ME 3560 Fluid Mechanics Chapter I. Introduction • Fluid Mechanics

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ME3560 – Fluid Mechanics
Summer 2016
ME 3560 Fluid Mechanics
Chapter I. Introduction
• Fluid Mechanics is the science that deals with the behavior of fluids at
rest or in motion, and the interaction of fluids with solids or other fluids
at the boundaries.
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Chapter I. Introduction
Summer 2016
ME3560 – Fluid Mechanics
1.1 Brief History of Fluid Mechanics
• One of the first engineering problems was the supply of water to cities
for domestic use and for the irrigation of crops.
•Roman aqueducts are a good example of water systems constructed at
the beginning of civilization.
•Roman aqueducts in
Segovia, Spain, built
around
the
1st
Century A.D.
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Chapter I. Introduction
ME3560 – Fluid Mechanics
Summer 2016
• ►It has been found that from 283 to 133 BC the Hellenistic city of
Pergamon (Turkey) built a series of pressurized led and clay pipelines,
up to 45 km long that operated at a pressure exceeding 1.7 Mpa (180 m
of head, 246.6 psi).
• ► The earliest contribution theory to fluid mechanics was made by
Archimedes (285–212 BC). He formulated and applied the buoyancy
principle.
•During the Middle Ages the application of fluid machinery expanded,
piston pumps were used for dewatering mines, water and wind mills
were perfected to grind grains and forge metal.
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Chapter I. Introduction
ME3560 – Fluid Mechanics
Summer 2016
•The figure shows a mine hoist powered by a
reversible water wheel (Georgius Agricola 1556).
•The development of fluid systems and machines
continued during the Renaissance. The scientific
method was perfected and adopted throughout Europe.
• Simon Stevin (1548–1617), Galileo Galilei (1562–1642), Edme
Mariotte (1620–1684), and Evangelista Torricelli (1608–1647) were
among the first to apply the scientific method to investigate hydrostatic
pressure distributions and vacuums.
•Blaise Pascal integrated and refined the work developed by these
scientists.
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Chapter I. Introduction
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• The Italian monk Benedetto Castelli (1577–1644) was the first person
to publish a statement of the continuity principle for fluids.
• ► Sir Isaac Newton (1643–1727) applied his laws to fluids and
explored fluid inertia and resistance, free jets, and viscosity.
• ► The Swiss engineer Daniel Bernoulli (1700–1782) and his associate
Leonard Euler (1707–1789) built upon Newton’s studies and defined the
energy and momentum equations.
•Bernoulli published in 1738 his treatise Hydrodynamica. This may be
considered the first fluid mechanics text.
•Jean d’Alembert (1717–1789) developed the idea of velocity and
acceleration components, a differential expression of continuity , and his
“paradox” of zero resistance to steady uniform motion.
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Chapter I. Introduction
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• ► Fluid mechanics theory up through the end of the 18th century had
little impact on engineering since fluid properties and parameters were
poorly quantified.
•The French school of engineering led by Riche de Prony (1755–1839)
along with Ecole Polytechnic and the Ecole Ponts et Chaussees were the
first to incorporate calculus and scientific theory to the engineering
curriculum. This brought a change to the engineering theory by making it
more practical and capable of solving real world problems.
•Scientists such as Antonie Chezy (1718–1798), Louis Navier (1785–
1836), Gaspar Coriolis (1792–1843), Henry Darcy (1803–1858)
contributed to fluid engineering and theory and were students and/or
instructors at these schools
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Chapter I. Introduction
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• ► By the mid 19th century the advances in fluid mechanics were
coming from different fronts:
•Jean Poiseuille (1799–1869) had accurately measured flow in capillary
tubesfor multiple fluids.
•Gotthilf Hagen (1797–1884) had differentiated between laminar and
turbulent flow in pipes.
•Osborn Reynolds (1842–1912) continued Hagen’s work and developed
the dimensionless number that bears his name.
• ► In parallel work Louis Navier and George Stokes (1819–1903)
completed the general equations of fluid motion with friction (Navier–
Stokes equations).
•James Francis (1815–1892) and Lester Pelton (1829–1908) pioneered
work in turbines.
