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Southern Methodist UniversityBobby B. Lyle School of EngineeringCEE 2342/ME 2342 Fluid Mechanics
Roger O. Dickey, Ph.D., P.E.
I. INTRODUCTIONB. Physical Characteristics of FluidsC. The Science of Fluid MechanicsD. Units and DimensionsE. Fluid Properties
Reading Assignment:
Chapter 1, following along with daily lecture material.
B. Physical Characteristics of Fluids
In engineering applications, matter exists in one of three physical states:
• Solid
• Liquid
• Gas
The term fluid encompasses both liquids and gases.
Fluids
Qualitative characteristics of a solid:
• Essentially has constant volume
• Maintains a fixed shape
• Molecules are held in a more less “rigid”
structure
Qualitative characteristics of a liquid:
• Molecules are relatively free to change
their positions with respect to each other
but they are restricted by cohesive forces
to a practically constant volume
• Assumes the shape of a container
Qualitative characteristics of a gas:
• Molecules are unrestricted by cohesive
forces
• Has no definite volume
• Expands to fill a container
Definitions of shear force/shear stress and
normal force/normal stress (pressure):
(i) A shear force on a surface is the
component of the total applied force that is
tangent to the surface. Shear stress is the
shear force per unit area of the surface.
(ii) A normal force on a surface is the
component of the total applied force that is
perpendicular to the surface. Normal stress
or pressure is the normal force per unit
area of the surface.
Consider a solid subjected to shear forces,
Initial solid shape in static equilibrium
Apply shear force
Deformation of the solid takes place until internal forces (spring-like forces due to molecular cohesion) build up to resist further deformation resulting in a deformed solid shape in static equilibrium
A fluid can be defined as follows:
Fluid a substance that continuously
deforms (i.e., flows) under the action of a
shear stress.
Consider a fluid element subjected to shear forces,
Initial fluid element shape in static equilibrium
Apply shear force
Continuous deformation occurs until the shear stress is removed
Fluid as a Continuum -
It is virtually impossible to evaluate the forces
acting on, and the behavior of individual
molecules when studying the behavior of fluids
because the number of molecules in a tiny
volume is astronomical (approximately 1021
molecules/mm3 for liquids and 1018
molecules/mm3 for gases).
In the fluid sciences, behavior of fluids is
characterized by considering the average, or
macroscopic value of physical quantities of
interest where the average is taken for an
extremely large number of molecules. These
macroscopic fluid properties can be perceived
by human senses, and measured by instruments.
In essence, the derivation and application of the
basic principles of fluid mechanics requires that
the actual molecular structure of fluids be
replaced by a hypothetical continuous medium
called the continuum.
C. The Science of Fluid Mechanics
The sub-disciplines of the science of fluid
mechanics are summarized on the following
flow chart:
Fluid Mechanics
Fluid Dynamics(Flow of Fluids)
Fluid Statics(Fluids at Rest)
Hydrodynamics(Incompressible Flow - liquids and low speed flow of gases)
Gas Dynamics(Compressible Flow - gases)
Aerodynamics
Hydraulics(Flow of Liquids in Pipes and Channels)
Examples of the practical importance of fluid
mechanics:
(1) Hydrology
• Flow of groundwater
• Flow of water in streams and lakes
• Ocean currents and tides
• Weather (flow of air in the atmosphere)
(2) Closed Conduit Flow
• Water supply transmission pipelines
• Water distribution networks
• Cold and hot water plumbing for buildings
• Oil and gas pipelines
• Air and oxygen distribution piping (e.g., in waste treatment systems, hospitals, etc.)
• Blood flow in human circulatory system
(3) Open Channel Flow
• Storm sewer networks and channels
• Sanitary sewer networks
• Wastewater plumbing in buildings (e.g.,
industrial wastewater and sewage
collection networks)
• Irrigation networks
(4) Turbomachinery
• Turbines (power generation)
• Pumps
• Compressors and blowers
(5) Aerodynamics
• Airframe design (lift and drag)
• Propulsion systems (propellers, jet
engines, rocket engines)
(6) Naval Architecture – design of boats and
ships
Consider the example of an automobile:
• Pneumatic tires
• Gasoline and engine coolant pumped through closed conduits
• Hydraulic shock absorbers
• Hydraulic brakes
• Aerodynamic body shapes increase fuel efficiency by reducing drag
Interest in fluid behavior dates back to ancient
civilizations based on the necessity for
development of water supply and irrigation
systems, design of boats and ships, and
propelling projectiles through the air (e.g., the
addition of feathers at the tail of an arrow,
creating drag to true the flight of the arrow).
The beginnings of the formalization of the
science of fluid mechanics began with the need
to control water in large irrigation systems in
ancient Egypt, Mesopotamia (Iraq), and India as
early as 3000 BC.
