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Title: all about elasticity
Description: This notes contains all you need to know about elasticity in very easy language. And with attractive diagrams.
Description: This notes contains all you need to know about elasticity in very easy language. And with attractive diagrams.
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1
PM1
ELASTICITY
The lion is the king of beasts
And husband of the lioness
...
But I think wherever the lion is
I'd rather be somewhere else
...
You will be introduced to the concepts of stress, strain and material strength
...
You will learn how to do calculations
involving simple situations of deformation
...
1
...
2
...
(ii) Use the relations between the three elastic moduli and stress and strain in simple
numerical problems
...
Recall that the elastic moduli have dimensions of force per area
...
Use a model of the microscopic structure of materials to explain elastic behaviour
...
(i)
Describe the experimental measurement of elastic moduli by direct determinations
...
6
...
PRE-LECTURE
1
...
However, no body
is completely rigid: forces also deform bodies
...
Remember that pressure is defined as force per area
...
Refer back to chapter FE3 and/or chapter FE5, where inter-molecular forces are discussed in
terms of the distance between molecules
...
The study of elasticity is concerned with how bodies deform under the action of pairs of applied
forces
...
The pairs of forces act in opposite directions along the same line
...
This is described in terms of strain
...
As a result of the deformation, internal forces are set up and these give rise to stresses
...
For these simple cases we
make the following definition
...
There are three particular cases we will consider
...
An oppositely directed pair of forces along a line extend the body in along that line
...
The stress and strain are then defined as follows:
F
Stress =
F/A
Strain =
e/L
L
A
e
F
Fig 1
...
Write the pressure as p, the original volume as V and the change in volume as DV
...
2 Definitions of stress and strain (uniform compression)
Shear
Demonstration
In the previous two deformations, either the length or volume of a body was changed
...
Shear occurs for example when oppositely directed
tangential forces are applied across opposite faces of a rectangular block of material
...
Write the force as F, the area across which the force is applied as A and the angle of
deformation (specified in the diagram) as q
...
Fig 1
...
PM1: Elasticity
In general, things are more complicated than this but can be resolved in terms of these basic
deformations
...
Fig 1
...
However, if the rod is
thin enough, one does not get a linear compression but rather the rod buckles
...
Demonstration
This was done for the case of linear extension using one of the testing machines in the Civil Engineering
Department
...
The complete stress-strain curve was as follows:
Fig 1
...
e
...
This region OP is
known as the elastic regime and the point P is called the elastic limit
...
From J to K the material flowed like a fluid; such behaviour is called plastic flow
...
Eventually (when the
point B was reached) the material fractured
...
e
...
The behaviour described above for mild steel is not typical of all materials
...
Demonstration
Other materials, such as concrete, do not flow plastically; such materials are called brittle
...
The latter is difficult to describe in a way which is easily applicable but in the former
the stress is proportional to the strain
...
Hence the ratio stress/strain is a constant; this constant is known as
the elastic modulus
...
Linear Extension
Stress
F/A
=
Strain
e/L
=
Y
named Young's modulus
Uniform Compression
Stress
Strain
=
-P
DV/V
=
pV
- DV
=
k
named bulk modulus
Shear
Stress
Strain
F/A
= q
=
F
Aq
=
n
named shear modulus or modulus of rigidity
<<
There is one other parameter which is necessary to describe the elastic behaviour or materials
...
When a body is linearly extended, it contracts in the direction at right angles
...
Fig 1
...
In the unstressed state the atoms or molecules are in equilibrium positions, such that if
PM1: Elasticity
they are pulled apart the forces between them are attractive and if they are pushed together the forces
are repulsive
...
Such calculations can and have been made,
particularly for crystals, where there is a regular array of atoms
...
1-4 EXPERIMENTAL MEASUREMENT OF ELASTIC MODULI
The elastic moduli can be determined in two basically different ways
...
An alternative method is to make use of the fact that the mechanical oscillations of bodies and
the characteristics of pressure waves propagating through them depend on the elastic moduli
...
Fig 1
...
2
...
8 Oscillations of a cantilever: Young's modulus
The oscillations of a cantilever are determined by its Young's modulus
...
Fig 1
...
This alternative method can be particularly useful when it is not possible to obtain a sample
suitable for the test machines
...
The propagation characteristics of a pressure wave are determined by the bulk modulus of
the material in which it is propagating
...
A table of the elastic moduli of various materials is included in the post-lecture material
...
The prime consideration in these applications is
that the materials be strong enough
...
In these latter
applications the materials must also be bio-compatible as well as strong enough
...
For some materials this breaking stress will be different under compression (compressive
strength) than under tension (tensile strength) or under shear (shear strength)
...
One such material
is "composite"
...
Its compressive strength is about 2
...
Since as well it looks like tooth enamel, it is
a very suitable material for anterior fillings
...
It is interesting that a material can fail at
stresses much less than this if the stress is applied and removed a large number of times
...
Demonstration
Dentures for example can fail by fatigue
...
as Structural Elements
The basic point in designing any element to withstand stress is to properly assess what the stresses
are
...
Weight bearing structures which occur in nature are of good design
...
These are basically columns and are in a state of compression due to their own
weight
...
PM1: Elasticity
This, however, is not the case
...
Failure
occurs when the tree's length becomes too great in comparison with its diameter
...
10 Bending of a column
To prevent this failure by bending the diameter should increase as the 3/2 power of length
...
Demonstration
Scaling, with this same relation, is also observed for bones of animals
...
Demonstration
For a given weight/unit length, beams of cross-section such as these
Fig 1
...
Fig 1
...
In others their good design leads to the bone being arranged
differently: it is all a matter of the nature of the stresses
...
Demonstration
No matter how well-designed bones are, they will fracture when the strength of the bone material is
exceeded
...
e
...
8
9
PM1: Elasticity
Fibres
The elastic properties of bone and timber are different in different directions
...
There are many fibrous materials in nature
...
The elastic properties of these fibres is obviously important in that they determine the
properties of the textiles made from them
...
D
Stress
B
O
C
A
Strain
Fig 1
...
This region is followed by a
linear region AB
...
If the stress is
removed in this region the strain returns to zero
...
It results from the long keratin molecules, of which the fibre is composed, changing
from a coiled shape to a more extended one
...
Arteries and the Lung
Strong fibrous materials, such as bone, are common in the body
...
The walls of the arteries fall into
this category
...
Demonstration
As the heart pumps, the pressure in the arteries increases and the artery walls stretch
...
The hardening of
the artery walls, which occurs with age, inhibits this process
...
Muscular effort is required in
inspiration to extend the lungs but expiration is mainly due to the relaxing of the stretched tissues
...
If the lungs are filled
with saline solution, a much lower pressure is required
...
If the lung is washed out with kerosene and the experiment of inflation with air is repeated,
it is found a much higher pressure is required than before
...
When surfactant is present it decreases the surface tension
during inspiration
...
)
10
PM1: Elasticity
Muscle and Skin
Demonstration
Other tissues in the body where stretchability is important are muscle and skin
...
The skin's elasticity
POST-LECTURE
1
...
units have been
used
...
That unit is used exclusively in
these notes
...
7 TABLES
The calculated values are based on microscopic models
...
TENSILE STRENGTH / 108 Pa
MATERIAL
THEORETICAL
rock salt (NaCl)
2
...
004
1
...
0
14
...
0
(bulk material)
(single crystal)
(bulk)
(single crystal)
(fibre)
For liquids and gases the shear modulus is zero; for liquids the bulk modulus is about the same
value as for solids but it is much smaller for gases
...
05
k/1010 Pa
n/1010 Pa
7
...
67
steel
19 - 21
16
...
1
7
...
9
glass (crown)
6
...
8
4
...
9
2
...
2
water
0
...
1
0
0
air (atmospheric pressure)
1
...
8 TENSILE AND COMPRESSIVE MODULI
A crystalline solid exhibits the same stress vs
...
On the other hand, bone and other biological materials show different behaviour
under tension and compression
...
10
...
1
The effective cross sectional area of a horse's femur (leg bone) is 7
...
3 ¥ 109 Pa
...
Q1
...
10 mm
5
...
14 Diagram for Q1
...
The distortion is
exaggerated, for clarity, in the diagram
...
The shear modulus of brass is 3
...
Q1
...
The bulk modulus of water is 2 ¥ 109 Pa
...
OBJECTIVES
Aims
In this chapter you will look at the behaviour of liquid surfaces and the explanation of that behaviour
both in terms of forces and in terms of energy
...
1
...
2
(i) Describe an experimental determination of the surface tension of a liquid by the
measurement of the force on a glass slide in contact with the liquid
...
