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Fluid Mechanics
Intoduction 1.1
Flow in pipes 1.2
Stresses
In fluid mechanics it is convenient to define a force per unit area
(=F/A), called a stress (same units as pressure).
• Normal stress acts perpendicular to the surface (F=normal force).
Tensile causes elongation Compressive causes shrinkage
(Pressure is the most important
example of a compressive stress)
F F F F A A
A
Fnormalnormal
Flow in pipes 1.3
Stresses
• Shear stress acts tangentially to the surface (F=tangential or shear
force). F
F A
A
Fshearshear
A fluid is defined as a substance that deforms continuously
when acted on by a shearing stress of any magnitude.
Intoduction 1.4
Introduction to Fluid Mechanics
• Fluid Mechanics is concerned with the behavior of fluids at rest and in
motion
• Distinction between solids and fluids:
– According to our experience: A solid is “hard” and not easily
deformed. A fluid is “soft” and deforms easily.
– Fluid is a substance that alters its shape in response to any force
however small, that tends to flow or to conform to the outline of its
container, and that includes gases and liquids and mixtures of solids
and liquids capable of flow.
– A fluid is defined as a substance that deforms continuously when
acted on by a shearing stress of any magnitude.
Intoduction 1.5
Course Organization
Textbook: deNevers “Fluid Mechanics for Chemical Engineers”
Introduction (Chapter 1) / Dimensions, Units Fluid statics: Fluid is at rest Fluid mechanics Fluid dynamics: Fluid is moving
Fluid statics (Chapter 2): Pressure, measurement of pressure, hydrostatic
forces, buoyancy
Fluid dynamics (Chapters 3-5, 7): Mass, energy and momentum balances
Applications in Engineering (Chapters 6, 9, 11, 12): Flow in pipes,
turbomachines, flow over immersed bodies, flow through porous media
Dimensional analysis and modeling (Chapter 13)
Motivation for Studying Fluid Mechanics
• Fluid Mechanics is omnipresent – Aerodynamics
– Bioengineering and biological systems
– Combustion
– Energy generation
– Geology
– Hydraulics and Hydrology
– Hydrodynamics
– Meteorology
– Ocean and Coastal Engineering
– Water Resources
– …numerous other examples…
• Fluid Mechanics is beautiful
Aerodynamics
Bioengineering
Energy generation
Geology
River Hydraulics
Hydraulic Structures
Hydrodynamics
Meteorology
Water Resources
Fluid Mechanics is Beautiful
Intoduction 1.17
TOPIC 1
Introduction (de Nevers 1.1-1.5.2, 1.8-1.10)
Intoduction 1.18
Dimensions and Units
In fluid mechanics we must describe various fluid characteristics in
terms of certain basic quantities such as length, time and mass
• A dimension is the measure by which a physical variable is expressed
qualitatively, i.e. length is a dimension associated with distance, width,
height, displacement.
Basic dimensions: Length, L
(or primary quantities) Time, T
Mass, M
Temperature, Q
We can derive any secondary quantity from the primary quantities
i.e. Force = (mass) x (acceleration) : F = M L T-2
• A unit is a particular way of attaching a number to the qualitative
dimension: Systems of units can vary from country to country, but
dimensions do not
Intoduction 1.19
Dimensions and Units
Primary
Dimension SI Unit
British
Gravitational
(BG) Unit
English
Engineering
(EE) Unit
Mass [M] Kilogram (kg) Slug Pound-mass
(lbm)
Length [L] Meter (m) Foot (ft) Foot (ft)
Time [T] Second (s) Second (s) Second (s)
Temperature [Q] Kelvin (K) Rankine (°R) Rankine (°R)
Force [F] Newton
(1N=1 kg.m/s2) Pound (lb) Pound-force (lbf)
Conversion factors are available in the textbook inside of front cover.
Intoduction 1.20
Units of Force: Newton’s Law F=m.g
• SI system: Base dimensions are Length, Time, Mass, Temperature
A Newton is the force which when applied to a mass of 1 kg
produces an acceleration of 1 m/s2.
Newton is a derived unit: 1N = (1Kg).(1m/s2)
• BG system: Base dimensions are Length, Force, Time, Temperature
A slug is the mass which produces an acceleration of 1 ft/s2 when
a force of 1lb is applied on it:
Slug is a derived unit: 1slug=(1lb) (s2)/(ft)
• EE system: Base dimensions are Length, Time, Mass, Force and
Temperature
The pound-force (lbf) is defined as the force which accelerates
1pound-mass (lbm), 32.174 ft/s2.