However, the first quantitative scientific laws in
the field of fluid mechanics were not developed
until 250 BC when the Greek scientist,
mathematician, and inventor Archimedes
discovered and recorded the principles of
hydrostatics and flotation.
The fundamental principles of hydrodynamics
were not set forth until the 17th and 18th centuries
with the greatest contributions made by Isaac
Newton, Daniel Bernoulli, and Leonhard Euler.
Refer to Table 1.9, pp. 28-29, in the textbook
for a chronological listing of some major
contributors to the science of fluid mechanics, an
excerpt is shown on the following two slides:
Table 1.9,pp. 28-29
Table 1.9(Continued)
D. Units and DimensionsBoth SI and British Gravitational (BG), also called U.S. Customary (USC), units are used:
Physical UnitsQuantity Dimension SI USCForce F N lbMass M kg slugLength L m ftTime T sec sec
A consistent set of physical units involves
either: (1) defining the units of F-L-T as the
basic units, then deriving the unit of M from
Newton’s Second Law of Motion (F = ma), or
(2) defining the units of M-L-T, and deriving
the unit of F from Newton’s Second Law.
Using SI units, M-L-T units are defined, and the
unit of F is derived:
2
2
secmkg 1 N 1
)m/sec kg)(1 (1 N 1
ma F
1 N is derived as the force required to accelerate a1 kg mass at 1 m/sec2
SI Force Dimensions:
22 or -MLT
TLM
Using USC units, F-L-T units are usually
defined, and the unit of M is derived:
ftseclb 1 slug 1
ft/sec 1lb 1 slug 1
aF m
2
2
1 slug is derived as the mass accelerated at 1 ft/sec2 by a force of 1 lb
USC Mass Dimensions:
122
or -LFTLTF
Reconsider the derived unit of force in the SI
system of units,
Multiply both sides of this equation by 1 sec2,
2secmkg 1 N 1
mkg 1 secN 1 2
Divide both sides of the equation by 1 m and
rearrange,
msecN 1 kg 1
2
These are the same physical dimensions derived for slugs, the unit of mass in the USC system of units:
122
or -LFTLTF
A comprehensive listing of dimensions for
common physical quantities encountered in
science and engineering is shown in Table 1.1
Dimensions Associated with Common
Physical Quantities, p. 5 in the textbook:
Table 1.1 (Continued)
Table 1.1 (Continued)
Homework No. 1 Dimensional consistency of
mathematical expressions describing fluid
phenomena.
E. Fluid Properties
Properties vary from fluid-to-fluid. Properties for
a given fluid frequently vary with temperature.
Refer to Tables 1.5 and 1.6, inside the front
cover of the textbook, as shown on the
following two slides:
Measures of Mass and Weight
(1) Specific Weight, ,
weight per unit volume
SI Units - N/m3
USC Units - lb/ft3
3LF
(2) Density, ,
mass per unit volume
SI Units - kg/m3
USC Units - slug/ft3
3LM
(3) Relationship Between and ,
Weight and mass are related by Newton’s Second Law,
F = ma
Weight, w, is the gravitational force exerted on a body of mass, m, by the earth and thus,
w = mg
where, g = gravitational acceleration
Divide both sides of the previous equation by
the volume, , of the body of weight w and
mass m yields,
By definition,
gVm
Vw
Vmρ and
Vwγ
V
Substituting,
Standard gravitational acceleration at mean sea
level (MSL) is,
g = 9.81 m/sec2
g = 32.2 ft/sec2
ρ gγ [Equation (1.6), p. 12]
(4) Specific Gravity, S,
S ratio of the density of a fluid to
the density of water at 4° C
S is dimensionless (one of many dimensionless
ratios encountered in fluid mechanics) and,
thus, the numerical value of S does not depend
on the system of units.
Viscosity
Contrary to the case of solid bodies, specific
weight and density are insufficient to uniquely
characterize the dynamic behavior of fluids
when acted upon by external forces. For
example, two fluids with similar density, such
as certain oils and water, can behave quite
differently when flowing.
There is need for an additional physical
property to describe the “fluidity” of gases and
liquids. This property is called viscosity.
Viscosity is the fluid property that offers
resistance to shear stresses.
Film Clip Textbook film, Segment V1.3: Viscous Fluids.
Isaac Newton made pioneering experimental
observations of fluids upon which shear forces
were applied. He discovered that the resulting
velocity gradient created within the fluid was
directly proportional to the applied shear stress.
His discovery is called Newton’s Law of
Viscosity, and for one-dimensional flow in the
x-direction it is written as,
where,
dydu
[Equation (1.9), p. 15]
= shear stress applied to the fluid [F/L2]
u = velocity in the x-direction [L/T]
y = distance along the y-axis [L]
(i.e., distance above the x-axis)
= velocity gradient perpendicular to the
direction of flow
= viscosity
dydu
LTL
22 or
FTL
LTF
is given the name viscosity but is also called
the absolute viscosity or dynamic viscosity.
