx

2

,,+=

Where I xx,C is the second moment of area about the x axis passing through the centroid

of the area and yc (the y coordinate of the centroid) is the distance between the two

parallel axes. Therefore, we could have

A g P y

I y y

oC

C xx

C P )]sin/([

,

θ ρ +

+=

If Po=0, which is usually the case when the atmospheric pressure is ignored. It

simplifies to

A y

I y y

C

C xx

C P

,

+=






1992.

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37

Therefore, the vertical distance of the center of pressure from the free surface is

determined from θ sin p p yh = .

The I xx,C for some common areas are given in figure 4.9. For these and other areas that

possess symmetry about the y axis, the center of pressure lies on the y axis directly

below the centroid. The location of the center of pressure in such area is simply the

point on the surface of the vertical plane of symmetry at a distance hp from the free

surface.

Figure 4.9 The centroid and the centroid moments of inertia for some common

geometries

C

b/2

b/2

a/2 a/2

A=ab; I xx,C =ab /12

Rectangle

C

R

A= 2 Rπ ; I xs,C = 4/

4 Rπ

Circle

a b

A= abπ ; I xs,C = 4/3

abπ

Elli se

C

C

a/2 a/2

2b/3

b/3

A=ab/2; I xx,C =ab /36 Triangle

C

R

π 3

4 R

A= 2/2

Rπ ; I xs,C =4

11.0 R

Semicircle

C

a

b

π 3

4b

A= 2/abπ ; I xs,C =3

11.0 ab

Semiellipse






1992.

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38

A B

C

Vertical projection

of the curved surface

Horizontal projection

of the curved surface

F H

F v

W

LECTURE 5

5.1 Hydrostatic forces on submerged curved surfaces

For a submerged curved surface, the determination of the resultant hydrostatic force is

more involved since it typically requires the integration of the pressure forces thatchange direction along the curved surface. It required to determine the horizontal and

vertical components FH and FV. this could be done by considering the free body

diagram of the liquid block enclosed by the curved surface and the two plane surfaces(one horizontal and one vertical) passing through the two ends of the curved surface

as shown in figure 5.1.

Figure 5.1 determination of hydrostatic force acting on a submerged curved surface.

Note that the vertical surface of the liquid block is simply the projection of the curved

surface on a vertical plane and the horizontal surface is the projection of the curved

surface on a horizontal plane. The resultant force acting on the curved solid surface is

then equal and opposite to the force acting on the curved liquid surface (Newton’s

third law). The force acting on the imaginary horizontal or vertical plane surface and

its line of action can be determined as discussed in the lass section.

Horizontal force component on curved surface: x H

F F =

Vertical force component on curved surface: W F F yV +=

The horizontal component of the hydrostatic force acting on the curved surface is

equal to the hydrostatic force acting on the vertical projection of the curved surface.

The vertical component of the hydrostatic force acting on a curved surface is equal to

the hydrostatic force acting on the horizontal projection of the curved surface, plus the

weight of the fluid block.

F R






1992.

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39

The magnitude of the resultant hydrostatic force acting on the curved surface is22

v H RF F F +=

H

V

F

F =α

The exact location of the line of action of the resultant force, distance can bedetermined by taking a moment about an appropriate point. The discussion above is

valid for all curved surfaces regardless of whether they are above or below the liquid.

If the curved surface is a circular arc (full arc or any part of arc), the resultant

hydrostatic force acting on the surface always pass through the center of the circle.

This because the pressure forces are normal to the surface, and all lines normal to the

surface of a circle pass through the center of the circle. Thus the pressure forces froma concurrent force system at the center which can be reduced to a single equivalent

force system at the center which can be reduced to a single equivalent force at that point.

Figure 5.2 Figure 5.3

Note, hydrostatic forces acting on a plane or curved surface submerged in a

multilayered fluid of different densities can be determined by considering different parts of surfaces in different liquids as different surfaces, finding the force on each

part, and then adding them using vector addition. As shown in Figure 5.3

Plane surface in a multilayered fluid: ∑∑ ==iici R R

A P F F ,,

Where icioic gh P P ,,

ρ += is the pressure at the centroid of the portion the surface in

fluid i and Ai is the area of the plate in that fluid. The line of action of this equivalent

force can be determined from the requirement that the moment of the equivalent force

about any point is equal to the sum of the moments of the individual forces about the

same point.

5.2 Buoyancy and stability

Buoyancy force is a force that tends to lift the body in a fluid which is mainly caused

by the increase of pressure in a fluid with depth. The common practices: the object

F R

a

b

FR1

FR2

Oil

Water






1992.

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feels lighter and weighs less in a liquid than it does in air. For example, the ship canfloat on water, the yachts park on water. Consider a thick plate in the water as shown

in figure 5.4:

Figure 5.4 A plate of uniform thickness h submerged in a liquid

The pressure force at the top and bottom surfaces are

ghA F f top ρ =

Ahh g F f bot )( Δ+= ρ

The difference between these two forces is a net upward force which is the buoyant

force,

gV hA g F F F f f topbot B ρ ρ =Δ=−=

The gV f ρ is simply the weight of the liquid whose volume is equal to the volume of

the plate. Thus we conclude: the buoyant force acting on the plate is equal to theweight fo the liquid displaced by the plate. This is independent of the location inside

the liquid and the density of the solid body. This relation and statement are valid for

any body regardless its shape. Consider an arbitrary shaped body submerged in a fluid

and compare it to a body of fluid of the same shape indicated by the dashed line:

Figure 5.5 comparison between the solid body and the same shaped liquid body

It is obvious that the buoyant forces are the same between the above two bodies since

the pressure distribution only relies on the depth. Therefore, the weight and the

buoyant force must have the same line of action to have a zero moment since the fluid

h

Δ h

F top

F bot

A

Liquid

Solid Liquid

FB FB






1992.

