1
PM1
ELASTICITY
The lion is the king of beasts
And husband of the lioness.
Gazelles and things on which he feasts
Address him as Your Lioness
There are those who admire that roar of his
In the African jungles and veldts.
But I think wherever the lion is
I'd rather be somewhere else.
OBJECTIVES
Aims
In this chapter you will see that all elastic deformations can be described in terms of linear, shear and
bulk changes. You will be introduced to the concepts of stress, strain and material strength. You
will apply these ideas to some real-world deformations. You will learn how to do calculations
involving simple situations of deformation.
Minimum Learning Goals
When you have finished studying this chapter you should be able to do all of the following.
1.
Explain, interpret and use the terms
elastic deformation, plastic flow, permanent set, ductile, brittle, compressive, tensile,
linear!extension, uniform compression, shear, pressure, fracture, stress, strain,
elastic!modulus, Young's!modulus, shear modulus, bulk modulus, strength, elastic limit.
2.
(i)
Recall and state Hooke's Law.
(ii) Use the relations between the three elastic moduli and stress and strain in simple
numerical problems.
3.
Recall that the elastic moduli have dimensions of force per area.
4.
Use a model of the microscopic structure of materials to explain elastic behaviour.
5.
(i)
Describe the experimental measurement of elastic moduli by direct determinations.
(ii)
Use mechanical oscillations to measure elastic moduli indirectly.
6.
Describe situations involving strength and/or deformation in the human body and in fibrous
materials.
PRE-LECTURE
1.
In mechanics, forces acting on an extended body are assumed to produce only translational
and/or rotational accelerations of the body: the body is assumed to be rigid. However, no body
is completely rigid: forces also deform bodies.
2.
Remember that pressure is defined as force per area.
3.
Refer back to chapter FE3 and/or chapter FE5, where inter-molecular forces are discussed in
terms of the distance between molecules.
2
PM1: Elasticity
LECTURE
1-1 STRESS, STRAIN AND THE BASIC DEFORMATIONS.
The study of elasticity is concerned with how bodies deform under the action of pairs of applied
forces. In this study there are two basic concepts: stress and strain.
The pairs of forces act in opposite directions along the same line. Thus there is no resulting
acceleration (change of motion) but there is a resulting deformation or change in the size or shape
of the body. This is described in terms of strain.
The strain is the relative change in dimensions of a body resulting from the external forces.
As a result of the deformation, internal forces are set up and these give rise to stresses. In
many simple cases, these stresses are simply related to the external forces, because when these two
are in balance the deformation will be maintained without further change. For these simple cases we
make the following definition.
The stress is the external force divided by the area over which this force is applied.
There are three particular cases we will consider.
Linear Extension
Demonstration
The first type is the linear extension.
An oppositely directed pair of forces along a line extend the body in along that line.
Write the magnitude of these forces as F , the cross-sectional area at right angles to F as A, the
original length as L and the extension as e. The stress and strain are then defined as follows:
F
Stress =
F/A
Strain =
e/L
L
A
e
F
Fig 1.1 Definitions of stress and strain (linear extension)
3
PM1: Elasticity
Uniform Compression
Demonstration
If the forces are applied uniformly in all directions, we have a deformation typified by that produced by a
uniform hydrostatic pressure.
Write the pressure as p, the original volume as V and the change in volume as DV. The stress
and strain are then defined as follows:
Stress =
p
Strain = - DV/V
(the significance of the
minus sign is that the
volume decreases as the
pressure increases)
p
Fig 1.2 Definitions of stress and strain (uniform compression)
Shear
Demonstration
In the previous two deformations, either the length or volume of a body was changed. In the shear
deformation, only the shape of a body is changed. Shear occurs for example when oppositely directed
tangential forces are applied across opposite faces of a rectangular block of material. These forces deform the
rectangular block into a parallelogram.
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. The stress and strain are then defined as follows.
Fig 1.3 Definitions of stress and strain (shear deformation)
These three deformations are the three basic types.
PM1: Elasticity
In general, things are more complicated than this but can be resolved in terms of these basic
deformations.
Demonstration
As an example of the more complicated behaviour one can get, consider a rod under the action of a
compressive force in the direction of the rod.
Fig 1.4 Buckling of a rod as a result of an applied linear compression
If there are no complications, this is merely the opposite of the linear extension. However, if the rod is
thin enough, one does not get a linear compression but rather the rod buckles.
1-2 ELASTIC AND NON-ELASTIC BEHAVIOUR
Let us now consider what happens to a body under the action of one of these types of deforming
force as the force is gradually increased from zero.