•Clemens Herschel (1842–1930) invented the Venturi meter
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Chapter I. Introduction
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• Irish and English engineers such as William Thompson, Lord Kelvin
(1824–1907), William Strutt, Lord Rayleigh (1842 –1919), and Sir
Horace Lamb (1849–1934) investigated problems such as dimensional
analysis, irrotational flow, vortex motion, cavitation, and waves.
• ► At the dawn of the 20th century the Wright brothers (Wilbur, 1867–
1912; Orville, 1871–1948) through application of theory and
experimentation perfected the airplane.
• ► In 1904, Ludwig Prandtl (1875–1953) showed that fluid flows can
be derived into a layer near the walls, the boundary layer, where the
friction effects are significant and an outer layer where such effects are
negligible and the simplified Euler and Bernoulli equations are
applicable.
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Chapter I. Introduction
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• ► By the mid 20th century the existing theories were adequate for the
tasks at hand and the fluid properties and parameters were well defined,
thus supporting a enormous expansion of the aeronautical, chemical,
industrial and water resources sector.
• ► In the late 20th century, Fluid Mechanics research was dominated by
the development of the digital computer. The ability to solve large
complex problems, such as global climate modeling or to optimize the
design of a turbine blade.
•The principles that we will study in this curse apply to flows ranging
from very small to extremely large scales.
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Chapter I. Introduction
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1.2 Definition of a Fluid
• A fluid is defined as a substance that deforms continuously when acted
on by a shearing stress of any magnitude.
•When common solids such as steel or other metals are acted on by a
shearing stress, they will initially deform (usually a very small
deformation), but they will not continuously deform (flow).
•Common fluids such as water, oil, and air satisfy the definition of a
fluid—that is, they will flow when acted on by a shearing stress.
•Some materials, such as slurries, tar, putty, toothpaste are not easily
classified since they will behave as a solid if the applied shearing stress
is small, but if the stress exceeds some critical value, the substance will
flow. The study of such materials is called rheology.
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•To describe the behavior of fluids at rest or in motion, we consider the
average, or macroscopic, value of the quantity of interest.
•The average is evaluated over a small volume containing a large
number of molecules.
•The volume is small compared with the physical dimensions of the
system of interest, but large compared with the average distance
between molecules.
•For gases at normal pressures and temperatures, the spacing is on the
order of 10−6 mm. For gases, the number of molecules per cubic
millimeter is on the order of 1018.
•For liquids it is on the order of 10−7 mm. For liquids, the number of
molecules per cubic millimeter is on the order of 1021.
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1.3 The Non–Slip Condition
• Fluid flow is often confined by solid surfaces, and it is important to
understand how the presence of solid surfaces affects fluid flow.
•.Consider the flow of a fluid in a stationary pipe or over a solid surface
that is nonporous. Experimental observation indicates that a fluid in
motion comes to a complete stop at the surface and assumes zero
velocity relative to that surface.
•That is, a fluid in direct contact with a solid “sticks” to the surface due
to viscous effects and there is no slip.
•This is known as the non–slip condition.
• The fluid property responsible for the
non–slip condition is the viscosity.
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Chapter I. Introduction
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Summer 2016
• A fluid layer adjacent to a moving surface has the same velocity as the
surface
•A consequence of the non–slip condition is that all velocity profiles
must have zero values with respect to the surface at the points of
contact.
•Another consequence of the non–slip condition is the surface drag,
which is the force a fluid exerts on a surface in the direction of the flow.
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Chapter I. Introduction
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1.4 Classification of Fluid Flows
• There is a wide variety of fluid flow problems and it is convenient to
classify them based on some common characteristics to group them.
Viscous versus Inviscid Regions of Flow
•When two fluid layers move relative to each other, a friction force
develops between them and the slower layer tries to slow down the
faster layer.
•This internal resistance to flow is quantified by the viscosity. The
viscosity is caused by cohesive forces between the molecules in liquids
and by molecular collisions in gases. There is no fluid with zero
viscosity.
•Flows in which the viscous effects are important are called viscous
flows.
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Chapter I. Introduction
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Summer 2016
• In many flows of practical interest, there are regions where the viscous
forces are negligibly small compared to inertial or pressure forces.
• Neglecting the viscous effects in such inviscid flow regions greatly
simplifies the analysis without much loss in accuracy.
• The development of viscous and inviscid
regions of flow as a result of inserting a flat
plate parallel to a fluid stream of uniform
velocity is shown in the picture.