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41

body is in static equilibrium and the net force is zero. This is known as the“Archimedes’ principle” and is expressed as:

The buoyant force acting on a body immersed in a fluid is equal to the weight

of the fluid displaced by the body, and it acts upward through the centroid of

the displaced volume.

It is now clear that the weight of the entire body must be equal to the buoyant force,

which is the weight of the fluid whose volume is equal to the volume of thesubmerged portion of the floating body. That is:

f

bodyave

total

subtotal bodyave sub f B

V

V gV gV W F

ρ

ρ ρ ρ

,

,

=→=→=

Therefore, the submerged volume fraction of a floating body is equal to the ratio of

the average density the body to the density of the fluid. When the density ratio equals

one, the floating body becomes completely submerged.

From these we could conclude for a body immersed in a fluid as shown in figure 5.6:

1, f ave ρ ρ = ; the body will remain at any point in the fluid

2, f ave ρ ρ > ; the body will sink to the bottom

3, f ave ρ ρ < ; the body will float on water

Figure 5.6 A solid body inside the fluid

From the above equation, we could find that the buoyant forces are proportional to the

density of the fluid and therefore, we might think that such forces exerted by the gas

such air can be negligible. This is certainly the case in general however there are

some significant exceptions. Take an example, a body with a volume 0.1 m3

and theair density is 1.19kg/m3, the buoyant force exerted by air on this body is

N gV F f B 2.11.0*81.9*19.1 === ρ

If the body is 65kg which is about 638 N, the measurement error is 0.2% which isnegligible in comparison to the weight of body.

But the buoyancy effect sometimes cannot be negligible and dominate someimportant natural phenomena such as the rise of warm air in a cooler environment and

f ave ρ ρ >

f ave ρ ρ =

f ave ρ ρ <

Sinking

body

Fluid

Suspended

body

Floating

body






1992.

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42

thus the onset of natural convection currents, the rise of the hot air or hydrogen balloons, the rise of the water vapor to high elevations.

Stability of immersed and floating bodies

An important application of the buoyancy concept is the assessment of the stability of immersed and floating bodies with no external attachments. This topic is of great

importance in the design of ships and submarines. Here we only provide some general

qualitative discussion on vertical and rotational stability.

When an immersed or floating body is in static equilibrium, the weight and the buoyant force acting on the body balance each other, and such bodies are inherently

stable in the vertical direction. If the immersed body is raised or lowered to a different

depth, the body will remain in equilibrium at that location. This also applies to the

floating body. If a floating body is raised or lowered somewhat by a vertical force, the

body will return to its original position as soon as the external effect is removed.

These circumstances are called neutrally stable since it does not return to its original position after a disturbance.

Figure 5.7 An immersed body is stable if the body is bottom-heavy and thus thecenter of gravity G is below the centroid B of the body

The rotational stability of an immersed body depends on the relative location of thecenter of gravity G of body and the center of buoyancy B, which is the centroid of the

displaced volume. An immersed body is table if the body is bottom heavy and thus

the point G is below the point B. A rotational disturbance of the body in such cases

produces a restoring moment to return the body to its original stable position.

According to this concept, the stable design for ships and submarines calls for the

engines and cabins for the crew to be located at the lower half in order to shift the

weight to the bottom as much as possible. Another example is the hot-air or helium balloons which is stable since the load is always at the bottom of the balloons. When

the G is above the B, the body is not stable and any disturbance will cause this body

to turn upside down. A body for which G and B coincide is neutrally stable. This is

BG

FB

W

Liquid

FB

G

W

BB

FB

W

(c) Neutral(b) Unstable(a) Stable




the case for bodies whose density is constant throughout. There is no tendency tooverturn.

The rotational stability criteria are similar for floating bodies. The stable situation is

the same as the immersed body. When the center of G is below of buoyancy B, it is

always stable. But a floating body may still be stable when G is above B. This is because the centroid of the displaced volume shifts to the side to point B’ during a

rotational disturbance while the center of gravity G of the body remains unchanged. If

the point B’ is sufficiently far, these two forces create a restoring moment and returnthe body to the original position. A measure of stability for floating bodies is the

metacentric height GM, which is the distance between the center of gravity G and themetacenter M, the intersection point of the lines of action of buoyant force before and

after rotation. If the M is above the G point, the floating point is stable. Otherwise it is

unstable. If the M is below the point G, it generates an overturning moment instead of

a restoring moment causing the body to capsize. The length of the metacentric height

GM above G is a measure of the stability: the larger it is, the more stable the floating

body will be.

Figure 5.8 A floating body is stable if the metacentre M is above the centre of gravity

G, and thus GM is positive, and unstable if the M is below G and thus the GM is

negative. From P.M. Gerhart, R. J. Gross, J. I. Hochstein, Fundamentals of fluid mechanics, 1992

Fluids Lecturenotes

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