Demonstration
This was done for the case of linear extension using one of the testing machines in the Civil Engineering
Department. A sample of mild steel was tested and the stress as a function of strain was recorded on a chart
recorder.
The complete stress-strain curve was as follows:
Fig 1.5 A stress/strain curve
As the stress was increased it was at first proportional to the strain; if, in this region, the stress were
removed, the strain would return to zero i.e. the body would return to its original length. This region OP is
known as the elastic regime and the point P is called the elastic limit.
4
5
PM1: Elasticity
As the stress was further increased, a point Y, known as the yield point, at which the stress rapidly
dropped, was reached. From J to K the material flowed like a fluid; such behaviour is called plastic flow.
After a region K to L of partial elastic behaviour, plastic flow continued from L to M . Eventually (when the
point B was reached) the material fractured.
It should be noted that once the material was taken out of the elastic regime (into the non-elastic regime,
where plastic flow occurred) the body suffered a permanent deformation or permanent set, i.e. removal of the
stress did not reduce the strain to zero.
The behaviour described above for mild steel is not typical of all materials. Materials that
behave approximately like this, showing elastic behaviour and plastic flow, are called ductile.
Demonstration
Other materials, such as concrete, do not flow plastically; such materials are called brittle.
1-3 HOOKE'S LAW and ELASTIC MODULI
As we have seen, when a material is stressed there are basically two different regimes: the elastic and
the non-elastic. The latter is difficult to describe in a way which is easily applicable but in the former
the stress is proportional to the strain.
This proportionality between stress and strain is known as Hooke's law; it applies to all of
the three basic deformations. Hence the ratio stress/strain is a constant; this constant is known as
the elastic modulus. There are three elastic moduli, one for each of the three basic deformations.
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. This is
Poisson's ratio. When a body is linearly extended, it contracts in the direction at right angles. Poisson's ratio, s,
is the ratio of the lateral strain to the longitudinal strain.
Fig 1.6 Poisson effect: Contraction of rod in direction transverse to the direction of the
applied stress
>>
A Microscopic Model
The values of the various parameters we have defined must depend on the microscopic structure of
the material. 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.
Where these forces as a function of distance between the atoms or molecules are known, one
could, in principle, calculate the elastic moduli. Such calculations can and have been made,
particularly for crystals, where there is a regular array of atoms. However, the values obtained are
always too high, due to the presence, in even the purest crystals, of imperfections such as
dislocations and impurity atoms.
1-4 EXPERIMENTAL MEASUREMENT OF ELASTIC MODULI
The elastic moduli can be determined in two basically different ways.
The most direct way is to use one of the engineering-type machines you have seen and to
measure the strain appropriate to different stresses.
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.
Demonstration
1.
Fig 1.7 Oscillations of a coiled spring: shear modulus
The frequency of oscillation of a coiled spring is determined by the shear modulus of the material of
which it is made.
2.
Fig 1.8 Oscillations of a cantilever: Young's modulus
The oscillations of a cantilever are determined by its Young's modulus.
6
PM1: Elasticity
7
3.
Fig 1.9 Torsional oscillations: shear modulus
The torsional oscillations of a rod are determined by its shear modulus.
This alternative method can be particularly useful when it is not possible to obtain a sample
suitable for the test machines. Investigations of possible changes in the elasticity of bones in the
body with age and disease have been made, for example, by setting the bones into oscillation and
measuring the oscillation frequencies.
The propagation characteristics of a pressure wave are determined by the bulk modulus of
the material in which it is propagating. This will be discussed in more detail in the lecture on sound
(chapter PM7).
A table of the elastic moduli of various materials is included in the post-lecture material.
1-5 APPLICATIONS
Repair of the Human Body
Many materials are used in the repair of the body. The prime consideration in these applications is
that the materials be strong enough. External to the body there are, for example, artificial limbs, and
internal to the body there are, for example, plates used for repairing fractures. In these latter
applications the materials must also be bio-compatible as well as strong enough.
The strength of a material is defined as that stress which causes the material to break.
For some materials this breaking stress will be different under compression (compressive
strength) than under tension (tensile strength) or under shear (shear strength).
Demonstration
An example of a material used for repair of the body is material used for filling teeth. One such material
is "composite". This is a polymer mixed with quartz and is quite strong in compression, as it must be, since
large compressive stresses are experienced in biting. Its compressive strength is about 2.5 ¥ 108 Pa which
compares favourably with that of tooth enamel viz 4 ¥ 108 Pa. Since as well it looks like tooth enamel, it is
a very suitable material for anterior fillings.
If the strength of a material is exceeded it will fail. 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. This
phenomenon is known as fatigue.
Demonstration
Dentures for example can fail by fatigue.