(i) Use a model of the microscopic structure of liquids to explain the phenomenon of
surface tension in terms of potential energy
...
(iii) Do simple numerical calculations associated with energy per area
...
(ii)
Demonstrate that these two descriptions are dimensionally equivalent
...
g
...
g
...
2T
(ii) Recall, explain and use the relationship h = rgr for capillary rise
...
7
Explain, by identifying the relevant forces and using scaling arguments, why insects can walk
on water but larger animals cannot
...
1 N
...
PM2: Surface Tension
13
PRE-LECTURE
Recall from earlier lectures, particularly chapters FE2 and FE3 the following facts about the general
nature of forces
...
There is a crossover point where the
force is zero - neither attractive nor repulsive
...
In
particular, the molecules of a substance tend to come together (pulled by the intermolecular
attraction) until on the average their distances apart correspond to the cross over point between
attraction and repulsion
...
(iii) Equilibrium can be discussed in terms of potential energy
...
For a simple two body system you can see this by considering the diagrams on pages 17 and
59 of the Forces and Energy book
...
Small insects can walk on water without getting wet
...
The strength of the membrane varies for different liquids, e
...
it is much less for soapy water than pure
water
...
However, they cannot swim on soapy water
...
]
2-2 MEASUREMENT AND DEFINITION OF SURFACE TENSION
The strength of the surface membrane can be imagined to arise from a set of forces acting on each
point of the surface, parallel to the surface, like the skin of a drum
...
1 Experimental measurement of surface tension
Note that because the water surface curves up near the glass slide the surface tension forces between the
glass and the water are vertical rather than horizontal
...
2 Shape of liquid meniscus
A first experiment yielded this result:
A certain amount of sand (weight, W) was needed to keep slide just in contact with water; when the water
was removed this amount of sand plus a 0
...
4 mN) was needed to have the slide in the same
position
The difference, 5
...
A second experiment tested whether the force depended on the length of the slide (recall that on the surface
of a drum, a bigger cut is harder to repair than a smaller one)
...
Force = T ¥ length
...
PM2: Surface Tension
15
Demonstration
In the second experiment the width of the slide was 1 mm, so the total length of the line of contact
between the glass and the water was (76 + 1 + 76 + 1)mm
...
06 N
...
[Most books of tables quote 0
...
m-1
...
Demonstration
A little detergent added to the water lowers it surface tension considerably
...
Its units in the S
...
system are N
...
2-3 MICROSCOPIC EXPLANATION AND SURFACE ENERGY
To understand why the phenomenon of surface tension arises, you must think of intermolecular
attraction as recalled in the pre-lecture material
...
In solids, this separation is fixed, whereas in gases, the random motion due to heat
predominates
...
Consider a fixed number of liquid molecules
...
If they have small surface
area, the average intermolecular separation is relatively low
...
A logical conclusion from this is that energy has to be added in order to increase the surface
area of a liquid
...
Associated
with the surface there is a potential energy that depends on the area of the surface
...
Since the equilibrium configuration of any system is that in which the potential energy is least,
a liquid left to itself will assume a shape which minimises surface area, thereby minimising the total
surface potential energy
...
LOOP OF
THREAD
CONTAINER
PURE WATER
WATER AND
DETERGENT
Shaded area here is greater than shaded area here
Fig 2
...
)
PM2: Surface Tension
energy
force
The dimensions of energy are force ¥ length, so area has the same dimensions as length
...
4 Effect of placing a drop of detergent inside a triangle of matches that are floating
on the surface of water
This is basically the same as the loop of thread demonstration, but it is easier to explain why
each match moved in terms of forces as thus for the match at the top of the diagram:
larger force
(water: higher
surface tension)
smaller force
(detergent: lower
surface tension)
Fig 2
...
h
WATER (DYED)
Fig 2
...
16
17
PM2: Surface Tension
Demonstration
Glass tube of narrow bore in water
...
g
...
In fact, the height varies inversely as r
...
7 The rise of water in a wedge between two flat glass sheets
(iii)
We would like to have shown that height decreased with increasing density, but we could not
find two common liquids with roughly the same surface tension and vastly different densities
...
50 mm and the measured rise was
28 mm
...
06 N
...
m-3 ¥ 9
...
s-2 ¥ 0
...
Specific Applications:
(i) Rise of water through soils
...
So the same kind of capillarity formula will apply
...
[Note water rises fastest in column with largest grains
...
]
(ii)
Chromatography
...
See post lecture material for a more
careful description
...
But certainly different liquids are attracted to different
solids in different degrees
...
Demonstration
Drops on solid surfaces
...
8 Water and mercury drops on glass and lead surfaces
Laboratory workers measure the intersurface forces in terms of the angle of contact defined
as follows
...
9 Definition of f, the angle of contact between a liquid and a solid surface
The concept of angle of contact is treated further in the post lecture
...
Water is said to wet glass completely (the angle of contact
is virtually zero)
...
Demonstration
Oil on glass will repel water
...
10 The presence of oil results in the water forming a drop rather than spreading over
the glass surface
Demonstration
Waterproofing of material (this usually involves coating fibres with oil or polymers)
...
Water birds spread oil on their feathers to make them water resistant
...
Some West Australian sands are virtually impervious to water as a result of fibrous material
between the grains making them water resistant
...
19
PM2: Surface Tension
Detergents
The properties of detergents arise from their complicated molecular structure
...
11 A detergent molecule
When detergent is put into water this happens:
Fig 2
...
The surface
tension is lower than that of pure water
...
grease
water
DETERGENT ADDED
Fig 2
...
Emulsification
...
Demonstration
Oil and water
...
Q2
...
This is not easy to prove in general but consider the following concrete example: assume that the length of
thread in the loop was 0
...
2
...
5 cm
3
...
3 cm
4 cm
3
...
2 cm
Fig 2
...
1
Q2
...
The following example illustrates this fact
...
The surface tension of the soapy water is T
...
15 Diagram for Q2
...
Since the film has two surfaces, the relation between F and T is
F
=
2Ú T
...
2-7 MORE ON CAPILLARITY
The law quoted in 2-4 can be derived theoretically as follows
...
Glass
molecules
Water molecules
Fig 2
...
Therefore, all
around the top of the water, the glass is exerting a force on the water
...
So that it why the water rises in the tube: because the glass is pulling it up
...
The next question is: why does not the water keep rising indefinitely?
The answer is that the higher the column the more the weight of the water in the column pulls it
back
...
The two forces are in equilibrium so
2prT
=
rπr2hg
and, therefore, for this situation, where the water wets the glass completely, the final height of the
water column can be written
2T
h
=
rgr
Q2
...
3 mm) after
a long time the water finally stopped rising at a height of ~ 150 mm
...
Apply the relation and find how much error is in fact introduced
...
tangent
ANGLE OF f
line
CONTACT
Fig 2
...
18 Angle of contact for water-glass contact
for mercury-glass, as in the next diagram, it is almost 180°
...
19 Angle of contact for mercury-glass contact
When the angle of contact is less than 90°, the liquid is said to wet the solid surface, while it is
said not to wet the surface if the angle of contact is greater than 90°
...
22
PM2: Surface Tension
23
Forces associated
with surface tension
Angle
f
Fig 2
...
16
Redrawing an earlier diagram in a more general way, we note that the force the liquid exerts on
the wall (and vice versa) is not vertical
...
There is a
vertical component, T cos f (which for f equal to 0˚ or 180˚ is T), which causes the liquid in a
capillary tube to rise
...
The general form is
2Tcosf
h =
rgr
For clean glass-water contacts f ª 0 and cos f ª 1
...
Q2
...
The formula for capillary height will
therefore have a minus sign in it
...
5
Why can insects walk on water, but larger animals (no matter how much water repellent material they put on
themselves) cannot?
Similarly, why will a needle float on water, but a much larger piece of metal of exactly the same shape will not?
Try to answer this question as follows:
(i)
Consider a nice simple geometric shape for the needle, say a rectangular bar
...
50 mm
...
8 ¥ 103 kg
...
(iii)
Now assume it is on top of the water with an angle, f, as shown
...
21 Needle "floating" on water
Calculate the total upward force (remember the force associated with surface tension acts right around the contact line
between the needle and the water)
...
Will its weight be supported by surface tension?
(vii)
See if you can use the kind of scaling argument which was employed in chapter FE8 to answer the original
question succinctly
...
useful in biological contexts
...
It is particularly
Paper Chromatography: Here a few drops of the chemical mixture are put onto a piece of filter paper and
allowed to dry
...
The solvent is sucked up into the filter paper (by capillary action), and as it flows past the dried
mixture, it dissolves out the chemical constituents and carries them along
...
e
...