Intoduction 1.21
Units of Force – EE system
To make Newton’s law dimensionally consistent we must include a
dimensional proportionality constant:
cg
gmF
where
2
f
mc
)s)(lb(
)ft)(lb(1740.32g
Intoduction 1.22
Example: Newton’s Law
• An astronaut weighs 730N in Houston, TX, where the local
acceleration of gravity is g=9.792 m/s2. What is the mass of the
astronaut? What is his weight on the moon, where g=1.67 m/s2?
• Redo the same problem in EE units. In EE units the astronaut weighs
164.1lbf, gHouston=32.13 ft/s2 and gmoon=5.48 ft/s2.
Intoduction 1.23
Dimensional Homogeneity
• All theoretically derived equations are dimensionally homogeneous:
dimensions of the left side of the equation must be the same as those
on the right side.
– Some empirical formulas used in engineering practice are not
dimensionally homogeneous
• All equations must use consistent units: each term must have the
same units. Answers will be incorrect if the units in the equation are
not consistent. Always chose the system of units prior to solving the
problem
Intoduction 1.24
Properties of Fluids
Fundamental approach: Study the behavior of individual molecules
when trying to describe the behavior of fluids
Engineering approach: Characterization of the behavior by considering
the average, or macroscopic, value of the quantity of interest, where the
average is evaluated over a small volume containing a large number of
molecules
Treat the fluid as a CONTINUUM: Assume that all the fluid
characteristics vary continuously throughout the fluid
Intoduction 1.25
Measures of Fluid Mass and Weight
• Density of a fluid, r (rho), is the amount of mass per unit volume of a
substance: r = m / V
– For liquids, weak function of temperature and pressure
– For gases: strong function of T and P
from ideal gas law: r = P M/R T
where R = universal gas constant, M=mol. weight
R= 8.314 J/(g-mole K)=0.08314 (liter bar)/(g-mole K)=
0.08206 (liter atm)/(g-mole K)=1.987 (cal)/(g-mole K)=
10.73 (psia ft3)/(lb-mole °R)=0.7302 (atm ft3)/(lb-mole °R)
)T,P(rr
(1.1)
Intoduction 1.26
Measures of Fluid Mass and Weight
• Specific volume: u = 1 / r
• Specific weight is the amount of weight per unit volume of a substance:
g = w / V = r g
• Specific Gravity (independent of system of units)
C4@OH2
SGr
r
Flow in pipes 6.27
Forces acting on a fluid
The forces acting on a fluid are divided into two groups:
• Body forces act without physical contact. They act on every mass
element of the body and are proportional to its total mass. Examples
are gravity and electromagnetic forces
• Surface forces require physical contact (i.e. surface contact) with
surroundings for transmission. Pressure and stresses are surface
forces.
28
Definition of Pressure
Pressure is defined as the amount of force exerted on a unit
area of a substance:
Pam
N
area
forceP
2
29
Direction of fluid pressure on boundaries
Furnace duct Pipe or tube
Heat exchanger
Dam
Pressure is a Normal Force
(acts perpendicular to surfaces)
It is also called a Surface Force
30
Absolute and Gauge Pressure
• Absolute pressure: The pressure of a fluid is
expressed relative to that of vacuum (=0)
• Gauge pressure: Pressure expressed as the
difference between the pressure of the fluid and
that of the surrounding atmosphere.
Usual pressure guages record guage pressure.
To calculate absolute pressure:
gaugeatmabs PPP
31
Absolute & Gauge Pressure: Schematic
+
- +
+
32
Units for Pressure
Unit Definition or
Relationship
1 pascal (Pa) 1 kg m-1 s-2
1 bar 1 x 105 Pa
1 atmosphere
(atm)
101,325 Pa
1 torr 1 / 760 atm
760 mm Hg 1 atm
14.696 pounds
per sq. in. (psi)
1 atm
Flow in pipes 6.33
Shear Flow
NO-SLIP CONDITION: The fluid “sticks” to the solid boundaries.
The velocity of the fluid touching each plate is the same as that of the
plate (Vo for the top plate, 0 for the bottom plate).
The velocity profile is a straight line: The velocity varies uniformly from 0
to Vo
Vo
= F/A
A
yo
Fluid
x
y
F
= F/A
o
o
o
o
y
V
dy
dV and y
y
V)y(V
Flow in pipes 6.34
Shear Flow
The force, F is proportional to the velocity Vo, the area in contact with
the fluid, A and inversely proportional to the gap, yo:
y
V AF
o
o
Recall, shear stress, = F / A
y
V
o
o
In the limit of small deformations the ratio Vo/yo can be replaced by the
velocity gradient dV/dy:
dy
dVg
Rate of shearing strain
or shear rate:
g
or
dy
dV
Flow in pipes 6.35
Newton’s law of Viscosity
Newton’s law of viscosity
dy
dV
[=N/m2 . s=Pa . s]: Viscosity
n = /r : Kinematic viscosity [=m2/s]
Newtonian fluids: Fluids which obey Newton’s law: Shearing stress is
linearly related to the rate of shearing strain.