Bones etc. as Structural Elements
The basic point in designing any element to withstand stress is to properly assess what the stresses
are. The element is then designed so as to withstand these stresses without being unnecessarily big.
Weight bearing structures which occur in nature are of good design. Of particular interest in
this regard are trees. These are basically columns and are in a state of compression due to their own
weight. One might think that their heights would be limited only by the requirement that the
compressive strength be not exceeded; thus no relationship between height and diameter would be
expected.
PM1: Elasticity
This, however, is not the case. A column fails not by compression but by bending. Failure
occurs when the tree's length becomes too great in comparison with its diameter.
Demonstration
Fig 1.10 Bending of a column
To prevent this failure by bending the diameter should increase as the 3/2 power of length.
This is observed on average for trees.
Demonstration
Scaling, with this same relation, is also observed for bones of animals.
Scaling is not the only good design feature found in bones.
Demonstration
For a given weight/unit length, beams of cross-section such as these
Fig 1.11 I-shape and tube-shape beams
are much stronger against bending than solid beams such as this.
Fig 1.12 Solid beams
Demonstration
Many bones indeed are of tubular shape. In others their good design leads to the bone being arranged
differently: it is all a matter of the nature of the stresses. For example, in the top of the femur, the bone is
arranged in thin sheets separated by marrow, the sheets being so arranged to give the greatest strength when the
bone is experiencing those forces to which it is normally subjected.
Demonstration
No matter how well-designed bones are, they will fracture when the strength of the bone material is
exceeded. This is most likely to happen when the bone is stressed in a direction other than usual i.e. when it
is stressed in a way for which it was not designed.
8
9
PM1: Elasticity
Fibres
The elastic properties of bone and timber are different in different directions. This is so because
these materials are fibrous. There are many fibrous materials in nature. One important class of
fibres are those used in making textiles: the natural fibres wool and cotton and the various synthetic
fibres.
The elastic properties of these fibres is obviously important in that they determine the
properties of the textiles made from them.
Demonstration
As an example of these properties, the stress-strain curve for a wool fibre in tension is given.
D
Stress
B
O
C
A
Strain
Fig 1.13 Stress-strain curve for wool fibre
The narrow region OA corresponds to the "crimp" in the fibre being removed. This region is followed by a
linear region AB. As the stress is further increased the curve flattens out into the region BC. If the stress is
removed in this region the strain returns to zero. Therefore this region does not correspond to a region of
plastic flow, as for steel. It results from the long keratin molecules, of which the fibre is composed, changing
from a coiled shape to a more extended one. With further stress, the curve again rises and finally the fibre
ruptures at the point D.
Arteries and the Lung
Strong fibrous materials, such as bone, are common in the body. There are other materials in the
body where strength is not the important thing but stretchability. The walls of the arteries fall into
this category. It is only because they are elastic that the blood flow is smooth.
Demonstration
As the heart pumps, the pressure in the arteries increases and the artery walls stretch. When the aortic
valve shuts and the pressure in the arteries drops, the walls relax maintaining the blood flow. The hardening of
the artery walls, which occurs with age, inhibits this process.
The elasticity of the lung tissues plays a very significant part in respiration. Muscular effort is required in
inspiration to extend the lungs but expiration is mainly due to the relaxing of the stretched tissues.
Demonstration
The stretching can be shown by measuring the pressure to fill the lungs with air. If the lungs are filled
with saline solution, a much lower pressure is required. This difference is because forces associated with
surface tension play a large part in the operation of the lung; when the lung is filled with saline solution these
forces do not act. 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. The kerosene washes out a chemical known as
"surfactant" which regulates the surface tension. When surfactant is present it decreases the surface tension
during inspiration. (This will become clearer after the surface tension lecture - PM2 - when more details will
be given.)
10
PM1: Elasticity
Muscle and Skin
Demonstration
Other tissues in the body where stretchability is important are muscle and skin.
decreases noticeably with age.
The skin's elasticity
POST-LECTURE
1.6 UNITS
You will have noticed that in the television lecture, on occasions, units other than SI. units have been
used.
The correct unit, as agreed by the International Conference on Weights and Measures, for
stress (or strength, or any of the elastic moduli) is the pascal (Pa). That unit is used exclusively in
these notes.
1.7 TABLES
The calculated values are based on microscopic models. The lack of correspondence is the result of
dislocations and impurity atoms.
TENSILE STRENGTH / 108 Pa
MATERIAL
THEORETICAL
rock salt (NaCl)
2.7
iron
46
cellulose
11
OBSERVED
0.004
1.0
2.0
14.0
1.0
(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.