Typical units are,
SI Units — (no special name)
USC Units — (no special name)
c-g-s — (named Poise, P)
2msecN
2ftseclb
2cmsecdyne
Consider an experiment where a fluid film of
thickness “b” is located between a moving top
plate having velocity U in the +x-direction, and
a parallel but fixed bottom plate as shown in
Figure 1.5, p. 15 in the textbook, slightly
modified on the following slide:
Figure 1.5, p. 15 –Modified
bU
dydu
x
1
The constant velocity of the top plate, U, is
induced by applying a constant horizontal force,
P, to the plate. For a plate having planar area A,
the shear stress (i.e., force per unit area)
applied to the fluid by the top plate is simply:
2
LF
AP
The fluid between the plates deforms continuously
under the action of the shear stress, . The fluid
motion may be conceptualized as many thin
horizontal layers sliding one over another at
differing rates yielding a velocity gradient ,
which is also the rate of angular deformation of
the fluid as illustrated in Figure 1.5. Hence, is
sometimes called the rate of shearing strain.
dydu
dydu
Experimental observations reveal that fluids
“stick” to solid boundaries due to molecular
adhesion forces. Thus, fluid particles in direct
contact with moving solid boundaries move with
the same velocity as the solid surface, and fluid
particles contacting stationary solid boundaries
have zero velocity. This is called the no-slip
condition for flowing fluids.
Film Clip Textbook film, Segment V1.4: No-Slip Condition.
Experimental observations for the geometry of the current experiment reveal that the fluid velocity u increases linearly when proceeding upward in the +y-direction, hence is constant, i.e.,
The no-slip condition establishes the boundary conditions: (i) u = 0, y = 0 at the bottom plate, and (ii) u = U, y = b at the top plate. Thus,
bU
dydu
bU
dydu
00
dydu
yu
dydu
Viscosity is also a constant, i.e., a characteristic
physical property of the specific fluid used in the
experiment. Therefore, according to Newton’s Law
of Viscosity, , the shear stress must
remain constant throughout the fluid, ,
being transmitted from one sliding fluid layer to
the next.
bU
dydu
Most fluids adhere to Newton’s Law of Viscosity
—shear stress varying as a linear function of
velocity gradient and having viscosity, μ, as the
constant of proportionality. These fluids are
termed Newtonian fluids. Fluids that do not
adhere to the law are classed as non-Newtonian
fluids. Figure 1.7, p. 16 in the textbook
graphically compares Newtonian and several
common classes of non-Newtonian fluids:
Figure 1.7, p. 16 –Modified
μ1
Film Clip Textbook film, Segment V1.6: Non-Newtonian Behavior – Shear-Thickening
It is sometimes convenient in fluid mechanics to
use the kinematic viscosity, , which is defined
as the viscosity divided by the density of the
fluid:
M
TLFLMLTF3
2
From Newton’s Second Law of Motion, force has
equivalent dimensions of which yields the
common dimensions for kinematic viscosity, ,
that is:
TL2
2TLM
TL
MTLTLM
MTLF 22
Common units for kinematic viscosity are,
SI Units - m2/sec (no special name)
USC Units - ft2/sec (no special name)
c-g-s - cm2/sec (named Stoke, St)
Viscosity does not vary significantly with pressure and, as such, is assumed solely a function of temperature. Figure 1.8, p. 17 in the textbook is reproduced on the following slide, showing that:
(i) Liquids – viscosity decreases dramatically with increasing temperature
(ii) Gases – viscosity increases slightly with increasing temperature
(iii) Viscosities for common liquids are several orders of magnitude higher than those of gases.
Figure 1.8,p. 17
Dyn
amic
Vis
cosi
ty, μ
(N·s
/m2 )
Temperature, C
Homework No. 2 Newton’s Law of Viscosity.
Vapor Pressure
Vapor pressure, pv , is defined as,
pv the absolute pressure at which a
liquid will boil at a given temperature
Vapor pressure increases as the temperature of the liquid increases, refer to Handout — I.E. Properties of Common Fluids. A liquid can be made to boil at low temperature by reducing the pressure.
In many common liquid flow situations (e.g., in
some hydraulic machinery such as pumps and
turbines, inside valves, and along the face of
dam spillways) pressures at or below the vapor
pressure of the liquid can occur. Under these
conditions, the liquid flashes to vapor forming
tiny bubbles or gas pockets.
As the bubbles are carried along by the flow,
they can enter zones of higher pressure where
the vapor condenses and the bubbles collapse.
As the liquid rushes into the vacuum left by the
collapsed bubble, high transient forces are
generated (like tiny hammer blows).