So if you remove the paper
from the solvent after a while the various chemical constituents of the original mixture will be at different positions
on the filter paper
...
) Here the solvent is put on top of the
mixture, and allowed to flow through a plug composed of grains of cellulose
...
In our experiment (which we filmed in the Department of Agricultural Chemistry with the help of Dr Bob
Caldwell) the final order of chemical constituents is
TOP:
Flavonoid
(Yellow)
Chlorophyll B
(Green)
Xanthophyll
(Yellow)
Chlorophyll S
(Green)
Pheophytin
(Purple)
BOTTOM
Carotenoids
(Yellow)
Only the two chlorophyll bands show up well on the TV screen
...
They are listed here
merely for the purpose of showing you what range the values of surface tension can
have
...
m-1
water (20°C)
0
...
059
alcohol
0
...
063
turpentine
0
...
513
2-12 REFERENCES
"Surface tension in the lungs"
Scientific American, p 120, Dec 1962
...
26
PM3
HYDRODYNAMICS
Some fish are minnows
Some are whales
...
Fish like scales
...
They don't get cold,
They don't get drowned
But every fish wife
Fears for her fish
...
OBJECTIVES
Aims
In this chapter you will look at the behaviour of fluids in motion and the explanation of that
behaviour both in terms of forces, energy and the continuity of the fluid
...
Minimum Learning Goals
When you have finished studying this chapter you should be able to do all of the following
...
Explain, interpret and use the terms:
thrust force, lift force, streamline, turbulence
...
(i) Explain why the description of mutual forces between a moving fluid and a stationary
object is identical to that for a stationary fluid and a moving object
...
(iii) Explain why it is preferable in discussing liquid flow, to consider the liquid as a
continuous substance rather than individual molecules
...
Describe how energy is dissipated in turbulent motion
...
(i)
Recall the definition of Reynolds number
vLr
R= h
and state how L is determined in different situations
...
(iii) Do simple calculations and interpretations involving Reynolds number
...
(i) Explain how, for streamline motion in a tube (or channel) of variable cross-section, the
flow speed depends on the cross-sectional area
...
)
(ii)
Give a quantitative description of the branching effect at pipe junctions
...
6
...
(Bernoulli's Principle
...
PM3: Hydrodynamics
27
PRE-LECTURE
Recall the following background information from earlier chapters, particularly chapters FE3, FE4
and FE5
...
This same pressure is exerted by each
part of the fluid on neighbouring parts
...
We will not in fact pursue kinetic theory any
further - but concentrate on experimentally observable laws concerning fluid pressure and its
effects
...
(b) The pressure within a fluid can vary from point to point; in a fluid at rest the pressure
varies with vertical height according to the law
...
It will be the concern of this lecture to establish how the pressure varies inside a fluid which is
in motion
...
Typically forces such as electromagnetic or gravitational are conservative and
frictional forces are dissipative
...
However since the origin of these effects has a more complicated microscopic
explanation, this classification is not always so straightforward
...
LECTURE
3-1 THRUST AND LIFT FORCES
The study of hydrodynamics involves the study of the interaction of fluids and solid bodies
...
(a) A moving fluid exerts a force on a stationary object because each molecule of the fluid, on
bouncing, is accelerated by the solid
...
Before collision
After collision
Force exerted on molecule
Figure 3
...
Force exerted on solid
Figure 3
...
3 Lift force exerted by horizontal wind on an inclined wing
Examples: Hovering birds, gliders, kites
...
e
...
Example : Fish tails
...
4 Thrust force on flipping fish tail
(This example is in fact too complicated to worry about too much for now; suffice it to say that
the backward and forward motion of the tail results in an average forward thrust
...
Yet the main conclusion is correct: if you
want to move up through a fluid, you must push the fluid down; if you want to move forward, you
must push the fluid backwards
...
Application
Aeronautical engineers can predict how an aeroplane will behave in flight by observing it at rest in a wind
tunnel (or even in a water tank)
...
When a stream of (gently) flowing fluid is diverted by the presence of a wall, the particles of fluid do
not all bounce off the wall, most bounce off other fluid particles
...
Figure 3
...
The origin of this force must be that the fluid molecules bounce off one another,
causing those next to the wall to bounce off it more violently
...
More of this later
...
It turns out that it is more helpful in describing fluid flow to think of the fluid as a continuous
substance rather than to concentrate on the motion of individual molecules
...
These stream lines can of course be curved or straight, depending on the
flow of the fluid
...
The tubes have elastic properties:
(a) A tensile strength, which means that the parts of the fluid along a particular streamline stick
together and do not separate from one another,
(b) zero shear modulus, which means that each streamline moves independently of any other
...
When the motion becomes
too violent, eddies and vortices occur
...
Demonstrations
Wakes of boats
Liquid tank demonstration
...
When a body is moved through a stationary fluid in streamline motion some kinetic energy is
given to the fluid, but only temporarily
...
But when turbulence is established, a net amount of kinetic energy is left in the fluid after the
body has passed
...
Air turbulence means increased fuel consumption in
aircraft, and many cunning and intricate devices are used to reduce turbulence
...
Demonstration
Shapes of marine animals, specially shaped corks
...
(i) Speed of flow - faster flow gets turbulent more easily
...
[Just what physical quantity is involved here is not obvious
...
The symbol for it is h/r
(see post lecture)
...
For water flowing at the same speed through narrow pipes, the flow becomes turbulent more
easily in the tube of larger radius
...
We define for any
system a number R, called the Reynolds number
vLr
R ≡
h
where v is a typical flow speed of the fluid, L is a typical length scale and h/r the kinematic
viscosity of the fluid
...
There is no theoretical explanation of this value of 2000, it is just found to be the case
...
The reason behind this is very easy to grasp
...
One sees many applications of this
...
Demonstrations
(i) In flowing rivers, when going from deep to shallow, the flow speed increases (often becoming
turbulent)
...
When a fluid flows past a Y-junction made up of pipes of the same diameter, the total crosssectional area after the branch is twice that before the branch, so the flow speed must fall to half
...
6 Y-junction with pipes of same diameter
Conversely, if it is important to keep the flow speed up, the pipes after the branch must have
half the cross-sectional area of those before
...
7 Y-junction with pipes of half the original cross-section
(Note: blood will clot if its speed falls too low
...
The bulk modulus of a gas, while lower than that of a solid, is still large enough for the
equation of continuity to describe its motion
...
Also the tube structure of the respiratory system is remarkably similar to that of the circulatory system
...
So
the equation of continuity says that where streamlines crowd together the flow speed must increase
...
8 Streamline pattern around an aerofoil
PM3: Hydrodynamics
3-5 BERNOULLI'S PRINCIPLE
Demonstration
An interesting effect which is easy to show is that, for a fluid (e
...
air) flowing through a pipe with a
constriction in it, the fluid pressure is lowest at the constriction
...
The reason is easy to understand
...
The changes in energy must result from work being done on
the fluid and the only forces in the tube that might do work on the fluid are the driving forces
associated with changes in pressure from place to place
...
9 Application of Bernoulli's Principle
The units of pressure, N
...
m-3; that is, pressure is dimensionally
equivalent to work/volume
...
e
...
In cases where the flow is not horizontal, we should add in the gravitational potential energy/volume
l
also: 2 rv2 + p
+ rgh = constant
...
For the very simple cases it says what we had before the fluid pressure is lowest where the flow speed is highest
...
You can support a weight this way:
Nozzle
Air
low
pressure
high
pressure
Figure 3
...
[Note in most automobiles,
petrol is sucked into the carburettor in this way
...
Nozzle: low pressure
High pressure
(Atmospheric)
Figure 3
...
{Fly
sprays always have a small air hole
...
There is a high pressure on one side (so a big force) and low pressure (small force) on the
other
...
This is one of the ways players can get cricket or
ping pong balls to swerve
...
POST-LECTURE
3-6 MORE ON REYNOLDS NUMBER
There are several points to note about the definition of the Reynolds Number
...
The quantities L and v are only typical values of size
and speed
...
For a body
moving through a fluid it might be either length or breadth or thickness - or any other dimension
you might think of
...
It is not until you learn more about the Reynolds Number that
you can really hazard an intelligent guess at which one you should use
...
It is not surprising then that the magic number
of 2000 is also only rough
...
There are many ways in which this "stickiness" or viscosity manifests itself
...
How easily the fluid becomes turbulent is related to this but to the
density (r) as well - or if you like, it defines a different measure of stickiness
...
h is measured in units of Pa
...
s
...
s-1][m][kg
...
s]
This will be understood when you come to see where the Reynolds Number comes from
...
34
PM3: Hydrodynamics
Q3
...