•The fluid sticks to the plate on both sides
due to the non – slip condition.
• Two zones are present, a viscous flow region (boundary layer) and an
inviscid flow region.
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Chapter I. Introduction
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Internal versus External Flow
•A fluid flow is internal or external depending on whether 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 in a pipe or a duct is internal flow if the fluid is completely
bounded by solid surfaces.
•The flow of liquids in a duct which is only partially filled is called
open channel flow. Flow of rivers is an example of this type of flows.
• Internal flows are dominated by the influence of viscosity throughout
the flow field.
• In external flows the viscous effects are limited to boundary layers
near solid surfaces and to wake regions downstream of bodies.
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Chapter I. Introduction
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Compressible versus Incompressible Flow
• A flow is classified as being compressible or incompressible,
depending on the level of variation of density during flow.
•Incompressibility is an approximation and a flow is said to be
incompressible if the density remains constant, that is, the volume of
every portion of fluid remains unchanged.
•The densities of liquids are essentially constant (incompressible).
•Gases on the other hand are highly compressible. However, gas flow
can often be considered incompressible if the density changes are under
5 percent, which is usually the case when Ma < 0.3
= =
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Summer 2016
• The speed of sound in air (room temperature, sea level) is c = 346 m/s.
Therefore, the compressibility effects in air can be neglected at speeds
under about 100 m/s (≈ 220 mi/hr).
Laminar versus Turbulent Flow
• Some flows are smooth and orderly while others are rather chaotic.
• The highly ordered fluid motion characterized by smooth layers of
fluid is called laminar.
• The highly disordered fluid motion that typically occurs at high
velocities and is characterized by velocity fluctuations is called
turbulent.
• Flow that alternates between being laminar and turbulent is called
transitional.
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Summer 2016
Natural (or Unforced) versus Forced Flow
• A forced flow is a flow in which the fluid is forced to flow over a
surface or in a pipe by external means such as pump or a fan.
• In natural flows any fluid motion is due to natural means such as the
buoyancy effect.
Steady versus Unsteady Flow
• The terms steady and uniform are used frequently in engineering.
• The term steady implies no change at a point with time. The opposite
of steady is unsteady.
•The term uniform implies no change with location over a specified
region.
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Chapter I. Introduction
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• The terms unsteady and transient are often used interchangeably,
however, in fluid mechanics unsteady applies to any flow that is not
steady, and transient applies to developing flows.
One–, Two–, and Three–Dimensional Flows
• The best way to describe a flow field is through the velocity
distribution, thus the flow can be one–, two–, or three –dimensional,
depending on the number of coordinate directions required to describe
the flow.
• In the most general case, a fluid flow is described by three–dimensions
[V(x, y, z) or V(r, θ, z)].
• In many instances, the variation of the velocity in certain directions
can be small relative to the variation in other directions and can be
ignored with negligible error. Thus the flow can be 1–D or 2–D.
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Chapter I. Introduction
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1.5 System and Control Volume
•A system is a collection of matter of fixed identity (always the same
atoms or fluid particles), which may move, flow, and interact with its
surroundings.
•A system is a specific, identifiable quantity of matter. It may consist of a
relatively large amount of mass or it may be an infinitesimal size.
•A system may interact with its surroundings by various means (by the
transfer of heat or the exertion of a pressure force, for example).
• A system may continually change size and shape, but it always
contains the same mass.
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•A control volume, is a volume in space (a geometric entity, independent
of mass) through which fluid may flow.
•In fluid mechanics, it is difficult to identify and keep track of a specific
quantity of matter.
• In several cases, the main interest is in determining the forces put on a
device rather than in the information obtained by following a given
portion of the air (a system) as it flows along.
• For these situations it is more adequate to use the control volume
approach.
•Identify a specific volume in space (a volume associated with the device
of interest) and analyze the fluid flow within, through, or around that
volume.
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Chapter I. Introduction
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• In general, the control volume can be a moving volume, although for
most situations we will use only fixed, non-deformable control volumes.
• The matter within a control volume may change with time as the fluid
flows through it.
• The amount of mass within the volume may change with time.
•The control volume itself is a specific geometric entity, independent of
the flowing fluid.
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Chapter I. Introduction
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•All of the laws governing the motion of a fluid are stated in their basic
form in terms of a system approach.
