chapter 1. introduction · concerned with the behavior of liquids and gases at rest (fluid statics)...
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Fluid MechanicsDM23834
Chapter 1. Introduction
Eunseop Yeom
School of Mechanical Engineering, Pusan National University
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1 Introduction Fluid Mechanics
• Fluid mechanics is that discipline within the broad field of applied mechanics
concerned with the behavior of liquids and gases at rest (fluid statics) or in
motion (fluid dynamics).
• This field of mechanics obviously encompasses a vast array of problems that
may vary from the study of blood flow in the capillaries to the flow of crude
oil across Alaska.
3
1 Introduction
Gas Liquids Statics Dynamics
Air, He, Ar,
N2, etc.
Water, Oils,
Alcohols,
etc.
0 iF
Viscous/Inviscid
Steady/Unsteady
Compressible/
Incompressible
0 iF
Laminar/
Turbulent
, Flows
Compressibility ViscosityVapor
Pressure
Density
PressureBuoyancy
Stability
Chapter 1: Introduction
Surface
Tension
FluidMechanics
Fluid Dynamics:
Rest of Course
Chapter 2: Fluid Statics
Fluid Mechanics Overview
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1 Introduction Types of Flows/Fluids
▪ Static vs. Dynamic
▪ Compressible vs. Incompressible
▪ Viscous vs. Inviscid
▪ Laminar vs. Turbulent
▪ Homogeneous vs. Heterogeneous
▪ Non-reacting vs. Reacting
▪ Newtonian vs. Non-Newtonian
Laminar Turbulent
Newtonian
Non-Newtonian
Homogeneous Heterogeneous
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1.1 Some Characteristics of Fluids• What is a fluid?
A fluid is defined as a substance that deforms continuously when acted on by
a shearing stress of any magnitude.
▪ The molecular structure of asolid has densely spaced
molecules with large
intermolecular cohesive forces.
▪ In case of a liquid, themolecules are spaced farther
apart, the intermolecular
forces are smaller than for
solids, and the molecules have
more freedom of movement.
Solid ice Liquid water Water vapor
A shear stress is defined as the component of stress parallel to the cross section.
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1.2 Dimensions, Dimensional Homogeneity, and Units
Qualitative Aspect
• Qualitative aspect serves to identify the nature, or type, of the characteristics
(such as length, time, stress, and velocity).
• Qualitative description is given in terms of certain primary quantities, such as
Length, L, time, T, mass, M, and temperature, θ. The primary quantities are
also referred to as basic dimensions(기본차원).
• These primary quantities can then used to provide a qualitative description of
any other secondary quantity:
For example, area≒L2,velocity ≒LT-1, density ≒ML-3, Force ≒ MLT-2.
Mass[M], Length[L], time[T], and Temperature[θ] MLT system
Force[F], Length[L], time[T], and Temperature[θ] FLT system
Force[F], Mass[M], Length[L], time[T], and Temperature[θ] FMLT system
System of Dimensions
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1.2 Dimensions, Dimensional Homogeneity, and Units
Table 1.1 Dimensions Associated with Common Physical Quantities
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1.2 Dimensions, Dimensional Homogeneity, and Units
Dimensionally Homogeneous (동차)
• All theoretically derived equations are dimensionally homogeneous
that is, the dimensions of the left side of the equation must be the same as
those on the right side, and all additive separate terms have the same
dimensions.
General homogeneous equation: valid in any system of dimensions
Restricted homogeneous equation : restricted to a particular system of dimension
V = V0 + at LT-1 ≒ LT-1 + LT-2 × T
2
2gtd d = 4.90 × t2
g = 9.81 m/sec2
Valid only for the system
of SI unit
d = 16.1 × t2
g = 32.2 ft/sec2
Valid only for the system
of English unit
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1.2 Dimensions, Dimensional Homogeneity, and Units
Systems of Units
• In addition to the qualitative description of the various quantities of interest, it
is generally necessary to have a quantitative measure of any given quantity.
• For example, if we measure the width of this page in the book and say that it is
10 units wide, the statement has no meaning until the unit of length is defined.
In 1960 the 11th General Conference on Weights and Measures,
the international organization responsible for maintaining precise uniform standards of
measurement, formally adopted the International System of Units as the international
standard.
This system, commonly termed SI, has been widely adopted worldwide.
English unit is widely used in the United States.