(i) A Sydney Harbour ferry
(ii) Household plumbing pipes
(iii) The circulatory system
...
2 m
...
]
(iv) Spermatozoa swimming
...
s-1
...
Consider a fluid flowing through an irregular tube like this
Speed = v 2
B
Speed = v 1
A
Area = A
Area = A
2
1
Figure 3
...
So the mass which flows past A is r1 A1v1Dt
...
Now, when the flow is steady all the material which goes past A must go past B in the same
time (or else it will continually piling up somewhere) so
r 1 A1v1Dt
...
r 1 A1v1 = r 2 A2v2
Then if the fluid is incompressible, its density does not change, so
A1v1
= A2v2
which is the result stated earlier
...
But notice also
that if the fluid is approximately incompressible, i
...
if its density never changes by very much, then
the equation of continuity, as we quoted it, is approximately true
...
It is given the name volume rate of flow, and is usually
denoted by the symbol q
...
Q3
...
PM3: Hydrodynamics
35
Figure 3
...
You cannot apply this to water flowing around a bend in the river
...
3-8 BERNOULLI'S EQUATION
If you really want a more careful derivation of Bernoulli's equation, you can look it up in another
book
...
Just remember that,
because you are using the equation of conservation of energy, it is important that there should be no
energy dissipation through turbulence
...
Nonetheless, provided there is not too much turbulence, the law will approximately apply
...
Q 3
...
2
Use the average speed for blood flow quoted above and a typical human blood pressure of 104 Pa to explain why
this is so
...
4 In section 2 above, you analysed streamlined flow round a corner
...
Can you reconcile this with the kind of
simple minded diagrams drawn for the lift force on wings drawn in figure 3
...
5 When you are in the dentist's chair, the dentist uses a device based on the venturi effect to suck saliva out of
your mouth
...
36
PM4
VISCOSITY
Come crown my brow tin leaves of myrtle
I know the tortoise is a turtle
...
I know to my profound despair
I bet on one to beat a hare
...
OBJECTIVES
Aims
In this chapter you will look at the effect of the application of shear stresses to fluids and the
associated phenomenon of fluid viscosity
...
Poiseuille's
equation, which describes the flow rate of viscous liquids through pipes is presented, discussed and
applied to a number of situations
Minimum Learning Goals
When you have finished studying this chapter you should be able to do all of the following
...
Explain and use the following terms: shear stress, velocity gradient, viscosity, newtonian
liquid
...
(i) Describe an experiment that shows qualitatively a relationship between shear stress and
velocity gradient
...
3
...
s
...
(i)
Recall that the coefficient of viscosity for water is about 1 ¥ 10- 3 Pa
...
(i)
Describe the different response of liquids and solids to an applied shear stress
...
5
...
(Poiseuille's equation
...
(iii) Describe three phenomena, including both water pipes and the human body, which relate
to Poiseuille's equation
...
Present the analogy between current in an electric circuit and fluid flow in a pipe system and
explain what is meant by resistance in fluid flow
...
(i)
Explain how energy is dissipated by viscosity
...
PM4: Viscosity
37
PRE-LECTURE
Keep in mind two particular points that have been made so far in these Properties of Matter lectures
...
In
last lecture we pointed out that this number told you whether or not a particular flow system
was likely to be turbulent or streamline
...
The basic reason for the existence of this number and why it takes
the form it does is perhaps one of the most important questions in the whole study of fluid
flow
...
As we have pointed out it is the behaviour of a substance
under shear which essentially distinguishes between a solid and a liquid
...
e
...
A liquid has a very, very small shear modulus
...
Nonetheless the sliding of one bit does
have an influence on the other, and this is what viscosity is all about
...
The flow of water
through pipes is an important part of this lecture - and obviously much the same kind of
mathematical reasoning can be used to talk about it, as was used to discuss D
...
circuits
...
If some quantity (say pressure) varies with
distance (x), being big at some point and small at another, we say a pressure gradient exists
...
In a fluid we might expect the flow speed to change from point to point, and we could describe
this variation by measuring the velocity gradient
...
Demonstration
Observe the flow of water, glycerine, oil, treacle, lava, pitch
...
experimental record is:
1920
Pitch poured in funnel
1938 (Dec
...
) Second drop fell
1954 (Aug
...
) Fifth drop fell
...
38
PM4: Viscosity
Demonstration
(i) The rate at which solids fall through liquids (this has already been discussed in chapter FE4)
...
(iii)
Smoke rings
...
These are all traceable (in the end) to molecular adhesion, but their explanation and connection
with one another is very complicated
...
We must select one
physical effect to measure, and try to understand the others in terms of it
...
When a fluid (e
...
air) flows past a stationary wall (e
...
table top), the fluid right close to the
wall does not move
...
So a velocity gradient
exists
...
1 Velocity gradient in a stream of fluid moving past a stationary wall
We will find that the magnitude of this gradient (how fast the speed changes with distance) is
characteristic of the fluid
...
Demonstration
Observe the velocity gradient in a tank of treacle
...
Before the experiment starts the weight of an attached pan is
adjusted so that the plate is neutrally buoyant - i
...
it does not tend to sink in the liquid or to rise
...
2 Experiment to measure coefficient of viscosity: static situation
39
PM4: Viscosity
An extra force is now applied to cause the plate to move through the liquid
...
The complete apparatus is
Pointer
Scale
Weights (to provide
known force)
Glycerine
Plate (of known size)
Fig 4
...
6
(ii) For the same force, the speed of the plate decreases as the area of the plate increases
...
5
5
5
7
3
We interpret these two sets of results as indicating that the speed of the plate increases with the
shearing stress (recall the definition of stress given in chapter PM1)
(iii) For a given speed, we change the velocity gradient by moving the plate closer to the wall
...
4 Increased velocity gradient when the plate is closer to the wall
Mass on pan
7g
Time (close to wall)
4
...
40
PM4: Viscosity
More careful experimentation, or more detailed theoretical analysis, will clarify these
conclusions into Newton's law of viscosity which says: When a shearing stress acts within a fluid
moving in a streamlined motion, it sets up in the liquid a velocity gradient which is proportional to
the stress
...
Water is obviously a lot less viscous than glycerine
...
Some common measuring devices are:
Demonstrations
(i) Commercial oil companies use simpler viscometers
...
The "grade" of an oil is the number of seconds it takes to pour a measured amount through a certain tap
...
g
...
(iii) Laboratories usually use a torsional viscometer, which is really a very refined version of the apparatus
we used above
...
5 A torsional viscometer
Of course the viscosity of a liquid can change
...
However, this kind of detail you can catch up on later, when you come to talk about viscous
effects in your own discipline
...
We have given Newton's statement, relating velocity gradient to shear stress (pressure)
...
Solids
When a shearing stress is applied to a solid it suffers a shear (i
...
a shear deformation)
x
A
C
y
B
Fig 4
...
When the shearing stress is
removed, if the solid is elastic the deformation recovers
...
However, so long as the shear is applied it continues to shear
...
The basic law of behaviour of elastic solids and viscous liquids are:
Elastic solids obey Hooke's law which says
shear stress µ
shear deformation
{Remember:
shear
=
Viscous liquids obey Newton's law which says
shear stress µ
length AC
length AB }
velocity gradient
...
This can be seen as follows, with reference to figure 4
...
So, Newton's law can be restated
shear stress µ rate of shear deformation
...
The most important one, which is the only one we will consider is flow through a long pipe
...
One thing however is obvious
...
Demonstration
Polystyrene chips on surface of a treacle tank
...
7 Velocity profile for moving treacle
[This technique of showing the velocity profile of the flow will become important later
...
The bigger the pressure difference, the faster will be
the flow;
(ii) the length of the pipe
...
(iii) the radius of the pipe
...
This dependence is very marked
...
(Theory says we should have got 16 times as much
...
Water flows much more easily than glycerine
...
r4 p
qv = lh 8
This is known as Poiseuille's law
...
See post lecture material
...
It is uneconomical to use spray irrigation too far from a river since the resistance of a
pipe increases with its length, and you need too big a pump
...
Here ∆p and l are fixed (by geography), and the volume rate of flow is
fixed by the requirements of the population of Sydney
...
[This is an
oversimplification, see post lecture
...
The resistance to flow is determined
primarily by the narrow tubes leading to the alveoli
...
(iv) Circulatory system: Two points are worth making
...
Blood pressure is highest when it
leaves the heart (through the aorta) and lowest when it returns (through the inferior vena cava)
...
Why?
(b)Any constriction of the tubes - for example a build up of cholesterol on the walls of the arteries increases the resistance and hence the pressure drop (it goes as r4 remember)
...
And at times of stress, when an increased flow rate is required, there can be a breakdown
...