Where the collapsing bubbles are in contact
with solid boundaries these “hammer blows”
can cause pitting and erosion of the solid
surface. The phenomena of vapor pocket
formation and collapse is called cavitation.
Cavitation
Worker inspecting cavitation damage to spillway
Cavitation – Glen Canyon Dam Spillway near Page, AZ
Workers repairing cavitation damage to dam spillway
Cavitation – Dworshak Dam Spillway near Ashaka, Idaho
Bulk Modulus of Elasticity
Bulk Modulus of Elasticity, Ev , often simply
called the Bulk Modulus, is a measure of the
elasticity of a liquid.
Let,
= volume of liquid [L3]
p = pressure [F/L2]
V
Then,
vEV
dpVd
Rate of change ofvolume with respect to pressure
The minus sign indicates that volume decreases with increasing pressure
Rearranging yields,
SI Units - N/m2 (called the Pascal)
USC Units - lb/ft2 which is frequently
converted to lb/in2 or psi
2
LF
VVddpEv
PressureDimensions
[Equation (1.12), p. 20]
Since a decrease in volume (i.e., ) for a
given mass, , causes an increase in
density (i.e., d > 0), the previous equation may
also be written:
2
LF
ddpEv
PressureDimensions
[Equation (1.13), p. 20]
0Vd
Vm
Values of Ev tend to be very large for common
liquids, e.g., approximately 300,000 psi for
water. For other examples refer to Handout —
I.E. Properties of Common Fluids,
emphasizing that liquids can be assumed
incompressible for the vast majority of applied
fluid mechanics problems.
However, there are a few engineering
applications involving extremely large pressure
changes where it becomes necessary to consider
the compressibility of liquids:
• Water hammer – rapid valve closure can
cause large pressure transients that
generate loud “bangs”, shaking, and even
rupture of piping.
• Pumping of groundwater from deep
confined aquifers.
Valve Closure Water Hammer
Water Hammer Damage
Water Hammer Shock Alleviation
Wells in Confined Aquifers
Surface Tension
At interfaces between liquids and gases,
between two immiscible liquids, and between
liquids and solids unbalanced cohesive forces
develop that cause the liquid surface to appear
and behave as if it were a “skin” or “membrane”
stretched over the fluid mass.
Consider a liquid-gas interface, e.g., the
interface between water in a drinking glass and
the surrounding air. Molecules within the
interior of the liquid mass experience balanced
cohesive forces because they are surrounded by
like molecules that are attracted to each other
equally.
Conversely, liquid molecules residing at the liquid-
gas interface are subjected to a net force toward the
interior of the liquid mass because the cohesive
forces between liquid molecules are much stronger
than the adhesive forces between the liquid
molecules and the overlying gas. The unbalanced
cohesive forces along the interface create the
appearance of a taut skin over the liquid surface.
A tensile force acts in the plane of the surface
“skin,” along any line in the surface, much like
the tensile force in a drum head. The intensity of
the tensile force per unit length is called surface
tension, σ [F/L].
Surface tension is a characteristic property of a given liquid, as illustrated in Handout — I.E. Properties of Common Fluids for various liquids in contact with air (e.g., 0.466 N/m for mercury compared to 0.022 N/m for gasoline). However, surface tension also depends on the other substance—liquid, gas, or solid—forming the interface with the liquid. Furthermore, surface tension varies with temperature, tending to decrease with increasing temperature.
Surface tension forces are usually negligible compared to inertial and viscous shear forces in flowing fluids. However, there are a number of interesting and/or important phenomena involving surface tension:
• Steel needles, razor blades, and water walking insects can float on water under the right conditions, because the tensile force in the taut skin balances their weight.
• Liquids can form tight, compact droplets when placed on smooth solid surfaces.
Film Clip Textbook film, Segment V1.9:
Floating Razor Blade.
Water Strider Insect Mercury Droplets
• Formation of liquid droplets and gas bubbles, e.g., break up of liquid jets into discrete droplets through fuel injectors and atomizing spray nozzles, and diffusing compressed air into aeration tanks.
• Formation of a meniscus and the associated capillary rise (wetting liquids) or fall (non-wetting liquids) in small diameter tubes; involves liquid-gas-solid interfaces
Atomizing Spray Nozzles Fuel Injectors
Ceramic Disk, Fine BubbleAir Diffuser
Figure 1.10, p. 25Effect of capillary action in small tubes. (a) Rise of column for a liquid that wets the tube, e.g., water in a glass tube. (b) Free-body diagram for calculating column height. (c) Depression of column for a non-wetting liquid, e.g., mercury in a glass tube.
• Formation of a capillary fringe above ground
water aquifers, where the tiny interstitial
spaces between soil grains (soil porosity)
create natural “capillary tubes.”
Refer to Handout I.E. Surface Tension –
Examples Problems.
Homework No. 3 Surface tension.