(6.6)
The viscosity of a fluid measures its resistance to flow under an
applied shear stress.
Flow in pipes 6.36
True or False?
• When a fluid is subjected to a steady shear stress, it will reach a state of equilibrium in which no further motion occurs
• Pressure and shear stress have identical units
• When a flowing fluid is in contact with a solid surface, the velocity of the portion of the fluid in direct contact with the surface is always equal to zero.
• Newton’s law of viscosity relates shear stress to rate of shearing strain
Flow in pipes 6.37
Example: Shear stress
The space between two plates, as shown in the figure, is filled with
water. Find the shear stress and the force necessary to move the
upper plate at a constant velocity of 10 m/s. The gap width is
yo=0.1 mm and the area A is 0.2 m2. The viscosity of water is 0.001
Pa.s.
Vo F
A
yo Water
= F/A
Flow in pipes 6.38
Non-Newtonian fluids
Non-Newtonian fluids: Fluids which do not obey Newton’s law: Shearing
stress is not linearly related to the rate of shearing strain.
•Bingham plastics
•Shear thinning
•Shear thickening
The study of these materials is the subject of rheology
Flow in pipes 6.39
Typical Viscosity Values (Pa-s)
Basim Abu-Jdayil - Chemical & Petroleum Eng. Dept.
• Asphalt Binder……..
• Polymer Melt ………
• Molasses …………..
• Liquid Honey ………
• Glycerol ……………
• Olive Oil ……………
• Water ………………
• Air ………………….
100,000
1,000
100
10
1
0.01
0.001
0.00001
The range of viscosity values one encounters
suggests the use of logarithmic scales when
plotting data.
40
General Flow Classifications
Basim Abu-Jdayil - Chemical & Petroleum Eng. Dept.
Shear Rate (1/sec)
She
ar S
tres
s (P
a)
Dilatant
Newtonian
Pseudoplastic
An example of dilatancy is
wet sand on the beach.
Most materials used
industrially are
pseudoplastic.
41
General Flow Classifications
Basim Abu-Jdayil - Chemical & Petroleum Eng. Dept.
Shear Rate (1/sec)
She
ar S
tres
s (P
a)
Dilatant
Newtonian
Pseudoplastic
An example of dilatancy is
wet sand on the beach.
Most materials used
industrially are
pseudoplastic.
42
Bingham Fluid
Basim Abu-Jdayil - Chemical & Petroleum Eng. Dept.
τ, P
a
g /s
Ideal Yield Stress (Bingham Yield)
43
Basim Abu-Jdayil - Chemical & Petroleum Eng. Dept.
Some Interesting Phenomena
• shear thinning
• shear thickening
• yield stress
• viscoelastic effects
– Weissenberg effect
– Fluid memory
– Die Swell
– Tubeless Syphon
44
Basim Abu-Jdayil - Chemical & Petroleum Eng. Dept.
World’s Longest Running Laboratory Experiment – The Pitch Drop Experiment
• Pitch – derivative of tar (a dark sticky substance obtained from tar and
used in the building trades, especially for waterproofing roofs)
– @room temperature feels solid and can be shattered with a blow
of a hammer
– This experiment shows that in fact at room temperature pitch is a
fluid!
45
Basim Abu-Jdayil - Chemical & Petroleum Eng. Dept.
World’s Longest Running Laboratory
Experiment – The Pitch Drop Experiment
1927 – Prof Parnell in Univ. of Queensland
Australia heated a sample of pitch and poured it
into a glass funnel with a sealed stem. Three years
where allowed for it to settle, after which the stem
was cut.
Examine the viscosity of the pitch by the speed at
which it flows from a funnel into a jar.
Only eigth drops has fallen in 80 years.
The viscosity is approximated as 100 billion times
that of water.
46
Weissenberg Effect (Rod Climbing
Effect)
Basim Abu-Jdayil - Chemical & Petroleum Eng. Dept.
The fluid does not flow outward
when stirred at high speeds
Ms. Garcia-Rodrigues is s t u d y i n g R h e o l o g y a t U . o f Wisconsin-Madison, USA. The fluid shown is a 2% aqueous polyacrylamide s o l u t i o n , a n d t h e r o t a t i o n a l s p e e d i s n o m i n a l l y 0 . 5 H z .
47
Fluid Memory
Basim Abu-Jdayil - Chemical & Petroleum Eng. Dept.
Conserve their shape over
time periods or seconds or
minutes
Elastic like rubber
Can bounce or partially
retract
Example: clay
49
Die Swell
Basim Abu-Jdayil - Chemical & Petroleum Eng. Dept.