You work out the relevant physics
...
There are a large number of important liquids which do not
obey these laws, and they are called non-newtonian liquids
...
Demonstration
flow of syrup
...
We are not talking about energy loss due to turbulence, but about energy
loss which occurs even when the flow is streamlined
...
The criterion whether or not much energy is lost in this way, is therefore whether or not there
is much of a velocity gradient throughout the whole of the liquid
...
System relatively large
System relatively small
energy mostly conserved
much energy dissipated
Fig 4
...
So now we can appreciate another reason why the Reynolds number is important
...
e
...
On the other hand viscous
effects will not be important in thin liquids or in large, fast flow systems
...
)
44
PM4: Viscosity
In general then, flow patterns will be different in systems with low and with high Reynolds
numbers
...
Demonstration
The method of swimming is quite different for fishes (R ~ 10,000) and spermatozoa (R ~ 0
...
Modes of boat propulsion which work in thin liquids (water) will not work in thick liquids (glycerine)
...
You cannot do this with water
...
If you feel you
cannot do without it, there are plenty of books you can look up
...
The retarding force comes from viscous drag, which acts to prevent shearing
...
The flow speed must therefore increase until the resisting force balances the driving force; and
when that happens you will get Dp on one side of the equation and l on the other
...
There is most fluid-on-fluid slipping toward the outside of the
tube (purely by geometry) so the velocity gradient is larges there, and is zero in the centre
...
Velocity
component
Distance from axis of tube
Fig 4
...
And so the
volume rate of flow (equal to the average speed • area of the tube) is likely to depend on r4
...
45
PM4: Viscosity
The only point in going through even so sketchy a "derivation" is to point out two facts:
(i) Poiseuille's law only applies to fluids that obey Newton's law, and
(ii) The assumption of streamlined flow is also built in to Poiseuille's law
...
(You will recall
that in the experiment done on screen glycerine flowed through the wide pipe more slowly than
would be predicted by Poiseuille's law
...
1
Why should turbulence mean that the volume rate of flow is less than in streamlined flow?
Q4
...
3
The experiment was done in chapter PM2 in which water rose, by capillary attraction, through two columns
of soils
...
Can you say now
why this is so?
4-7 ELECTRICAL ANALOGUE
Ohm's Law says :
V
=
Poiseuille's Law says :
Dp
=
R I
8 lh
p r 4 ¥qv
The comparison is obvious, and hence it is most convenient to talk about flow through any
kind of tube in terms of resistance defined thus:
8l h
R ≡
p r4
Note that the unit of this resistance is: kg
...
s-1
...
It only means that Poiseuille's
equation is not valid, and you cannot use this explicit formula for the resistance
...
Once you appreciate this, then you can use all the mathematical techniques of circuit analysis;
in particular the rules for adding resistances in series and parallel
...
4
Consider water flowing along a l
...
0 Pa
A
750 mm
B
C
Fig 4
...
4
If you measured the fluid pressure at point B, what value would you get?
pressure
1
...
5
Consider these two different streamlined flow systems,
B
A
C
(a)
B
A
C
(b)
Fig 4
...
5
The lengths of the two pipes in the section BC are equal to the lengths AB in both cases
...
In both
cases also, the pressure at A is 4 Pa, and at C is 1 Pa
...
And, by the same arguments used in 4-5 of chapter PM3,
this must be equal to the pressure drop
...
6
If you were designing a circulatory system for the human body, where a prime requirement is that as little
energy as possible should be dissipated, in order not to require the heart to pump any harder than absolutely
necessary, what Reynolds number would you aim for?
Compare this with the Reynolds number for blood, which is somewhere between 1000 and 2000
...
s
...
You can easily convert by remembering that
1 Pa
...
s
water (20°C)
1
...
3 ¥ 10-3
alcohol
1
...
5
mercury
1
...
8 ¥ 10-5
48
PM5
RHEOLOGY
Elephants are useful friends
Equipped with handles at both ends
...
If you think the elephant preposterous
You've probably never seen a rhinosterous
...
This is called non-newtonian behaviour
...
Minimum Learning Goals
When you have finished studying this chapter you should be able to do all of the following
...
Explain and use the following terms: rheology, non-newtonian flow, pseudo-plasticity,
dilatancy, plasticity, thixotropy, visco-elasticity, creep, stress relaxation
...
Describe, in terms of shear stress, shear, rate of shear, and time, the behaviour of the following
materials: starch solution, cornflour solution, wet sand, toothpaste, quickclay, wool fibres,
pitch, wet soil, blood
...
Describe an experiment showing the different effect of shear stress on a newtonian fluid (e
...
glycerine-water mixture) and a non-newtonian fluid (e
...
starch solution)
...
Refer back to chapter PM1 which discusses the behaviour of purely elastic materials
...
e
...
2
...
Note in particular that for simple fluids such as water the shear stress is
proportional to the velocity gradient, a relationship which is known as Newton's law
...
LECTURE
5-1 NON-NEWTONIAN FLUIDS
Rheology is the study of how bodies behave under the action of deforming forces
...
Most materials do not fall into these two extreme categories - in
fact strictly speaking none do if the relevant parameters are varied widely enough
...
To start we consider the fluids that do not obey Newton's law - the non-newtonian fluids
...
Basically, they fall into two different classes
...
49
PM5: Rheology
Demonstration
A solution of starch is pseudoplastic
...
(The newtonian fluid used was glycerine and water
...
In explaining this experiment we note that
(i)
the shear stress is related to the pressure difference driving the liquid
(ii)
the rate of shear is related to the flow rate
...
(iv) for the non-newtonian liquid (the starch solution) the viscosity increased as the shear stress
decreased (that is, as the head of liquid in the burette decreased)
...
(v)
the amount of glycerine in the solution was chosen so that the (constant) viscosity was
originally greater but later less than that of the starch solution
...
1 Variation of viscosity coefficient with shear rate
The second type of non-newtonian fluid is the so-called dilatant fluid which has the reverse
behaviour to a pseudo-plastic fluid viz its viscosity increases as the shear rate increases
...
Other examples of dilatant fluids are printing inks, vinyl resin pastes and suspensions at high solid content
such as wet beach sand which shows its dilatancy through the fact it stiffens when trodden on
...
Once this stress has been exceeded, the viscosity either remains
constant or decreases as the shear rate increases
...
Examples
An example of a plastic material is toothpaste
...
Some other examples of plastic materials of which there are many, are good brushing paints and sewage
sludge
...
2 Variation of shear rate with shear stress for newtonian and non-newtonian fluids
Another way in which the non-newtonian nature of a fluid can show itself is in its radial
velocity profile as it flows through a narrow tube
...
For non-newtonian fluids the radial velocity profile is not parabolic
...
For plastic materials there is a completely flat
region in the centre where the shearing stress is less than the yield value
...
Before leaving non-newtonian fluids, one further complication needs to be mentioned viz
...
There are two types of these
...
Further, after being sheared
at high rates and left at rest, the fluid does not recover its higher viscosity behaviour until after a
certain characteristic time has elapsed which may be as long as several hours
...
It is important to realise that there is basically no difference between a thixotropic and a
pseudo-plastic material
...
Demonstration
Thus it is that if a thixotropic varnish which is like jelly after being left at rest for a long time is mixed
up, it becomes quite liquid and stays like that for many minutes after the mixing has ceased
...
Demonstration
This was shown by placing a paste of plaster of Paris on an inclined plane
...
On stopping the vibration,
however, the flow ceased instantaneously
...
This would then have been a
demonstration of what happens in certain earthquakes where buildings are destroyed because they were built on
"quick-clay"
...
Whether these behave as solids or liquids depends on how long the stress
is applied
...
A liquid will continue to deform as long as the stress is applied, but for a solid
there will be an instantaneous deformation and this will then remain constant with time
...
Demonstration
If we take, for example, a copper wire and stress it, we do get an instantaneous deformation but if we keep
the stress constant and observe over a period of a day or so, we find that if the stress is large enough then the
wire will continue to deform appreciably over this period - it will have flowed slowly but continuously like a
liquid
...
[Since it flows one can of course ascribe a viscosity to such a solid
...
s
...
5 mm
...
For a solid a certain stress is required to produce the strain, and to maintain the strain the
stress must continue at this value
...
Demonstration
But if we take a wool fibre for example and carry out this experiment, we find that the stress does not
remain constant nor drop instantaneously to zero
...
3 Stress-strain curve behaviour for a wool fibre
...
The time for the stress to drop by e-1, (e is
the exponential function) is known as the stress relaxation time, t
...
The materials in between are the visco-elastic
materials
...
Demonstrations
A good example is the material known as "silly" putty
...
