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ME 305 Fluid Mechanics I
Part 1
Fundamental Fluid and Flow Properties
These presentations are prepared by
Dr. Cneyt Sert
Mechanical Engineering Department
Middle East Technical University
Ankara, Turkey
[email protected]
Please ask for permission before using them. You are NOT allowed
to modify them.
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High School Definition of a Fluid
Three most common phases of matter are solid, liquid and
gas.
Liquids and gases are together called fluids.
Intermolecular Attraction Forces
Molecules Volume and
Space
Solid Strong Relative positions are
rather fixed
Definite volume
Definite shape
Liquid Medium Free to change their
relative positions
Definite volume
Indefinite shape
Gas Weak Practically unrestricted Indefinite volume
Indefinite shape
Later well give another definition for fluid, based on its
behavior under shear forces.
Exercise : What is the difference between gases and vapors, from
a technical point of view?
Exercise : What are the fourth and fifth phases of matter? Do a
research on them.
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Branches of Science that Study Fluids
Mechanics deals with both stationary and moving bodies under the
influence of forces.
Statics is the branch of mechanics that deals with bodies at
rest.
Dynamics is the branch of mechanics that deals with bodies in
motion.
Fluid mechanics deals with the behavior of fluids at rest (fluid
statics) and in motion (fluid dynamics).
Hydrodynamics studies liquids (incompressible flow) in
motion.
Hydraulics studies liquids flowing in pipes, ducts and open
channels.
Gas dynamics studies compressible flow of gases with high
density changes.
Aerodynamics is similar to gas dynamics, but also covers low
speed flows. It focuses on air flow.
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Application Areas of Fluid Mechanics
Too many to list all of them. Some examples are
Household appliances : refrigerator, vacuum cleaner, dish
washer, washing machine, water meter, natural gas meter, air
conditioner, radiator, etc.
Turbomachines : pump, turbine, fan, blower, propeller, etc.
Military : Missile, aircraft, ship, underwater vehicle,
dispersion of chemical agents, etc.
Automobile : IC engine, air conditioning, fuel flow, external
aerodynamics, etc.
Medicine : Heart assist device, artificial heart valve,
Lab-on-a-Chip device, glucose monitor, controlled drug delivery,
etc.
Electronics : Convective cooling of generated heat.
Energy : Combuster, burner, boiler, gas, hydro and wind turbine,
etc.
Oil and Gas : Pipeline, pump, valve, offshore rig, oil spill
cleanup, etc.
Almost everything in our world is either in contact with a fluid
or is itself a fluid.
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At the microscopic scale, fluids are composed of molecules.
The Concept of Continuum
Question : Is it possible to keep track of all these molecules
?
Answer : Practically impossible and not necessary for most
engineering problems.
Rather, we study most engineering problems at the macroscopic
scale.
That is we treat fluids as continuum and do not concern with the
behavior of individual molecules.
1 m3 air
~ 2 x 1025 molecules ~ 1025 molecules
A glass of water
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The Concept of Continuum
Microscopic level: Each fluid molecule shown below moves at a
different speed in a different direction.
Macroscopic level: The speed at point A is 60 km/hr. The
direction of air flow at point A is as shown below.
A
60 km/hr is the average speed of molecules in the small volume
surrounding point A. We can say that the fluid particle located at
point A is moving with a speed of 60 km/hr.
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Continuum assumes that fluid and flow properties, such as
density, velocity, pressure, temperature, etc. vary continuously
throughout the fluid.
In continuum, the smallest element of a fluid is NOT a fluid
molecule, but rather a fluid particle, which contains enough number
of molecules to make meaningful statistical averages.
Question: Is continuum a reasonable assumption?
Practical answer: Yes, in many engineering problems.
Detailed answer: Depends on the Knudsen number.
= ( )
Continuum is known to be valid for < 0.01.
In this course we will always treat fluids as continuum.