– as a polymer exits a die, the diameter of liquid stream
increases by up to an order of magnitude
– caused by relaxation of extended polymer coils, as stress is
reduced from high flow producing stresses present within the
die to low stresses, associated with the extruded stream
moving through ambient air
50
Tubeless Siphon
Basim Abu-Jdayil - Chemical & Petroleum Eng. Dept.
When the siphon tube is lifted out of
the fluid, the Newtonian liquid (N)
stops flowing; the macromolecular
fluid (P) continues to be siphoned.
51
Application of Rheology
Basim Abu-Jdayil - Chemical & Petroleum Eng. Dept.
• Polymer melts, solutions including composites • Rubber • Lubricants • Paints • Printing inks • Paper and pulp • Food • Biological fluids • Concrete and Clay • Cosmetics • Pharmaceuticals • Smart materials – ER-fluids
52
Importance of Rheology
Basim Abu-Jdayil - Chemical & Petroleum Eng. Dept.
Measuring rheological properties helps us bridge the
gap between molecular structure and product performance.
Rheology is important in industrial processing and design of a wide
range of complex chemical products.
53
Yield Stress
Basim Abu-Jdayil - Chemical & Petroleum Eng. Dept.
Rheological modifiers are used to control the
yield behavior of fluids.
Why modify the yield behavior?
* to avoid sedimentation and increase the shelf live
* to reduce flow under gravity
* to stabilize a fluid against vibration
54
Basim Abu-Jdayil - Chemical & Petroleum Eng. Dept.
Emulsion Yield Stress
0 20 40 60 80 100 120 140 160 -20
shear rate [1/s}
0
5
10
15
20
25
30
str
ess [P
a]
yield stress determined in a stress ramp
Requirements are:
minimum yield stress (10 Pa) to retard sedimentation low shear viscosity of 500 Pas to impede flocculation shear thinning to 2 Pas at 10'000 s-1 to allow fast application on the hair
Formulation of a conditioning shampoo
55
Toothpaste
Basim Abu-Jdayil - Chemical & Petroleum Eng. Dept.
Toothpaste consists of solid particles
suspended in an aqueous solution of
various polymers.
A good toothpaste should be a Bingham
fluid, so that it can easily be squeezed out
of the tube but will not drip off the
toothbrush the way water or honey would.
56
Thixotropy
Basim Abu-Jdayil - Chemical & Petroleum Eng. Dept.
The thixotropy characterizes the time dependence of
reversible structure changes in complex fluids. The
control of thixotropy is important to control:
sagging and leveling and the related gloss
of paints and coatings, etc..
process conditions for example to avoid
structure build up in pipes at low pumping
rates i.e. rest periods, etc.... 58
Paints
Basim Abu-Jdayil - Chemical & Petroleum Eng. Dept.
A good paint should be a
thixotropic with a yield stress fluid,
So in the can it will be very
viscous and the pigment will not
settle to the bottom,
But when it is stirred, it will
become less viscous and can easily
be brushed onto a surface.
In addition, the brushing should
temporarily reduce the viscosity so
that the paint will flow sideways
and fill in the brush marks (called
leveling in the paint industry); then,
as it stands, its viscosity should
increase, so that it will not form
drops and run down the wall.
59
Basim Abu-Jdayil - Chemical & Petroleum Eng. Dept.
Viscosity ranges of coatings
The two coatings show the same consistancy after formulation,
but they exhibit very different application performance
0.1 1 10 100 1000 10000 100000 10
-3
10 -2
10 -1
10 0
10 1
10 2
10 3
10 4
HSV MSV LSV
Roling Brushing Spraying
Mixing Pumping Consistency Appearence
Leveling Sagging Sedimentation
At Rest Processing Performance
Vis
cosity h
[P
as]
shear rate g [1/s]
60
Engine Oil
Basim Abu-Jdayil - Chemical & Petroleum Eng. Dept.
Motor oil, or engine
oil, is an oil used for
lubrication of various
internal combustion
engines
Base oil is Newtonian
Polymers are added to
give eth engine oil the
shear thinning behavior
61
Shear Thickening
Basim Abu-Jdayil - Chemical & Petroleum Eng. Dept.
They filled a pool with a mix of cornstarch and water
made on a concrete mixer truck. It becomes a non-
newtonian fluid. When stress is applied to the liquid it
exhibits properties of a solid.
63
Flow in pipes 6.64
Effect of temperature on viscosity
Viscosity is very sensitive to temperature
• The viscosity of gases increases with temperature:
n
oo T
T
ST
T C 3/2
o
Power-law
Sutherland equation
• The viscosity of liquids decreases with temperature:
-B/Te D