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1.3 Analysis of Fluid BehaviorsAnalysis of any problem in fluid mechanics necessarily includes statement of the basic
laws governing the fluid motion. The basic laws, which applicable to any fluid, are:.
▪ Conservation of mass
▪ Newton’s second law of motion
▪ The principle of angular momentum
▪ The first law of thermodynamics
▪ The second law of thermodynamics
• NOT all basic laws are required to solve any one problem. On the other hand,
in many problems it is necessary to bring into the analysis additional relations
that describe the behavior of physical properties of fluids under given
conditions.
• Many apparently simple problems in fluid mechanics that cannot be solved
analytically. In such cases we must resort to more complicated numerical
solutions and/or results of experimental tests.
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1.4 Measures of Fluid Mass and Weight Density
The density of a fluid is defined as mass per unit volume.
• Different fluids can vary greatly in density
• Liquids densities do not vary much with pressure and temperature
• Gas densities can vary quite a bit with pressure and temperature
• Density of water at 15 °C : 1000 kg/m3
• Density of air at 4 °C : 1.20 kg/m3
Specific Volume(비체적)
• The specific volume, v, is the volume per unit mass
• It is the reciprocal of the density, used in thermodynamics
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1.4 Measures of Fluid Mass and Weight Specific Weight (비중량)
The specific weight of fluid is its weight per unit volume.
• Specific weight characterizes the weight of the fluid system
• Specific weight of water at 4° C : 9.80 kN/m3
• Specific weight of air at 4° C : 11.9 N/m3
g = local acceleration of gravity, 9.807 m/s2
Specific Gravity (비중)
• Gases have low specific gravities
• A liquid such as Mercury has a high specific gravity, 13.5
• The ratio is unitless.
• Density of water at 4°C : 1000 kg/m3
ρg
The specific gravity of fluid is the ratio of the density of the fluid to
the density of water at 4°C.
C4O@H2
ρ
ρSG
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1.5 Ideal Gas Law Ideal Gas Law
• p = absolute pressure
• T = temperature (K)
• R = Ru/M, where Ru = universal gas constant (8.31J/K∙mol), M = average molecular weight
1. Atmospheric air
2. Cotton candy
3. Exhaust gas from jet engine
4. Fuel spray in internal combustion engine cylinder
RT
pρ
▶Which of the following can be approximated as ideal gas?
Gases are highly compressible in comparison to fluids, with changes in gas density
directly related to changes in pressure and temperature through the equation.
• Ideal gas model tends to fail at lower temperatures or higher pressures, when
intermolecular forces and molecular size become important. It also fails for most heavy
gases, such as many refrigerants, and for gases with strong intermolecular forces, notably
water vapor.
1. Atmospheric air
TnRVp u RTM
TR
m
TnRpv uu
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1.6 Viscosity
Viscosity is a measure of the resistance of a fluid which is being deformed by
either shear stress or extensional stress.
Shearing Flow
Sharing stress:
A
P
• The bottom plate is rigid fixed, but the upper plate is free to move.
• If a solid, such as steel, were placed between the two plates and loaded with
the force P, the top plate would be displaced through some small
distance, δa.
• The vertical line AB would be rotated through the small angle, δβ, to the
new position AB’.
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1.6 Viscosity
• The molecules in two neighbouring fluid layers move in the same direction.
Between the fluid layers there is an imaginary separation plane. It is assumed that
all molecules of the same layer move with the same velocity. The molecule velocities in
two layers are different. Since the separation plane is permeable, molecule exchange
between the fluid layers occur through diffusion.
• Molecules which move from the faster layer to the slower layer transfer a velocity
surplus and thus an impulse surplus (impulse = mass × velocity difference). During the
transition from the slower to the faster layer they transfer an impulse deficit.
• An impulse surplus produces an accelerating force and an impulse deficit a retarding
force in the fluid layer (force = impulse per time). The deceleration and acceleration
force are the cause of the shear stress arising in the separation plane between the fluid
layers (tension = stress per area).
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1.6 Viscosity
• When the force P is applied to the upper
plate, it will move continuously with a
velocity U.
• The fluid “sticks” to the solid boundaries and
is referred to as the NON-SLIP conditions.
• The fluid between the two plates moves with
velocity u = u(y) that would be assumed to
vary linearly, u = Uy/b.
• In such case, the velocity gradient is du /
dy=U / b.
Real Fluids
In more complex flow situations, velocity
gradient is not du / dy=U / b.