Continuum (contd)
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Exercise : Is it easier to break the continuum assumption for a
gas or a liquid?
Exercise : Air at standard atmospheric conditions has a mean
free path of 8 x 10 8 m. What will be the limiting characteristing
length that will break the validity of the continuum assumption for
an application using standard atmospheric air ? Search on the web
for micro and nano scale flows to see if there are applications at
such small scales. ( Ans : 8 microns )
Exercise : How much the mean free path of air should be
increased so that we can start questioning the validity of
continuum for flow around a missile with a characteristing length
of 10 m ? Is it possible to have such high mean free path values at
the outermost regions of the atmosphere ? ( Ans : 0.1 m )
Exercise : Do a research on rarefied gas dynamics. Give examples
of applications where it is encountered.
Exercise : Direct Simulation Monte Carlo (DSMC) is a numerical
method that can be used to study high flows. Do a research on
general Monte Carlo methods and DSMC is specific.
Continuum (contd)
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Properties of the Continuum
It is common to use and to fix the thermodynamic state. Then all
other properties can be expressed as a function of these two
= (, ) , = (, ) , = (, ) , etc.
Common thermodynamic properties of a fluid are
Pressure Pa , atm , bar , mmHg
Density kg/m3
Temperature K or oC
Internal energy J/kg
Enthaply J/kg
Entropy J/(kg K)
Specific heat , J/(kg K)
Viscosity Pa s , poise
Thermal conductivity W/(m K)
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Density
( ) [ kg/m3 ]
Mass contained in a unit volume of a fluid. =
Density determines the inertia of a unit volume of fluid and
hence its acceleration when subjected to a given force. Gases are
easier to accelerate than liquids.
Density also determines the amount of gravitational force
(weight) acting on a fluid body. Weight of gases are neglected more
often than that of liquids.
Fluids have a very wide range of density.
Hydrogen Gas
Methane (Natural Gas)
Air Water Mercury
Density [kg/m3]
0.084 0.67 1.2 998 13600
(at standart conditions)
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Density
Add heat and allow piston to go up to keep constant
: increases : decreases
: increases : increases
Push down
Remove heat to keep constant
Density in general is a function of and , i.e. = (, )
If a fluids density is a function of pressure only (not
temperature) it is called a barotropic fluid, a simplification
mostly used in meteorology.
Following processes demonstrate how density of an ideal gas ( =
) changes with temperature and pressure
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Density (contd)
Density variation of water (blue) and air (green) with
temperature at 1 atm
Density variation of water with pressure and temperature
Wat
er d
en
sity
[kg
/m3]
Air
de
nsi
ty [
kg/m
3]
1.4
1.2
1.0
0.8
Temperature [oC]
0 20 40 60 80 100
1000
990
980
970
960
Temperature [oC]
0 5 10 15 20 25 30
Wat
er d
en
sity
[kg
/m3]
1010
1008
1006
1004
1002
1000
998
996
Adapted from engineeringtoolbox.com
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Density (contd)
Changes in and results in changes in , which can be formulated
as
=
+
Divide both sides by
=
1
+1
Exercise : As can be read from the 2nd graph of the previous
page, at 20 oC density of water changes from 998.2 kg/m3 at 1 atm
to 1002.7 kg/m3 at 100 atm, a change of 0.5 %. Calculate the bulk
modulus of elasticity of water at 20 oC. ( Ans: 2.2 x 104 atm or
2.2 GPa )
Reciprocal of bulk modulus of elasticity ( Ev )
Negative of coeff. of thermal expansion ( b )
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Density (contd)
Exercise : At 1 atm density of water changes from 998 kg/m3 at
20 oC to 975 kg/m3 at 75 oC, a change of 2.3 %. Calculate the
coefficient of thermal expansion of water at 1 atm. ( Ans: 4.2 x
10-4 K-1 )
Exercise: Bulk modulus of solids is too high. For iron it is
about 160 GPa. How much pressure is required to decrease the volume
of an iron block by 0.5 % ? ( Ans: 0.8 GPa or 8000 atm )
As seen in previous examples water (and in general all liquids)
compress a little, but in many engineering problems changes in
their density can be neglected.