If, however, a stress is applied over a long time, it will flow - it behaves as a
viscous liquid
...
This will flow but it also shows elastic properties in that it recoils if the
flowing stream is broken or cut
...
<<
An interesting property of visco-elastic materials is the phenomenon known as the Weissenberg
effect
...
This is
observed when cake mix is mixed with a rotating beater
...
These materials are quite complicated
...
The clays are particularly complex in that their properties
depend markedly on the presence of electrolytes
...
At low concentrations of water, these materials behave as solids, and if one does have a flow
problem it can be considered as the flow of the water through the soil rather than as the flow of a
composite material
...
Water
Dye
lines
Sand
Fig 5
...
The dye lines show the velocity
is fastest where the cross-section is narrowest, and thus illustrate the equation of continuity
...
Clay for
example can become either plastic or pseudo-plastic or indeed thixotropic and at all but the highest
concentrations of water, is visco-elastic
...
53
PM5: Rheology
Model
building
Level
gradually
raised
Water
Sand
Sand
Fig 5
...
They have found widespread application
particularly in the ceramic industry
...
Demonstrations
A clay-water mixture suitable for pouring into a mould or piping throughout a factory has to have quite a
high concentration of water
...
This process is known as deflocculation
(the aggregation of the clays particles is greatly affected by the electrolyte)
...
As the water leaves, the wall of the casting gradually builds up
...
5-4 BLOOD
Another material of obvious importance is blood
...
Blood is a complex fluid, consisting of a plasma in which are suspended a variety of cells, the
predominant ones being the red cells
...
Measurement show that blood is thixotropic
...
The viscosity for people with certain diseases such as myocardial infarction and
thrombosis is much higher, particularly at low shear rates
...
s
INFARCTION AND
THROMBOSIS
NORMAL MEN
1
0
...
01
0
...
1
1
10
100
0
...
1
1
10
100
Rate of shear / s-1
Fig 5
...
1
...
These aggregates can,
and do, form when the shear rate is low, but at higher shear rates they break up, giving a lower
viscosity
...
If the red cells were rigid particles, when their concentration reached 65%, blood would have
the consistency of concrete
...
Blood is still very fluid even at 99% red cell concentration
...
Thus it is that blood can flow in the small capillaries - the cells
deform as they flow
...
Demonstration
An important consequence of the rheological nature of blood is that when it is artificially pumped, as it is
in certain types of heart surgery where the heart is by-passed, special pumps have to be used
...
They pump the blood so that the red cells are not damaged by too high shear rates but yet at a rate
sufficiently great so that aggregation does not occur
...
These in general affect
the flow just by dilating or contracting the blood vessels
...
This can be seen by measuring the blood flow in the vessels of the ear lobe by
passing a light beam through it
...
PM5: Rheology
55
POST-LECTURE
5-5 NOMENCLATURE
Though most materials can be classified into broad categories such as those mentioned above it
should be emphasised that some materials are very complex and don't neatly fit into such categories
...
Further
there are no sharp dividing lines between the categories, particularly as regards their time behaviour
...
5-6 PROBLEM
Q5
...
Categorise them and say why you think they fall into these categories
...
Today , p 23, April 1968
...
Amer
...
"The flow of matter"
Reiner, Sci
...
, p 122, December 1959
...
Educ
...
56
PM6
FRICTION
At midnight in the museum hall
The fossils gathered for a ball
...
Pterodactyls and brontosauruses
Sang ghostly prehistoric choruses
...
Cheer up, sad world, he said, and winked it's kind of fun to be extinct
...
The effects of naturally occluding
surface layers on of lubricants on friction are discussed
...
1
Explain and use the following terms : friction, normal force, real area of contact, coefficient of
friction, adhesion, cold-welding, lubrication, hydrodynamic lubrication, boundary
lubrication, elasto-hydrodynamic lubrication
...
(ii)
Do simple calculations based on the second of these laws
...
3
(i)
Using a microscope model, explain the laws of friction
...
4
Describe and explain how surface layers alter the value of the coefficient of friction
...
6
State the differences among three types of lubrication (hydrodynamic, boundary and
elasto-hydrodynamic) and give one example of each type
...
Remind yourself of the concept of a shear stress (Chapter PM1)
...
Recall that when a metal is subjected to increasing stress it normally goes through an elastic
regime where the stress is proportional to strain and then into a plastic regime where there is
flow of the metal at essentially a constant stress (Chapter PM1)
...
Remind yourself of the basic concepts of the flow of a viscous liquid and in particular of
Newton's law of viscosity (Chapter PM4)
...
The heavier the object the greater the effort
...
1 Sliding an object
Demonstration
It is a lot easier to move a heavy object by rolling it than by sliding it
...
2 Rolling an object
We can express these facts by saying that there is a friction force resisting the motion
...
Demonstrations
The force of sliding friction depends markedly on the surfaces involved
...
Friction plagues us in many contexts
...
Even more important than the power loss
is the wear which results from friction
...
Early applications were the use of animal fats to
lubricate chariot wheels
...
Demonstrations
Friction of a rope on something or on itself as in a knot enables large loads to be easily controlled
...
Traction of cars, bikes, trains etc
...
Traction in walking depends on friction
...
They are empirical laws which give the dependence of the friction force on the
relevant parameters
...
The
board is tilted until the object slides
...
(See post lecture material
...
3 Sliding down an incline
The laws were, however, determined using the following apparatus:
Fig 6
...
Weights were added to the carrier until the block was just on
the point of sliding
...
Friction force with 1 block = 4
...
The
weight was the same as before
...
The friction force Fr is independent of the area of contact
...
The friction force F is proportional to the normal component of the contact force N
...
Often
two values of this coefficient are given
...
The other is called the kinetic coefficient and
corresponds to the force required to keep the object sliding at a constant velocity
...
Demonstration
This was shown by repeating the measurement for 3 blocks on a sheet of aluminium
...
Even with this
apparatus, complications are evident - the tendency of the object to stick again after it has started to
slide
...
As well, the dependence
of friction on velocity can be investigated
...
The spring balance measures the friction force
...
5 Using a spring balance to measure the frictional force
This apparatus made evident the fluctuations which occur in the friction force and showed that
whereas at low velocities the friction force was essentially independent of the velocity, it did decrease
when the velocity became high
...
Firstly, a variety of techniques show that even when the surfaces look smooth, they are
microscopically rough
...
The oblique sectional technique
...
Fig 6
...
But the sliding of such surfaces over each other is non-dissipative
...
Once it is realised the surfaces are rough, it is apparent however that the real area of contact
must be small - the surfaces must only touch at a few points
...
It was further shown that
the real area of contact increased as the load increased
...
7 The "contact" between two plates, viewed at a microscopic level
Demonstration
The increase in area of contact with load was also shown by measuring the voltage drop across two surfaces
in contact in the following circuit
...
Surfaces
in "contact"
Digital
voltmeter
V
Ammeter
A
Fig 6
...
This is so since even at the smallest loads the stresses at the contact
points are large enough to make the metals deform plastically
...
The points of contact are in effect cold-welded forming a continuous solid
...
Demonstrations
That strong junctions are formed can be shown by oblique sections of the friction tracks formed when one
material is slid on another
...
This transfer can also be shown by sliding a radio-active metal on a non-radioactive metal
...
[As an aside, it should be noted that the transfer of metal which occurs when one metal is slid
on another (which can occur in screwing or hammering) is of relevance in bone surgery
...
used
to repair the bones
...
]
Bringing the various pieces of experimental information together, an explanation of the laws of
friction can be given
...
This is the
second law (see post-lecture material for further details)
...
This is the first law
...
There is also a
contribution associated with the "ploughing" of the hills of one surface through the other surface
...
PM6: Friction
61
6-4 OTHER FRICTIONAL BEHAVIOUR
The above picture of sliding friction for metals is incomplete
...
Indeed, it is only
because surface layers exist that metals can be slid on each other
...
Demonstration
The existence of these strong forces was shown using two accurately plane gauge blocks which were first
slid on each other to break down the surface layers
...
This can happen if the layers are such that they remain intact at low loads but break down at
higher loads
...
Layers of soft metal are placed on the surfaces of bearings to reduce friction
...
Generally, however, the coefficient of friction is much more load-dependent
...
For other materials such as plastics this behaviour arises because they are viscoelastic
...
This has a very low coefficient of
friction of 0
...
1 arising from the nature of its molecular structure which is 'streamlined"
...
Finally, a few words about rolling friction
...
The other is "free" rolling
...
In "free" rolling, the coefficient of friction is very low, less than 0
...
The mechanism for frictional energy loss is quite different from
that for sliding friction
...