No-slip boundary condition:
Fluid velocity remains the same as that of the
wall on the boundary
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1.6 Viscosity
b
tan
b
tUδ
δt
δβ
0δt lim
dy
du
b
U
Through continuation of experiment, shear stress is in direct proportion to velocity gradient.
Rate of shearing strain
dy
du
dy
du
The constant of proportionality; μ
Greek symbol (mu) is called
absolute viscosity, dynamic viscosity,
or simply the viscosity.
τ = [N/m2] = [(m/s)/m] = [s-1]dy
duμ = [N∙s/m2]
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1.6 Viscosity
• The viscosity depends on the particular fluid,
and for a particular fluid the viscosity is also
dependent on temperature.
• The gradient of the straight line corresponds
to the dynamic viscosity of the fluid.
• 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.
• Most common fluids such as water, air, and
gasoline are Newtonian fluid under normal
conditions.
Non-Newtonian Fluids
• Fluids for which the shearing stress is not
linearly related to the rate of shearing strain are
designated as non-Newtonian fluids.
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1.6 Viscosity
Non-Newtonian Fluids
•
The viscosity decreases with increasing
shear rate - the harder the fluid is sheared, the
less viscous it becomes. Many colloidal
suspensions and polymer solutions are shear
thinning. Latex paint is example.
•
The viscosity increases with increasing shear
rate - the harder the fluid is sheared, the more
viscous it becomes. Water-corn starch mixture
water-sand mixture are examples.
•
Neither a fluid nor a solid. Such material can
withstand a finite shear stress without motion,
but once the yield stress is exceeded it flows
like a fluid. Toothpaste and mayonnaise are
common examples.
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1.6 Viscosity Temperature Dependence
• For liquid, the viscosity decreases with an increase in temperature.
• For gases, an increase in temperature causes an increase in viscosity.
Why? molecular structure.
The liquid molecules are closely spaced, with strong cohesive
forces between molecules, and the resistance to relative motion between adjacent layers is related to these intermolecular force.
As the temperature increases, these cohesive force are
reduced with a corresponding reduction in resistance to motion.
Since viscosity is an index of this resistance, it follows that viscosity
is reduced by an increase in temperature.
Andrade’s equation μ = DeB/T (D and B are constant)
In gases, the molecules are widely spaced and intermolecular
force negligible.
The resistance to relative motion mainly arises due to the exchange
of momentum of gas molecules between adjacent layers.
As the temperature increases, the random molecular activity
increases with a corresponding increase in viscosity.
Sutherland equation μ = CT3/2/(T+S) (C and S are constant)
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1.6 Viscosity Kinematic Viscosity (동점성계수)
• Defining kinematic viscosity ν = μ/ρ.
• It is usually denoted by the Greek letter nu (ν).
• Since units for μ and ρ are N∙s/m2 and kg/m3, unit of kinematic viscosity is m2/s.
For example, water at 20 °C has a viscosity of 1.002 mPa·s (10-3 Pa·s).
The cgs(Centimetre–gram–second system of units) physical unit for dynamic viscosity is the poise (P),
named after Jean Poiseuille.
1 P = 1 g∙cm-1∙s-1 = 0.1 kg∙m-1∙s-1
It is more commonly expressed, particularly in ASTM standards, as centipoise (cP).
A centipoise (cP) is one one-hundredth of a poise, and one millipascal-second (mPa·s) in SI units.
(1 cP = 10−2 P = 10−3 Pa·s = 1 mPa·s)
N = [kg∙m/s2]
Physical Unit of Viscosity
μ = (N·s)/m2 = (Pa·s)= (N/m2)·s
= ((kg∙m/s2)·s)/m2= kg/(m·s)
∴ 1 Pa·s = 10 P
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1.7 Compressibility of Fluids Bulk Modulus (체적계수)
• Liquids are usually considered to be incompressible,
whereas gases are generally considered compressible.
• A property, bulk modulus Ev, is used to characterize compressibility of fluid.
• Measure of how pressure compresses the volume/density.
• Units of the bulk modulus are N/m2 (Pa).
• Large values of the bulk modulus indicate incompressibility.
• Incompressibility indicates large pressures are needed to compress the volume slightly
• It takes 3120 psi to compress water 1% at atmospheric pressure and 15.5°C.
• Most liquids are incompressible for most practical engineering problems.
dρ
dp
V/Vd
dpEυ
p is pressure, V = volume and ρ is the density.