•For example, “the mass of a system remains constant,” or “the time rate
of change of momentum of a system is equal to the sum of all the forces
acting on the system.”
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Chapter I. Introduction
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1.6 Dimensions, Dimensional Homogeneity, and
Units.
•The study of fluid mechanics requires to develop a system for
describing the fluid characteristics qualitatively and quantitatively.
•The qualitative aspect serves to identify the nature, or type, of the
characteristics (such as length, time, stress, and velocity).
V =& LT −1
a =& LT −2
F =& MLT −2
•The quantitative aspect provides a numerical measure of the
characteristics. The quantitative description requires both a number and
a standard (unit) by which various quantities can be compared.
m
V 
s
m
a 2 
s 
F[ N ]
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Chapter I. Introduction
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Chapter I. Introduction
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Chapter I. Introduction
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•All theoretically derived equations are dimensionally homogeneous,
and all additive separate terms must have the same dimensions.
For example, the equation for the velocity, V, of a uniformly accelerated
body is
V = V0 + at
where V0 is the initial velocity, a the acceleration, and t the time interval.
In terms of dimensions the equation is
LT −1 =& LT −1 + LT −1
and thus this equation is dimensionally homogeneous.
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Chapter I. Introduction
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1.6.1 Systems of Units
•In addition to the qualitative description of the various quantities of
interest, it is necessary to have a quantitative measure of any given
quantity.
•We will consider three systems of units that are commonly used in
engineering.
- International System (SI)
Quantity
Unit
Length
Meter (m)
Time
Second (s)
Mass
Kilogram (kg)
Temperature
Kelvin (K)
K = oC + 273.15
Chapter I. Introduction
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ME3560 – Fluid Mechanics
Quantity
Unit
Force
Work
Power
Newton (N)
Joule (J)
Watt (W)
1N = 1kg ⋅1m/s
1J = 1N ⋅ m
Summer 2016
2
1W = 1J/s = 1N ⋅ m/s
W = mg;g = 9.81m/s
2
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Chapter I. Introduction
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-British Gravitational (BG) System.
o
Quantity
Unit
Length
Time
Mass
Temperature
Force
Work
Power
Foot (ft)
Second (s)
Slug (slug)
Rankine (oR)
Pound (lb)
R = o F + 459.67
1lb = 1slug ⋅ ft/s 2
W = mg;g = 32.2ft/s 2
lb⋅ft
lb⋅ft/s
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-English Engineering (EE) System.
•In the EE system, units for force and mass are defined independently.
o
Quantity
Length
Time
Mass
Temperature
Force
Work
Power
Unit
Foot (ft)
Second (s)
Pound mass (lbm)
Rankine (oR)
Pound (lb)
R = o F + 459.67
1lb = 1lbm ⋅ 32.2ft/s 2
1slug = 32.2lbm
W = mg;g = 32.2ft/s 2
lb⋅ft
lb⋅ft/s
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Chapter I. Introduction
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1.7 Modeling in Engineering
•An engineering device can be studied either experimentally or
analytically.
•The experimental approach is advantageous because it deals with the
actual physical system and the desired quantity is determined by
measurement.
•The experimental approach is expensive, time consuming and often
impractical. Additionally the system to be studied might not exist.
•On the other hand, the analytical approach (including numerical
approach) has the advantage of being fast and inexpensive.
•The results obtained are subject to the accuracy of the assumptions,
approximations, and idealizations made in the analysis.
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Chapter I. Introduction
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• The description of most scientific problems involve equations that
relate the changes in some key variables to each other.
• Usually, the smaller the increment chosen in the changing variables, the
more general and accurate the description.
•Therefore, differential equations are used to investigate a wide variety
of problems in engineering and sciences.
• However, may problems can be studied without the need of using
differential equations.
• In this class we will learn different tools, additional to the solution of
differential equations to study problems in Fluid Mechanics
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Chapter I. Introduction
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1.8 Continuum
• Matter is made up of atoms that are widely spaced in the gas phase.
•However, it is very convenient to disregard the atomic nature of a
substance and treat it as a continuous, homogeneous matter with no
holes, that is, a continuum.
• The continuum idealization allows the treatment of properties as point
functions and to assume that properties vary continuously in space
without discontinuities.