Incompressible fluid is an idealization used mostly for liquids
that have a constant
density over the range of working conditions.
In this course well consider all liquids to be incompressible
(constant density) unless otherwise mentioned.
In this course well take = 1000 kg/m3 (constant) unless
otherwise mentioned.
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Density (contd)
Exercise : Although density changes in liquid flows are small,
they may result in an interesting phenomena called water hammer. Do
a research on this subject. What is it? When does it happen?
Exercise : Gases are much more compressible than liquids. First
show that for an ideal gas bulk modulus of elasticity is equal to
the pressure of the gas. Compare the bulk modulus of air with that
of water found previously ( Ans: Air is about 20000 times more
compressible than water).
Although gases are much more compressible than liquids they can
also be safely treated as incompressible (constant density) in many
engineering applications.
In this course well take = 1.2 kg/m3 unless otherwise
mentioned.
Mach number () is the nondimensional parameter that can be used
to check the importance of compressibility in gas flows. It is the
ratio of fluid speed to the speed of sound.
Flows with < 0.3 can safely be studied as incompressible.
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Density (contd)
Exercise : In a wind tunnel test air is blown around a car at a
speed of 120 km/hr. Calculate Mach number of the flow and decide if
compressibility affects are negligible or not. ( Ans: Ma = 0.1, air
can be treated as incompressible )
Hydrometer is a device used to measure density of a fluid based
on Archimedes principle. Its working principle will be studied in
Fluid Statics chapter and youll use it in the first experiment of
ME 305.
Specific gravity (Relative density): ( ) [ unitless ]
Ratio of density of a substance to the reference density of
water at 4 oC.
Specific weight: ( ) [ N/m3 ]
Weight per unit volume of a substance.
=
1000 /3
=
2 (at 4 )
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Velocity Field
Fluid velocity: ( ) [ m/s ]
In different coordinate systems velocity vector components
are
Cartesian : = + +
Cylindrical : = + +
In general velocity field of a flowing fluid is too complicated
to be expressed as an equation of space and time.
Visualization of flow over an airfoil
http://www.youtube.com/watch?v=gmSKGMSfOcs&NR=1
Visualization of flow over an F1 race car
http://www.ecourses.ou.edu
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Velocity Field (contd)
There are also simpler flows with easy to express velocity
fields.
Velocity profile at a cross section of fully developed, laminar,
steady, uni-directional pipe flow.
No-slip boundary condition is an important feature of fluid
flows that affects the velocity field. Its an experimental
observation that says A fluid in contact with a solid surface does
not slip it has the same velocity as the surface.
If the solid surface is not moving, than the velocity of fluid
particles adjacent to the surface are zero.
No temperature jump boundary condition is similar to no-slip BC.
It says that Temperature of fluid particles adjacent to a solid
wall is the same as the temperature of the wall.
www.ecourses.ou.edu
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Force Acting on a Fluid Body
Force: ( ) [ N = kg m/s2 ]
Body forces are distributed over the volume of a fluid. They
arise from action at a distance.
They result when a fluid is placed in a gravitational, magnetic,
electrostatic or electromagnetic force field. In this course well
consider only gravitational force.
Gravitational body force per unit mass is the gravitational
acceleration .
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Forces (contd)
Surface forces act on the boundaries of a fluid body by the
surroundings through direct contact. Fluids also apply a surface
force to their surroundings.
A surface force can be decomposed into a normal force acting
perpendicular to the surface and tangential (shear) force acting
parallel to the surface.