When objects slide on each other, kinetic energy is
converted to heat resulting in increase in the temperature of the surfaces
...
Fire can be produced by the high speed rubbing of one piece of wood on another
...
Demonstration
These local hot spots are basic to the polishing process
...
Obviously, a high melting
point polishing agent is necessary for efficient polishing
...
The high temperature enhances the plastic flow, and if it is high enough a layer of
essentially liquid metal is produced which acts as a lubricant
...
A lubricating layer of water is produced
...
6-6 LUBRICATION
The reduction of friction between two surfaces by placing another material between them is
known as lubrication
...
Hydrodynamic lubrication
The type of lubrication in which the surfaces are completely separated by a thin film of fluid is
known as hydrodynamic lubrication
...
001 and completely eliminates wear
...
A complete film is formed if the load is not too high
and the speed of the rotating shaft is great enough
...
9 Lubricant squeezed between a rotating shaft and its bearing (hydrodynamic
lubrication)
In this type of lubrication, the frictional energy loss is due only to the viscous forces in the
lubricant (see post-lecture material)
...
62
PM6: Friction
63
Boundary lubrication
When metal contact begins to occur as can happen if the speed of the journal is decreased, a
continuous film of fluid no longer exists
...
Lubrication under these conditions is known as
boundary lubrication
...
10 A very thin film of lubricant, with molecules illustrated (boundary lubrication)
In this type of lubrication, the coefficient of friction does not depend on the viscosity of the
lubricant but rather on its chemical nature
...
A layer is then
formed which acts as a lubricating film, and if it can be easily sheared than the friction is low
...
1 are obtained and the wear is slight
...
This can be explained neither in terms of hydrodynamic nor boundary layer lubrication
...
In the synovial joints the cartilage elasticity is such as to allow this
process
...
This theory of synovial joint lubrication is supported by the fact that
synovial fluid from rheumatoid arthritis cases is newtonian, and that wear in the joint occurs when
hard calcifying material is deposited on the cartilage
...
)
PM6: Friction
POST LECTURE
6-7 PROBLEMS
Q 6
...
Fig 6
...
1
Derive an expression for the coefficient of friction in terms of this angle q
...
2
...
Thus it has been stated in the lecture that friction cannot be understood in
terms of intermeshing surfaces sliding over each other since this is a non-dissipative process
...
6-8 AN EXPLANATION OF THE SECOND LAW OF FRICTION
The second law of friction can be explained in terms of the shearing of cold-welded junctions
...
Further, since plastic flow occurs at the junctions the normal force N is related to the area of
contact by the expression
N = Ap
where p is the yield pressure, the stress at which plastic flow occurs
...
If s and p are taken as those of the softer material, this expression predicts reasonable values
for m
...
64
65
PM6: Friction
6-9 MORE PROBLEMS
Q6
...
Explain how this energy dissipation occurs
...
4
A shaft of radius r is rotating with angular velocity w in a bearing
...
Stationary
bearing
Rotating
shaft
r
Lubricant
d
Fig 6
...
4
(a)
What is the velocity gradient in the film assuming that this gradient is uniform throughout the thickness of
the film?
(b)
Determine the shear stress in the lubricant using Newton's law of viscosity
...
(This result shows, as mentioned in the lecture, that to reduce the resistance to motion, the coefficient of viscosity
has to be decreased
...
)
66
PM7
SOUND
Some claim that pianists are human,
And quote the case of Mr Truman
...
Ape-like they are, he said, and simian,
Instead of normal men and wimian
...
You will find that the speed of sound
depends on the stiffness and the density of the medium it propagates through
...
The physics involved in the ear and in hearing is
discussed
...
1
Explain and use the following terms: acoustic impedance, specific acoustic impedance,
impedance matching, Fourier analysis, Fourier synthesis
...
3
(i) Recall how the specific acoustic impedance of a medium depends on the bulk modulus
and the density of the medium
...
(iii) Describe an experiment to verify these laws
...
5
Describe an experiment to analyze the frequency components present in a musical note or
other sound
...
(ii) Draw a schematic diagram of the cochlea identifying the basilar membrane and the nerve
cells
...
(iv) Describe how ageing limits the frequency response of the ear
...
(i) The definition of the Bulk Modulus, and Hooke's Law as it applies to substances
which are deformed by volume compression stresses
...
It
concerns the problem of getting energy from a source (battery) to somewhere it can be used (load)
...
If either one of these two is
much bigger than the other, then only a small fraction of the available energy appears in the load,
most is dissipated in the source
...
In many, many cases involving the transport of energy from one place to another, the same kind
of reasoning will be found to apply
...
Then, unless the impedance of the source is roughly
the same as the impedance of the receiver, energy will not readily get from one to the other
...
For a mass oscillating at the end of a spring, the force is given by Hooke's law pointing to the fact
that this kind of oscillation is essentially an elastic phenomenon
...
Sound consists of elastic vibrations also (pressure oscillations in fact) and you would
therefore expect that a mathematical analysis would yield a similar result: viz
...
(iv) Also in lecture FE7 you met the concept of fourier analysis, the breaking down of a
complicated oscillation into a sum of simpler sinusoidal oscillations
...
The inverse problem, that of starting with simple oscillations and combining them into a more
complicated shape is called fourier synthesis
...
We will
concentrate on two aspects only: its propagation and its analysis
...
PM7: Sound
7-1 ACOUSTIC IMPEDANCE
The ear may be sketched schematically thus:
Fig 7
...
There the vibrations of air pressure are translated into mechanical oscillations,
which are carried with slight mechanical advantage due to lever action, by the ossicles (D) to another
membrane, the oval window (E)
...
To understand the structure of the outer ear, we must talk in general terms about the propagation of
pressure waves through an elastic medium
...
2 A mechanical model of an elastic medium
The speed at which a disturbance will travel down this chain can be seen to depend on
(i) the mass of the object
...
ii) the strength of the springs
...
Generalizing to a three dimensional rather than a linear medium, we might expect the speed of
sound to increase as the density decreases, and as the bulk modulus increases
...
)
c= k/r
68
PM7: Sound
69
Consider now the boundary between two media
...
3 A disturbance “propagating” through a row of billiard balls
...
[Why?]
If any one ball is heavier or lighter than the others, then the disturbance will result in some
reflection as well as propagation of kinetic energy
...
A disturbance will propagate with no net motion in the medium, only if all cars and all springs are
identical
...
Demonstration
However, an increase in mass can, in part, be compensated for by a decrease in spring strength
...
Generalizing to three dimensions, we define a quantity called the specific acoustic impedance, z,
by the equation
z ≡ kr
to serve as an index to tell us whether sound energy can efficiently be transferred from one medium
to another
...
A particular acoustic device
will be described by a quantity called the acoustic impedance, which depends on both the shape
and size of the device
...
It may be helpful in understanding this subject to realize that acoustic
impedance plays a role analogous in some respects to resistance; and in the same way specific
acoustic impedance is analogous to resistivity
...
Since both the density and bulk modulus of skin are much greater
than that of air, the specific impedance of the eardrum is vastly different from that of the outside air
...
The rest is simply reflected back
...
As we said before, acoustic impedance depends also on the
geometry of the device; and the narrowing of the auditory canal plays a most important role
...
Crudely speaking, a narrow column of air is more difficult to get moving than completely free
air, because of viscous effects at the side of the tube
...
Fig 7
...
)
Nevertheless, the impedance of the air near the drum, and that of the drum itself are still badly
mismatched, and most of any sound wave is reflected away
...
This gives the impedance of the drum (relative to that of air [see post-lecture]
...
70
PM7: Sound
71
Since you know the elasticity and density of skin and bone, it should be possible to work out the
acoustic impedance of a normal ear and how it is affected if you change the pressure, for example
...
By this means, it is
possible to pick up certain specific defects in the drum or the ossicular chair - for example a
perforated drum or a calcified ossicular chain
...
From this point of view, the
ear is in an inefficient hearing device
...
7-2 FOURIER ANALYSIS
Beyond the oval window in the inner ear is the cochlea, a snail like structure which, if it could
be straightened out, might look in principle like this:
Fig 7
...
Because of
its geometrical shape, different parts of this membrane resonate with oscillations of different
frequencies
...
Fig 7
...
The increased
amplitude of vibrations thus set up, are made to cause a gas jet to flicker up and down
...
PM7: Sound
Demonstration
This apparatus is in principle capable of identifying all the harmonies present in any musical note
...
Modern spectrum analysis is all done electronically
...
You interpret the
output thus: for a reasonably pure note you might get something like
Fig 7
...
That is why you see such a slow rate of scan
...
]
*With the aid of a spectrum analyzer you can determine what gives the human voice or various
musical instruments their distinctive sounds
...