•This assumption is valid as long as the size of the system considered is
large relative to the space between the molecules. This will be the case in
all problems we analyze in this course of Fluid Mechanics
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Chapter I. Introduction
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•To describe the behavior of fluids at rest or in motion, we consider the
average, or macroscopic, value of the quantity of interest.
•The average is evaluated over a small volume containing a large
number of molecules.
•The volume is small compared with the physical dimensions of the
system of interest, but large compared with the average distance
between molecules.
•For gases at normal pressures and temperatures, the spacing is on the
order of 10−6 mm. For gases, the number of molecules per cubic
millimeter is on the order of 1018.
•For liquids it is on the order of 10−7 mm. For liquids, the number of
molecules per cubic millimeter is on the order of 1021.
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1.9 Measures of Fluid Mass and Weight
1.9.1 Density
•The density of a fluid is defined as its mass per unit volume.
m  kg   slug 
ρ=
,
3 
3 

V  m   ft 
•The specific volume of a fluid is defined as the ratio of the volume
occupied by the volume to its mass.
V  m 3   ft 3   ft 3 
v = =  , 
, 
ρ m  kg   slug   lbm 
1
•The density of liquids is assumed to be constant –incompressible fluids
•The density of gases depends on the temperature and pressure of the
system. For example, for ideal gases:
p = ρRT
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1.9.2 Specific Weight
•Specific weight (γ) is weight per unit volume.
γ = ρg
1.9.3 Specific Gravity
•Specific Gravity (SG) is defined as the ratio of the density of the fluid
to the density of water at 4 °C (39.2 °F): ρ=1.94 slugs/ft3=1000 kg/m3.
SG =
ρ
ρH [email protected]
2
=
o
C
γ
γ H [email protected]
2
o
C
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1.10 Viscosity
•If P is applied to the upper plate, it will move continuously with a
velocity, U.
•The fluid in contact with the upper plate moves with a velocity, U.
•The fluid in contact with the bottom fixed plate has a zero velocity.
•The fluid between the two plates moves with velocity u=u(y)=Uy/b
•A velocity gradient du/dy=U/b, develops in the fluid between the plates.
•The experimental observation that the fluid “sticks” to the solid
boundaries is referred to as the no-slip condition.
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•It can be experimentally determined that
P U
∂u
α → τα
A b
∂y
•τ shear stress in a fluid in motion
•∂ u/ ∂ y. Rate of shearing strain
(Velocity gradient)
• For a large number of fluids the relation between shear stress and
velocity gradient is linear:
∂u •µ Absolute (dynamic) viscosity
τ =µ
∂y
Dynamic viscosity is property that relates shearing stress and fluid
motion
M ⋅L 1
⋅
2
Shear Stress
M
T
L
Dimensions :
=
=
L 1
Velocity Gradient
LT
/ 
T L
N ⋅s
kg
= Pa ⋅ s •1poise = 0.1 N⋅s/m2
Units : 2 =
m
m ⋅s
2
Chapter I. Introduction
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•Fluids for which the
shearing stress is linearly
related to the rate of
shearing strain (also referred
to as rate of angular
deformation) are designated
as Newtonian fluids.
•µ = µ (T)
•For gases µ increases as T
does.
•For liquids µ decreases as T
does.
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•Fluids for which the shearing
stress is not linearly related to
the rate of shearing strain are
designated as non-Newtonian
fluids.
•Quite often viscosity appears
in fluid flow problems
combined with the density in
µ
the form
υ=
ρ
•ν kinematic viscosity
•The dimensions of ν are L2/T
• BG units are ft2/s
• SI units are m2/s.
•CGS units are cm2/s = St (stoke)
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1.11 Vapor Pressure (pv)
• Vapor pressure (saturation pressure) is a thermodynamic property and
it is the pressure at which phase change from liquid to gas (boiling)
occurs.
•Under certain circumstances in flowing fluids low pressures can be
generated such that cavitation may occur.
http://www.youtube.com/watch?v=GpklBS3s7iU&feature=related
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1.12 Compressibility of Fluids
1.12.1 Bulk Modulus
•A property that is commonly used to characterize compressibility is the
bulk modulus, Eν, defined as
dp
dp
Ev = −
=
dV / V dρ / ρ
•The bulk modulus has dimensions of pressure, FL−2.
•The units for Ev are lb/in.2 (psi) and N/m2 (Pa).