SUBSTANCE
aluminium
Y/1010 Pa
7.05
k/1010 Pa
n/1010 Pa
7.46 2.67
steel
19 - 21
16.4 - 18.1
7.9 - 8.9
glass (crown)
6.5 - 7.8
4.0 - 5.9
2.6 - 3.2
water
0.2
0
mercury
2.1
0
0
air (atmospheric pressure)
1.4 ¥ 10-5
1.8 TENSILE AND COMPRESSIVE MODULI
A crystalline solid exhibits the same stress vs. strain relation whether it is under tension or
compression. On the other hand, bone and other biological materials show different behaviour
under tension and compression.
11
PM1: Elasticity
1.10. PROBLEMS
Q1.1
The effective cross sectional area of a horse's femur (leg bone) is 7.0 ¥ 10-4 m2 and the Young's modulus of
this bone is 8.3 ¥ 109 Pa.
Calculate the strain that occurs in the femur when the horse (mass ~ 600 kg) puts its full weight on one leg.
Q1.2
0.10 mm
5.0 mm
F
500 mm
500 mm
F
Fig 1.14 Diagram for Q1.2
The square brass plate shown is sheared to the position of the dotted lines by the forces F. The distortion is
exaggerated, for clarity, in the diagram. Calculate the magnitude of these forces.
The shear modulus of brass is 3.5 ¥ 101 0 Pa.
Q1.3
By what fraction does the density of water at a depth where the pressure is 4 ¥ 105 Pa increase over the
surface density.
The bulk modulus of water is 2 ¥ 109 Pa.
12
PM2
SURFACE TENSION
The swan can swim while sitting down,
For pure conceit he takes the crown,
He looks in the mirror over and over,
and claims to have never heard of Pavlova.
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. The principle of minimum potential energy can be
invoked to explain many surface phenomena
Minimum Learning Goals
When you have finished studying this chapter you should be able to do all of the following.
1.
Explain, interpret and use the terms
intermolecular forces, capillarity, angle of contact, wetting.
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.
(ii)
3
Perform simple numerical calculations associated with such a determination.
(i) Use a model of the microscopic structure of liquids to explain the phenomenon of
surface tension in terms of potential energy.
(ii) Extend this argument to explain why liquids tend to assume a shape which minimises the
surface area of the liquid.
(iii) Do simple numerical calculations associated with energy per area.
4
(i) Explain how the surface tension of a liquid can be measured either in terms of force per
length or of energy per area.
(ii)
Demonstrate that these two descriptions are dimensionally equivalent.
5
(i) Explain how the phenomenon of capillarity results from forces between solid (e.g. glass)
and liquid (e.g. water) molecules.
2T
(ii) Recall, explain and use the relationship h = rgr for capillary rise.
6
Give examples of how the wetting characteristics of surfaces can be altered.
7
Explain, by identifying the relevant forces and using scaling arguments, why insects can walk
on water but larger animals cannot.
8
Recall that the surface tension of water has a magnitude of 0.1 N.m-1.
PM2: Surface Tension
13
PRE-LECTURE
Recall from earlier lectures, particularly chapters FE2 and FE3 the following facts about the general
nature of forces.
(i) The molecules of any substance - solid, liquid or gas - attract one another if they are far apart;
at short distances, the intermolecular force is repulsive. There is a crossover point where the
force is zero - neither attractive nor repulsive.
(ii) When a system is in equilibrium, then the sum of all the forces acting on the system is zero. 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. This means the normal state of a substance is an average kind of
equilibrium.
(iii) Equilibrium can be discussed in terms of potential energy. The equilibrium configuration is
one in which the potential energy is least.
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.
LECTURE
2-1 PHENOMENON OF SURFACE TENSION
The surface of any liquid behaves as though it is covered by a stretched membrane.
Small insects can walk on water without getting wet.
Demonstration
The membrane used is obviously quite strong: it will support dense objects, provided they are small and of
the right shape:
a needle,
a small square of aluminium sheet (weighted),
a container made of fine wire gauze.
The strength of the membrane varies for different liquids, e.g. it is much less for soapy water than pure
water.
Demonstration
Ducks swim on water without getting very wet. However, they cannot swim on soapy water. [There are
cases on record where ducks have drowned in farmyard ponds into which washing water was emptied, or in
streams polluted with non degradable detergents.]
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.
14
PM2: Surface Tension
Demonstration
The easiest way to measure these forces is with the following apparatus
BALANCE
ADJUSTABLE
WEIGHT
AND
SAND
GLASS
SLIDE
FIXED
COUNTER
WEIGHT
WATER
Fig 2.1 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.
SLIDE
MENISCUS
WATER
Fig 2.2 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 0.55 g (extra weight 5.4 mN) was needed to have the slide in the same
position
The difference, 5.4 mN, is a measure of the force due to the pull of the water on the slide.
