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,
1992.
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1
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
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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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,
1992.
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2
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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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,
1992.
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3
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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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,
1992.
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4
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
LECTURE 4 FLUID STATICS
4.1 Pressure measurement - application to Manometer 4.2 Hydrostatic forces on plane surface
LECTURE 5 FLUID STATICS
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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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,
1992.
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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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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,
1992.
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6
Lecture notes
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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,
1992.
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7
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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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,
1992.
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8
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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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,
1992.
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9
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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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,
1992.
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10
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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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,
1992.
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11
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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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,
1992.
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12
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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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,
1992.
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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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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,
1992.
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14
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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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,
1992.
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15
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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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,
1992.
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16
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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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,
1992.
th
17
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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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,
1992.
th
18
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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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,
1992.
th
19
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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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,
1992.
th
20
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
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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,
1992.
th
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
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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,
1992.
th
22
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
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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,
1992.
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23
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
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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,
1992.
th
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
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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,
1992.
th
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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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,
1992.
th
26
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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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,
1992.
th
27
(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
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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,
1992.
th
28
Figure 3.6 U-tube manometer
A
h1
h2
B
Fluid
density2
ρ
Fluid
density1
ρ
B’
Open to
atmosphere
A’
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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,
1992.
th
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
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McGraw Hill, 1999.; P.M. Gerhart, R. J. Gross, J. I. Hochstein, Fundamentals of fluid mechanics,2rd edition,
1992.
th
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
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Fluid Mechanics – MECH 2403
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McGraw Hill, 1999.; P.M. Gerhart, R. J. Gross, J. I. Hochstein, Fundamentals of fluid mechanics,2rd edition,
1992.
th
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
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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,
1992.
th
32
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.
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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,
1992.
th
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
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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,
1992.
th
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 +=== ∫
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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,
1992.
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35
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
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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,
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
,
+=
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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,
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
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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,
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
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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,
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
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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,
1992.
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40
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
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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,
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
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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,
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
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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