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1.12.2 Compression and Expansion of Gases
•When gases are compressed (or expanded), the relationship between
pressure and density depends on the nature of the process.
•Isothermal process
p
ρ
= cons; Ev = p
•Isentropic Process (frictionless compression (expansion), no heat is
exchanged with the surroundings) p
ρ
k
= cons; Ev = kp
k = c p / cv ; R = c p − cv
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1.12.3 Speed of Sound
•The velocity at which small disturbances propagate in a fluid is called
the acoustic velocity or the speed of sound, c
dp
c=
=
dρ
Ev
ρ
•For gases (isentropic process)
c=
kp
ρ
= kRT
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1.13 Ideal Gas Law
•The equation for ideal or perfect gases known as equation of state for
an ideal gas is:
•p is the absolute pressure
p
ρ=
•ρ the density
RT
•T the absolute temperature
•R the gas constant
pabs = p gage + patm
patm
lb
= 101.33 kPa = 14.7 2 (psi)
in
• This equation closely approximates the behavior of gases under normal
conditions when the gases are not approaching liquefaction.
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1.14 Surface Tension
•At the interface between a liquid and a
gas, or between two immiscible liquids,
forces develop in the liquid surface which
cause the surface to behave as if it were a
“membrane” stretched over the fluid mass.
•These types of surface phenomena are due
to the unbalanced cohesive forces acting
on the liquid molecules at the fluid surface.
• Molecules in the interior of the fluid mass
are surrounded by molecules that are
attracted to each other equally. However,
molecules along the surface are subjected
to a net force toward the interior.
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•As a result of this unbalanced force a hypothetical membrane is created
at the interface.
•A tensile force may be considered to be acting in the plane of the
surface along any line in the surface.
•The intensity of the molecular attraction per unit length along any line
in the surface is called the surface tension (σ). σ = F/l.
•For a given liquid the surface tension depends on temperature and the
other fluid it is in contact with at the interface.
•The dimensions of surface tension are FL−1. With units of lb/ft and N/m.
• The value of the surface tension decreases as the temperature increases.
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Determination of the Pressure inside a Drop
•If the spherical drop is cut in half (as shown), the force developed
around the edge due to surface tension is 2πRσ.
•This force must be balanced by the pressure difference, ∆p, between the
internal pressure, pi, and the external pressure, pe, acting over the circular
area, πR2.
2πRσ = ∆pπR 2
2σ
∆p = pi − pe =
R
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Chapter I. Introduction
ME3560 – Fluid Mechanics
Summer 2016
•Another phenomena associated with surface tension is the rise (or fall)
of a liquid in a capillary tube.
•If a small open tube is inserted into water, the water level in the tube
will rise above the water level outside the tube. In this situation we have
a liquid–gas–solid interface.
•In this case, there is an attraction (adhesion) between the wall of the
tube and liquid molecules which is strong enough to overcome the
mutual attraction (cohesion) of the molecules and pull them up the wall.
• Hence, the liquid is said to wet the solid surface
51
Chapter I. Introduction
Summer 2016
ME3560 – Fluid Mechanics
•The height, h, is a function of σ, R, γ, and the angle of contact, θ,
between the fluid and tube.
•An equilibrium analysis yields the following relations
2π Rσ cos θ = γπ R 2 h
2σ cos θ
h=
γR
•The angle of contact is a function of both the liquid and the surface.
•For water in contact with clean glass θ ≈ 0°.
•h is inversely proportional to R.
52
Chapter I. Introduction
ME3560 – Fluid Mechanics
Summer 2016
•If adhesion of molecules to the solid surface is weak compared to the
cohesion between molecules, the liquid will not wet the surface and the
level in a tube placed in a nonwetting liquid will actually be depressed.
•Mercury is a good example of a nonwetting liquid when it is in contact
with a glass tube.
•For nonwetting liquids the angle of contact is greater than 90°, and for
mercury in contact with clean glass θ ≈ 130°.
53
Chapter I. Introduction
Summer 2016
ME3560 – Fluid Mechanics
Pe
54
Chapter I. Introduction
Summer 2016
ME3560 – Fluid Mechanics
Read Sections:
1.7 Compressibility of Fluids
1.7.1 Bulk Modulus
1.7.2 Compression and Expansion of Gases
1.7.3 Speed of Sound
1.9 Surface Tension
55
Chapter I. Introduction
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