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).
Length of slide used in first experiment:
38 mm
Length of slide used in second experiment: 76 mm
Result of second experiment: the force due to the pull of surface increases to 10 mN
Deduction: The force which a liquid surface exerts on any body with which it is in intimate
contact (as described above) is directly proportional to the length of the line of contact.
Force = T ¥ length.
The constant of proportionality, T, is called the surface tension of the liquid.
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. These values give a value for the surface tension
of water of 0.06 N.m-1.
[Most books of tables quote 0.07 N.m-1.]
Other liquids have different surface tensions (see post lecture material).
Demonstration
A little detergent added to the water lowers it surface tension considerably.
As defined here the dimensions of surface tension are force per length. Its units in the S.I.
system are N.m-1.
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.
Molecules of any substance want to pack together so that their average separation is low.
In solids, this separation is fixed, whereas in gases, the random motion due to heat
predominates. In liquids, there is some random motion but, on the average, the molecular separation
is low.
Consider a fixed number of liquid molecules. If they are packed so that they have a large
surface area, their average intermolecular separation is relatively high. If they have small surface
area, the average intermolecular separation is relatively low. Their total potential energy is lower in
the latter case.
A logical conclusion from this is that energy has to be added in order to increase the surface
area of a liquid. The bigger the change in surface area, the more energy has to be put in. Associated
with the surface there is a potential energy that depends on the area of the surface. This means that
an alternative approach is to consider surface tension as an energy per surface area.
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.
Demonstration
Drops of water are spherical
Loop of thread on water; detergent added inside loop; loop takes a circular shape.
LOOP OF
THREAD
CONTAINER
PURE WATER
WATER AND
DETERGENT
Shaded area here is greater than shaded area here
Fig 2.3 Effect of placing a drop of detergent inside a loop of string that is floating on the
surface of water
(Surface tension of detergent and water is much lower than that of water.)
PM2: Surface Tension
energy
force
The dimensions of energy are force ¥ length, so area has the same dimensions as length .
Sometimes it is easiest to explain surface phenomena in terms of energy considerations,
sometimes in terms of force considerations
Demonstration
Three matches on water:
CONTAINER
MATCHES
DETERGENT
ADDED
becomes
PURE
WATER
Fig 2.4 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 2.5 The net force acting on the match pushes it away from the detergent
2-4 CAPILLARITY
A consequence of the phenomenon of surface tension is that many liquids will "creep up" tubes, an
observation made readily with glass tubes of very narrow bore.
h
WATER (DYED)
Fig 2.6 Capillary rise
The height of the water in the capillary above the level of the liquid in the surrounding liquid,
as indicated by h in the diagram, is called the capillary rise.
16
17
PM2: Surface Tension
Demonstration
Glass tube of narrow bore in water.
It can be demonstrated that:
(i) the capillary rise is larger for liquids of higher surface tension than of lower surface tension (e.g. larger
for pure water than for water and detergent) ;
(ii) the height increases as the radius of the bore of the tube gets smaller.
In fact, the height varies inversely as r.
Demonstration
Glass wedge in water:
ELEVATION
PLAN
HYPERBOLIC
!! !SHAPE
TWO GLASS SHEETS
WIDE
END
NARROW
END
RUBBER BAND
WATER (DYED)
Fig 2.7 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.
The relation between capillary rise, surface tension and density (see post lecture) is
2T
h
= rgr
The tube used in the demonstration had a bore of radius 0.50 mm and the measured rise was
28!mm. For a tube of this radius, the calculated rise is
2!¥ !0.06!N.m-1
h
=
1!¥ !103 !kg.m-3!¥ !9.8!m.s-2!¥ !0.50!¥ !10-3!m
= 2 cm.
Specific Applications:
(i) Rise of water through soils.
Demonstration
Although water rising in a column of soil is not rising through a tube of uniform bore it is moving
through spaces roughly the same size as the soil grains. So the same kind of capillarity formula will apply.
A consequence is that water rises highest in column with finest grains.
[Note water rises fastest in column with largest grains. We return to this in the post lecture of chapter
PM4.]
(ii)
Chromatography.
Demonstration
This is a method of chemical analysis which can be done by eye. See post lecture material for a more
careful description.
18
PM2: Surface Tension
2-5 WETTING
A question we have skimmed over is: why is there an attractive force between water and glass
causing the rise of water in a glass capillary tube? This is a question about intermolecular forces
which only chemists can answer properly. But certainly different liquids are attracted to different
solids in different degrees. For example, the level of mercury will fall in a glass capillary tube.