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1.7 Compressibility of Fluids Compression of Gases
RTp Ideal Gas Law:
• p is pressure, ρ is the density, R is the gas constant, and T is Temperature
constantρ
p
Isothermal Process (constant temperature):
pE
Isentropic Process (frictionless, no heat exchange):
constantkρ
p kpE
k is the ratio of specific heats, Cp (constant pressure) to Cv (constant volume),
R = Cp – Cv.
• If we consider air under at the same conditions as water, we can show that air is
15,000 times more compressible than water.
• However, many engineering applications allow air to be considered incompressible.
Cdρ
dpρp C
dρ
dpEυ C
kρp C1C kρk
dρ
dp 1C k
υ kE
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1.7 Compressibility of Fluids• When a gas undergoes a reversible process, the
process frequently takes place in such a manner that
a plot of log P versus log V is a straight.
• A process PVn is a constant called a polytropic
process.
nVd
Pd
ln
lnd ln P + nd ln V = 0
Polytropic index Relation Effects
n = 0 PV0=P Isobaric process (constant pressure)
n = 1 PV=mRT Isothermal process (constant temperature)
n = ∞ PV∞ Equivalent to an isochoric process (constant volume)
n = k (CP/CV ) PVk Isentropic process (adiabatic and reversible).
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1.7 Compressibility of Fluids Speed of Sound
• A consequence of the compressibility of fluids is that small disturbances introduced at
a point propagate at a finite velocity. Pressure disturbances in the fluid propagate as
sound, and their velocity is known as the speed of sound or the acoustic velocity, c.
ordρ
dpc
ρ
Ec v
Isentropic Process (frictionless, no heat exchange because):
kpc
Ideal Gas and Isentropic Process:
kRTc
1C kρkdρ
dpkρp Cρ
ρk
k
C
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• Evaporation occurs in a fluid when liquid molecules at the surface have sufficient
momentum to overcome the intermolecular cohesive forces and escape to the
atmosphere.
• Vapor Pressure (증기압) is that pressure exerted on the fluid by the vapor in a
closed saturated system where the number of molecules entering the liquid are the
same as those escaping. Vapor pressure depends on temperature and type of fluid.
• Boiling occurs when the absolute pressure in the fluid reaches the vapor pressure.
Boiling occurs at approximately 100 °C, but it is not only a function of temperature,but also of pressure. For example, in Colorado Spring (altitude of 1,840m), water boils
at temperatures less than 100 °C.
1.8 Vapor Pressure Evaporation and Boiling
Cavitation is a form of
Boiling due to low pressure
locally in a flow.
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1.9 Surface Tension Surface Tension
• At the interface between a liquid and a gas or two immiscible liquids, forces develop
forming an analogous “skin” or “membrane” stretched over the fluid mass which
can support weight.
• This “skin” is due to an imbalance of cohesive forces. The interior of the fluid is inbalance as molecules of the like fluid are attracting each other while on the interface
there is a net inward pulling force.
• Surface tension is the intensity of the molecular attraction per unit length along any
line in the surface.
• Surface tension is a property of the liquid type, the temperature, and the other fluid atthe interface.
• This membrane can be “broken” with a surfactant which reduces the surface tension.
Water strider
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1.9 Surface TensionReal Fluid Drops Mathematical Model
• R is the radius of the droplet, σ is the surface tension, ∆p is the pressure difference
between the inside and outside pressure.
• The force developed around the edge due to surface tension along the line:
• This force is balanced by the pressure difference ∆p:
RFsurface 2
2RpFpressure
Now, equating the Surface Tension Force to the Pressure Force, ∆p = pi – pe:
Rp
2
This indicates that the internal pressure in the droplet is greater
that the external pressure since the right hand side is entirely
positive.
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1.9 Surface Tension Capillary Action
• Capillary action in small tubes which involve a liquid-gas-solid interface is caused
by surface tension. The fluid is either drawn up the tube or pushed down.
“Wetted” “Non-Wetted”
Adhesion
CohesionAdhesion
Cohesion
• h is the height, R is the radius of the tube, θ is the angle of contact.
• The weight of the fluid is balanced with the vertical force caused by surface tension.
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1.9 Surface Tension
Equating the two and solving for h:
• For clean glass in contact with water, θ 0°, and thus as R decreases, h increases,giving a higher rise.
• For a clean glass in contact with Mercury, θ 130°, and thus h is negative or there isa push down of the fluid.