P P
: Force acting by the fluid on the wing at point P
: Surface normal at point P
, : Normal and shear components of
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Stress Acting on Fluid Body
( , ) [ Pa = N/m2 ]
Normal stress at point P : = lim0
Shear stress at point P : = lim0
Stress field at a point is a tensor quantity. Complete
definition of it requires nine components.
Cartesian stress tensor :
z
x
y
Direction of surface normal
Direction of force
P
Stresses acting on the top surface of a fluid element
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Pressure
( ) [ Pa = N/m2 ]
is the normal component of the force acting on an area divided
by that area.
It is also a thermodynamic property for a fluid in thermodynamic
equilibrium.
Pressure always acts as a compressive force perpendicular to the
surface.
For a fluid at rest pressure at a point is independent of
direction. This will be proved in Fluid Statics chapter.
Atmospheric pressure : 1 atm = 101.3 kPa = 14.7 psi = 760 mmHg 1
bar
Exercise : On a size 7 basketball ball it says Inflate to 8 psi,
but this is lower than the atmospheric pressure. What should you do
?
Exercise : To visualize how large atmospheric pressure is, watch
interesting experiments at
http://hyperphysics.phy-astr.gsu.edu/hbase/kinetic/patm.html#atm
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Pressure (contd)
Exercise : Water is flowing up inside a pipe against gravity as
shown below. Show all the forces acting on the fluid inside the
dashed lines.
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Viscosity ( ) [Pa s]
Measure of a fluids resistance to shear or angular
deformation.
It is about the fluidity of a fluid. It shows a fluids
resistance to change shape.
Experiment: Consider a solid block firmly attached to two
parallel plates.
The block deforms elastically if a force F is appiled to the
upper plate.
Shear stress Angular deformation
If the force is increased the block will break apart at some
point.
A
B
Fixed plate A
B F
Movie : Viscous fluids
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Viscosity (contd)
Perform a similar experiment with a layer of fluid between two
parallel plates.
Vertical fluid element AB will deform continuously as long as
the shear force is applied by moving the top plate.
F
A
B t0
First observation: After an initial transition, velocity of the
top plate will be constant (0) and the velocity profile within the
fluid will be linear.
t1
B t2
B
Movie : Shear deformation
0 = 0
= 0
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Viscosity (contd)
Angular deformation rate of line AB is
Exercise : Show that this deformation rate is equal to
, which is also equal to
Second obervation: Shear stress acting on a surface parallel to
the flow (such as the surface of the top plate) will be
proportional to deformation rate
Shear stress Rate of angular deformation or
A
B 0 0 +
B
=
= 0
=
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Viscosity (contd)
y
x coefficient of viscosity
absolute viscosity
dynamic viscosity
viscosity
or
is observed for fluids known as Newtonian fluids.
For Newtonian fluids, shear stress on a surface tangent to the
flow direction is proportional to the rate of shear strain or to
the velocity gradient on the surface (change of velocity in a
direction normal to the surface).
This is known as Newtons Law of Viscosity.
Proportionality constant of the above relations is known as
viscosity.
=
=
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Viscosity (contd)
Exercise : What is the sign convention for stress ? Are the
stresses shown in the previous figure negative or positive ? How
did we decide on the arrows of the stresses ?
Exercise : For the problem shown on the right, is the shear
stress negative or positive on the lower and upper fluids ? Show
the direction of shear forces acting by the fluids on the plates
?
Exercise : Consider two concentric cylinders with fluid in
between. Inner cylinder is fixed and outer one is rotating. Write
the proper form of Newtons law of viscosity for the stress acting
on the inner cylinder. You need to think about the correct indices
and velocity components to be used in the equation.
y x
r
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Viscosity (contd)
Exercise : A concentric cylinder viscometer may be formed by
rotating the inner cylinder of a pair or closely fitting cylinders.
A torque of 15 Nm is required to turn the inner cylinder at 10
rad/s while keeping the outer cylinder fixed. Determine the
viscosity of the Newtonian fluid in the clerance gap of the
viscometer. Assume that the fluid pressure is constant and velocity
distribution is linear.