Demonstration
Drops on solid surfaces.
WATER
MERCURY
WATER
GLASS
MERCURY
LEAD
Fig 2.8 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.
tangent
ANGLE OF f
line
CONTACT
Fig 2.9 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.
This phenomenon is called wetting. Water is said to wet glass completely (the angle of contact
is virtually zero).
The wetting characteristics of surfaces can be changed by putting a layer of a different material
on the surface.
Demonstration
Oil on glass will repel water.
WATER
OIL
GLASS
Fig 2.10 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).
Demonstration
Preening of birds.
Water birds spread oil on their feathers to make them water resistant.
Demonstration
Water resistant sands.
Some West Australian sands are virtually impervious to water as a result of fibrous material
between the grains making them water resistant. This leads to bad run off conditions in vast areas of
the state.
19
PM2: Surface Tension
Detergents
The properties of detergents arise from their complicated molecular structure. This can be illustrated
schematically thus:
This end is repelled by water
molecules [hydrophobic] and is
This end is attracted to water
attracted to oils, fats [lipiphilic]
molecules [hydrophilic]
H
(i)
H
H
H
H
H
H
H
H
H
H
H
C
C
C
C
C
C
C
C
C
C
C
H
H
H
H
H
H
H
H
H
H
H
O
C
O-
Fig 2.11 A detergent molecule
When detergent is put into water this happens:
Fig 2.12 Detergent molecules in water (schematic)
Note that along the surface there are water molecules and hydrophobic ends. The surface
tension is lower than that of pure water. It is easier to pull this surface apart than it is to pull a
surface of pure water apart
(ii)
In washing up water the following sequence occurs as the water is stirred up.
grease
water
DETERGENT ADDED
Fig 2.13 Stirring of soapy water during "washing up"
STIRRED
20
PM2: Surface Tension
The particles of organic matter are rendered soluble by being coated with detergent molecules:
lipophilic ends stick to the particles and hydrophilic ends point outwards.
Emulsification.
Many organic substances which are insoluble in water (DDT is a good example) can be mixed into
an emulsion with water by the addition of a little detergent.
Demonstration
Oil and water.
POST-LECTURE
2-6 UNITS AND DIMENSIONS
A couple of statements were made (or implied) in 2-3 above, which may not be all that obvious.
Q2.1
The loop of thread changed its shape to a circle because a circle is the geometrical shape which has
maximum area for a fixed circumference.
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.l!m and work out which, of the following possible shapes the loop could have, has
the largest area.
2.5 cm
1 cm
2.5 cm
3.3 cm
3.3 cm
4 cm
3.3 cm
3.2 cm
Fig 2.14 Diagram for Q2.1
Q2.2
Energy/area is the same as force/length. The following example illustrates this fact.
Imagine you are increasing the area of a rectangular soap film; as indicated the original dimensions of the film are a
and Ú. The surface tension of the soapy water is T.
a
Ú
F
Fig 2.15 Diagram for Q2.2
PM2: Surface Tension
21
Suppose that to stretch the film at a constant speed a uniform force F equal (and opposite) to the force associated with
surface tension is applied.
Since the film has two surfaces, the relation between F and T is
F
=
2Ú T .
Calculate the total work done in increasing the distance a by an amount b, and show it is proportional to the change
in area of the soap film.
2-7 MORE ON CAPILLARITY
The law quoted in 2-4 can be derived theoretically as follows. Ask yourself first, why should water
rise up inside the tube? It is an effect of the surface tension at the top of the water column,
particularly where it meets the glass wall.
Glass
molecules
Water molecules
Fig 2.16 Interaction of water and glass molecules
Each water surface molecule exerts forces on those near it Since there is equilibrium the last
water molecule must also have a force exerted on it by the glass molecule near it. Therefore, all
around the top of the water, the glass is exerting a force on the water. Because is so happens that
water wets glass so well, this force is a vertical force.
So that it why the water rises in the tube: because the glass is pulling it up. The length of the
line of contact between the water and the glass is 2p times the radius of tube, so the magnitude of the
upward force is:
=
T ¥ (2p radius of tube)
=
2 p rT.
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. Thus there is a downward force equal to r (pr2h) g.
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
In the experiment with soil, we found that for the coarse grained soils (radius of soil grains ~ 0.3!mm) after
a long time the water finally stopped rising at a height of ~ 150 mm.
Although soil is by no means a series of uniform bore capillary tubes, it cannot be too bad an approximation to
apply the above relation. Apply the relation and find how much error is in fact introduced.