*In order to get a feeling for the relationship between a sound and its spectrum, just listen to the
sounds as they are produced and see if you can correlate the most prominent features of the spectrum
you see with what you hear
...
And indeed the spectrum of one wind instrument for example, is very
like that of another
...
]
To sum up: the cochlea is a device for translating a series of pressure vibrations into a coded set of
electrical signals which the brain uses as a sensory input
...
That this is a true representation of the ear can be confirmed by two observations
...
In the way this technique is usually used, the pattern the child
has to reproduce is NOT the frequency spectrum but simply the pressure-time variations
...
[This technique is still not fully developed or accepted; and one suspects that it will not be until it is
carried out with complete spectrum analyzers rather than simple oscilloscopes, that it will prove most
effective
...
72
73
PM7: Sound
For example:
Fig 7
...
9 Another example of Fourier analysis with a minor change from Fig 7
...
And though these
two pressure-time patterns are quite different, they have the same fourier spectrum - the various
harmonics present have the same amplitudes
...
*The new breed of musical instruments based on this simple electronic organ principle are given the
generic name synthesizers
...
Demonstration
The new breed of musical instruments based on this simple electronic organ principle are given the
generic name synthesizers
...
POST LECTURE
7-3 REFLECTION AND TRANSMISSION
Actually to calculate how much energy is transmitted or reflected when a sound wave encounters an
impedance mismatch, like for example at the ear or a microphone, is very complicated, because it is
the mismatch of impedances of the device itself, and the small part of the air right next to it which
must be analyzed
...
Then we need to consider only the
bulk property of each medium - the specific acoustic impedances
...
It can be appreciated immediately that complete transmission (i
...
ar = 0) will only occur when the
two media have exactly the same specific acoustic impedances
...
1: Why can you hear small sounds so much more clearly under water than in air?
[ANS 9]
PM7: Sound
7-4 IMPEDANCE MATCHING
When there is an impedance mismatch between two media, it is possible to take steps to
increase the transfer of energy between the two
...
For large quantities of the media, this can be done by intervening a third medium
between the two
...
Perfect transmission of energy occurs, in theory, when
z3 = z1 z2
(where z3 is the specific impedance of the intervening medium) and where the thickness of
the medium is a quarter wavelength
...
7-5 FREQUENCY RESPONSE OF THE EAR
Because of physical limitations, the ear will not respond to all frequencies
...
If you remember what you have learnt about
resonance, then a solid body can resonate with a sound wave if its size is roughly similar to the
wavelength of the sound (in that material)
...
10 A veryschematic representation of the basilar membrane
The lowest frequency it can pick up well will correspond to the width of the big end, and the
highest frequency will correspond to the width of where exactly the last nerve cell is located at the
small end
...
This determines its ability to
follow a very high frequency vibration
...
For young people, the upper range is about 20 - 30 kHz, but in middle age, it is found that the upper
limit of hearing can drop by 80 Hz every six moths
...
Also the state of the joints in the ossicular chain clearly influence frequency response, since
these too must vibrate at the same frequency as the drum
...
2: In general how would you expect that the frequency range of the ear would vary with the size
of the animals? [Ans 25]
7-6 REFERENCES
Bekesy “The Ear” Scientific American, August 1957, p 66
...
74
75
PM8
ULTRASONICS
Puccini was Latin, and Wagner Teutonic,
And birds are incurably philharmonic
Suburban yards and rural vistas
Are filled with avian Andrews Sisters
...
That's what shepherds listen to in Arcadia
Before somebody invented the radia
...
Much of the lecture is concerned with the applications of
ultrasonics using techniques of echoscopy and the Doppler technique
...
1
...
2
...
3
...
4
...
(ii) State two applications of this property
...
(iii) State and describe two applications of the Doppler effect with ultrasonic waves
...
(i) Describe what is meant by the Doppler effect
...
(iii) State and describe two applications of the Doppler effect with ultrasonic waves
...
Describe and distinguish the techniques of radar, sonar and echoscopy
...
Describe and explain one example of the use of echoscopy
...
PM8: Ultrasonics
LECTURE
8-1 INTRODUCTION
Sound experienced by human falls in the frequency range 0 - 20 kHz
...
It can be detected by some animals
...
V
...
Ultrasound has many important applications some of which will be discussed in this lecture
...
Though the use of ultrasound by man is of relatively recent origin, BATS have always used it
...
They use this ultrasound for navigational purposes and
also for locating their prey
...
The elapsed time gives the bat information on how far the object reflecting the pulse of
ultrasound is from it
...
1 Production of sound using a tuning fork
Ultrasound is produced in the same way but to get ultrasound we have to make a vibration at
ultrasound frequencies
...
These have the property that when a voltage is applied in a certain direction, the dimension of the
material in that direction increases, and if the sense of the voltage is revesed then the dimension
decreases
...
76
PM6: Friction
77
Fig 8
...
These transducers can detect ultrasound by using them in reverse
...
Since ultrasound is at high frequency, its wavelength is short and so it can be focussed into
small regions
...
Strongly focussed high power ultrasound can be used for example to kill microorganisms, to study
cells by splitting them open, to produce small lesions in the brain, to treat a disease of the inner ear
known as Meniere's disease, to remove hard deposits on the valves of the heart and so on
...
Demonstration
Ultrasound at high power is also used extensively for clearning
...
The high intensity ultrasound causes
negative pressures in the liquid and as a result bubbles called "cavitation" bubbles are produced
...
If the object is kept too long in the cleaner,
particularly if it is thin, it can be extensively damaged
...
This is known as the DOPPLER EFFECT
...
The effect also applies to a wave reflected from a moving object
...
(NOTE: It is difficult to
upset a speeding conviction based on a radar trap, because you can demonstrate the device is working
properly just by using a tuning fork)
...
The "red shift" i
...
the shift to lower frequencies, of light from other
galaxies is interpreted to mean that the galaxies are moving away from each other and hence the
concept of the expanding universe
...
The frequency of the
detected sound can be measured with a digital frequency meter
...
Since the change in frequencyu is proportional to the velocity, a
measurement of this change can be used to determine the velocity of the source
...
In this latter form there are many uses of
Doppler techniques with ultrasound in medicine
...
Demonstration
In clinical medicine, the technique is used to detect blood flow in arteries and veins by means of
an external probe
...
A
paste is applied between the skin and probe to improve impedance matching and so lessen power loss
by reflection
...
A probe such as this shows quite different sounds for arteries and veins
...
In veins, it is more like a
wind-storm which cycles with the respiration
...
(N
...
The medical term "patent" which is used in describing this technique means "unblocked"
...
Demonstration
Small probes are placed around arteries during an operation
...
In an experiment, the leads are connected to a telemetering device carried in a
package on the animal
...
This technique has been used for example on dogs which
have been made hypertensive
...
In a proposed experiment it is to be used on baboons to study the
effect of diet on coronary disease
...
This technique is called the "pulse-echo" technique and with electro,magnetic waves is of course radar
...
It is of
course the technique used by bats for navigation
...
In medicine, the technique is used and is then known as echoscopy
...
How much energy is reflected depends on the sound power reflection coefficient, which in turn
depends on the specific acoustic impedances of the two media If these are almost the same, little
energy is reflected; if they are widely different, much energy is reflected
...
Thus the pulse-echo technique can be used to obtain
two-dimensional cross-sectional views through various organs in the body
...
The time delays and amplitudes of the signals from
the different boundaries make up the two dimensional picture called an echogram
...
The latter is a three dimensional view compressed into the two dimensions of the x-ray
plate
...
Demonstration
One use of this technique is in a machine designed to scan the eye
...
It should be noted that the
acoustic impedances of the transducer and the eye are matched in this machine by having the
transducer in water contained in a plastic membrane to which the eye, rubbed with a paste, is pressed
...
Demonstration
This technique is also used in obstetrics to scan the pregnant uterus
...
The technique is of majeor importance in this field for unlike x
rays, ultrasound appears to be completely safe
...
The technique can also be used for examining the non-pregnant
abdomen for picking up tumours
...
Material
Velocity/m
...
m– 3
Specific acoustic impedance/106 kg
...
m– 2
water
1530
1
...
53
blood
1534
1
...
59
fat
1440
0
...
40
brain
1510
1
...
55
liver
1590
1
...
64
muscle
1590
1
...
64
bone
3360
2
...
62
air
340
0
...
1 Why is the frequency of ultrasound which bats use so high?
[Ans 6]
Q8
...
[Ans
Title: all about elasticity
Description: This notes contains all you need to know about elasticity in very easy language. And with attractive diagrams.
Description: This notes contains all you need to know about elasticity in very easy language. And with attractive diagrams.