0.2 m 0.5 mm
0.1 m
0.5 mm
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Viscosity (contd)
Definition of fluid : A fluid deforms continuously under the
application of a shear (tangential) force, no matter how small the
force is.
It is more difficult to deform highly viscous fluids.
Viscosity can be measured using
capillary tube viscometer
falling sphere viscometer
concentric cylinder viscometer
Saybolt viscometer that youll be using in the first experiment
of ME 305
Kinematic viscosity: ( ) [m2/s] =
Air Water SAE 30 oil Glycerin Thick Molasses
50 15,000 75,000 375,000
Movie : Capillary tube viscometer
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Viscosity (contd)
Exercise : Laminar, fully developed, pressure driven flow inside
a constant diameter pipe has the shown paraboloic velocity profile.
For the flow of a Newtonian fluid with viscosity and centerline
velocity of , calculate the force exterted by the fluid on the pipe
wall over a pipe section of length L .
Comment on the dependency of the calculated force on the pipe
radius.
= (1 2/2) 2
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Viscous Behavior of Fluids Fluids
Inviscid (ideal)
Viscous
Newtonian Non-Newtonian
Time independent Time dependent
Pseudoplatic (shear thinning)
Bingham plastic
Dialatant (shear thickening)
Thixotropic Rheopetic
Pseudoplastic
Dialatant
Bingham plastic
Inviscid (ideal)
Elastic solid
Newtonian
1
Movie : Non-Newtonian fluid pool
http://www.youtube.com/watch?v=f2XQ97XHjVw
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Newtonian behavior simple (it is linear). Common fluids such as
water, air, oil
behave as Newtonian.
Inviscid (ideal) fluids do not exist in real world. They have m
= 0. This might be a
useful simplification for analytical analyses.
Dialatant fluids become thicker under increased shear stress.
(printing ink).
Bingham plastics do not flow below a certain amount of shear
stress. (toothpaste).
Pseudoplastics become thinner under increased shear stress.
(wall paint, blood).
For thixotropic fluids viscosity decreases with time
(lipstick).
For rheopetic fluids viscosity increases with time (betonite
solution).
Viscous Behavior of Fluids (contd)
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Affect of pressure on viscoity is small and often neglected.
Viscosity of liquids decrease with increasing temperature.
Viscosity of gases incresae with increasing temperature.
General formula for liquids: Andrades equation =
General formula for gases: Sutherlands equation =1.5
+
Exercise : As mentioned above, viscosities of liquids and gases
change with temperature in different ways due to two different
mechanisms that affect their viscosities. What are these mechanisms
?
Viscosity (contd)
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Surface Tension
( ) [N/m]
Surface tension is due to the asymmetric cohesive forces acting
on the molecules of a free surface (interface between a liquid and
a gas).
This asymmetry will result in a hypothetical skin (membrane) all
around the surface
Surface tension exists whenever there is a density discontinuity
between a liquid and another liquid, gas or solid.
Exercise : Read more about surface tension at
http://www.funsci.com/fun3_en/exper2/exper2.htm
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Surface Tension (contd)
Exercise : Surface tension is known to create a pressure
difference across a curved interface of two fluids. Consider a
spherical liquid droplet in a gas. Compute the pressure difference
in terms of the uniform surface tension () on droplets surface and
the diameter of the droplet ().
Gas Liquid
Capillary rise (or drop) observed when an open-end tube is
immersed into a liquid is also related to surface tension. Itll be
studied in Fluid Statics chapter.
Glass tube
water mercury
Exercise : Search for the following two surface tension related
research papers on Google Scholar and read them.
Surface-Tension-Dominant Powerless Nano/Micro Fluidic Systems
Characterization of the Surface Tension and Viscosity Effects on
the Formation
of Nano-Liter Droplet Arrays by an Instant Protein Micro
Stamper