PM2: Surface Tension
2-8 ANGLE OF CONTACT
The angle of contact is defined to be the angle between the surface of the liquid and the solid
surface at the point of contact.
tangent
ANGLE OF f
line
CONTACT
Fig 2.17 Angle of contact for a liquid that does not "wet" the solid surface
You will observe that for a water-glass contact, as in the next diagram, the angle of contact is
much smaller;
angle of
tangent
contact
line
small
f
Fig 2.18 Angle of contact for water-glass contact
for mercury-glass, as in the next diagram, it is almost 180°.
angle of
tangent
contact
line
large
f
Fig 2.19 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°.
When the angle of contact is not 0° or 180°, the angle explicitly enters those equations which
directly or indirectly involve the force exerted by a solid on a liquid due to surface tension.
22
PM2: Surface Tension
23
Forces associated
with surface tension
Angle
f
Fig 2.20 Close up of part of 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 horizontal component, T sin f (which for f equal
to 0˚ or 180˚ is zero), which results in a usually imperceptible distortion of the wall. 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.
So the equation for capillary rise that we wrote is not complete. The general form is
2Tcosf
h =
rgr
For clean glass-water contacts f ª 0 and cos f ª 1. So the equation was suitable for water in
a clean glass tube.
Q2.4
For mercury-glass we saw f ª 180° and we know that cos 180° = -1. The formula for capillary height will
therefore have a minus sign in it. Does this mean that if you put a glass tube in mercury the level of the
surface would be lower inside the tube?
PM2: Surface Tension
2-9 SCALING QUESTIONS
Q2.5
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. Take the length to be 40 mm
and the width 0.50 mm.
(ii)
Calculate its weight (the density of iron is 7.8 ¥ 103 kg.m-3).
(iii)
Now assume it is on top of the water with an angle, f, as shown.
Needle
f
Fig 2.21 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).
(iv)
Can the weight of the needle be supported?
(v)
How does the angle of contact depend on the weight?
(vi)
Now assume the "needle" is 4 m in length and 5 cm thick.
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.
24
25
PM2: Surface Tension
<< 2-10
CHROMATOGRAPHY
Chromatography is a technique for separating out the chemical constituents of mixtures.
useful in biological contexts. There are two commonly used forms.
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. Next the paper is touched to a reservoir of some solvent which will dissolve the chemical substance
you hope to detect. 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. However, different chemical substances
adhere more or less strongly to the paper (i.e. the surface tension between the surface of the solution and the fibres of
the paper differs) and so different chemical substances are carried along at different rates. 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.
Colour Chromatography
(This is the experiment we filmed.) Here the solvent is put on top of the
mixture, and allowed to flow through a plug composed of grains of cellulose. Again, the adhesion between the
chemical constituents of the sample (spinach leaf) and the cellulose grains is different and they all sink at different
rates. 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.
>>
2-11 VALUES OF SURFACE TENSION
Here are the values of surface tension of some common liquids. They are listed here
merely for the purpose of showing you what range the values of surface tension can
have
.
Liquid
Surface Tension/N.m-1
water (20°C)
0.073
water (100°C)
0.059
alcohol
0.022
glycerine
0.063
turpentine
0.027
mercury
0.513
2-12 REFERENCES
"Surface tension in the lungs"
Scientific American, p 120, Dec 1962.
"Synthetic detergents"
Kushner & Hoffman, Scientific American, p 26, Oct 1951.
26
PM3
HYDRODYNAMICS
Some fish are minnows
Some are whales.
People like dimples.
Fish like scales.
Some fish are slim,
And some are round.
They don't get cold,
They don't get drowned
But every fish wife
Fears for her fish.
What we call mermaids
and they call merfish.
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. The distinction between
smooth and turbulent flow is investigated.
Minimum Learning Goals
When you have finished studying this chapter you should be able to do all of the following.
1.
Explain, interpret and use the terms:
thrust force, lift force, streamline, turbulence.
2.
(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.
(ii)
Draw a diagram showing the origin of thrust and lift forces in such situations.
(iii) Explain why it is preferable in discussing liquid flow, to consider the liquid as a
continuous substance rather than individual molecules.
3.
Describe how energy is dissipated in turbulent motion.
4.
(i)
Recall the definition of Reynolds number
vLr
R= h
and state how L is determined in different situations.
(ii)
Recall that it is experimentally found that turbulent flow occurs if R ≥ 2000.
(iii) Do simple calculations and interpretations involving Reynolds number.
5.
(i) Explain how, for streamline motion in a tube (or channel) of variable cross-section, the
flow speed depends on the cross-sectional area. (Equation of continuity.)
(ii)
Give a quantitative description of the branching effect at pipe junctions.
(iii) Explain why flow speed must increase where streamlines are crowded together.
6.
(i) Use an energy argument to explain why, for constricted streamline flow, the fluid
pressure decreases as the flow speed increases. (Bernoulli's Principle.)
(ii) Describe and explain the following phenomena : the venturi effect, the chimney effect, the
working of an atomiser.
PM3: Hydrodynamics
27
PRE-LECTURE
Recall the following background information from earlier chapters, particularly chapters FE3, FE4
and FE5.
(i) All fluids (liquids and gases) exert a pressure on the walls of any container which contains
them - pressure being defined as force per unit area. This same pressure is exerted by each
part of the fluid on neighbouring parts. The kinetic theory is a school of thought which
seeks to understand how this pressure arises through the collision of individual molecules of
the fluid with the walls and with each other. We will not in fact pursue kinetic theory any
further - but concentrate on experimentally observable laws concerning fluid pressure and its
effects.
(ii) The simply established laws concerning fluid pressure are these:
(a) The pressure at any point in a fluid is the same in all directions (Pascal's Principle).
(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.
p = constant + rgh.
It will be the concern of this lecture to establish how the pressure varies inside a fluid which is
in motion.
(iii) In general, mechanical forces can be classified as either dissipative or conservative forces,
according to whether or not they result in the dissipation of energy (usually as conversion into
thermal energy). Typically forces such as electromagnetic or gravitational are conservative and
frictional forces are dissipative.
Workers in hydrodynamics (or aerodynamics) try to classify pressures and fluid forces
similarly. However since the origin of these effects has a more complicated microscopic
explanation, this classification is not always so straightforward. The basic criterion employed
is whether or not the equation of conservation of mechanical energy is obeyed.
LECTURE
3-1 THRUST AND LIFT FORCES
The study of hydrodynamics involves the study of the interaction of fluids and solid bodies. Three
apparently different kinds of interaction can be distinguished:
(a) moving fluids with stationary objects
(b) stationary fluids with moving objects
and (c) moving fluids with moving objects
From work you have done already you can understand in a general way where the forces of
interaction come from.
(a) A moving fluid exerts a force on a stationary object because each molecule of the fluid, on
bouncing, is accelerated by the solid. The solid exerts a force on the fluid.
Before collision
After collision
Force exerted on molecule
Figure 3.1 Collision of fluid molecules with a solid surface
28
PM3: Hydrodynamics
and, by the fact that forces occur in pairs, the fluid exerts an equal and oppositely directed
force on the solid.
Force exerted on solid
Figure 3.2 Force exerted by fluid molecules on solid surface
Wind direction
!WING
RESULTANT
LIFT FORCE
Deflected
air stream
Figure 3.3 Lift force exerted by horizontal wind on an inclined wing
Examples: Hovering birds, gliders, kites.
(b) A moving solid exerts a force on a stationary fluid by exactly the same mechanism, by
giving a velocity to (i.e. accelerating) each molecule of the fluid.
Example : Fish tails.
Tail
Motion
of tail
Wake
direction
! RESULTANT
THRUST FORCE
Pivot
Figure 3.4 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.)
From all this we want to draw two simple conclusions:
(1) This way of analysing things is too simplistic. 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.
(2) The physics of what happens is the same whether it is the fluid or the solid or both which
is moving.
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).
PM3: Hydrodynamics
29
3-2 STREAM LINES AND TURBULENCE
The preceding analysis is obviously too simplistic as can be seen from a very easy observation.
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.
Demonstration
Glycerine solution flowing in a flow tank.
Figure 3.5 Streamlines for fluid flowing past a solid obstacle
Since the stream is diverted (accelerated) the wall must be exerting a force on the fluid, and the
fluid on the wall. 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. That means the fluid pressure must
increase near the corner. More of this later.
Whatever the means whereby force is exerted on the wall, it is clear that for some parts of their
motion particles of the fluid do not travel in straight lines but in curved paths.
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. Particles of this
continuous fluid can be considered to travel along these smooth continuous paths which are given
the name streamlines. These stream lines can of course be curved or straight, depending on the
flow of the fluid.
This continuous substance can be regarded as being made up of bundles or tubes of
streamlines. 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.
Streamline motion is not the only possible kind of fluid motion. When the motion becomes
too violent, eddies and vortices occur. The motion becomes turbulent.
Demonstrations
Wakes of boats
Liquid tank demonstration.
Turbulence is important because it is a means whereby energy gets dissipated.
When a body is moved through a stationary fluid in streamline motion some kinetic energy is
given to the fluid, but only temporarily. When the body has passed, the fluid is still again; no net
energy has been given to it.
But when turbulence is established, a net amount of kinetic energy is left in the fluid after the
body has passed.
Application
This is very important in aeronautical engineering. Air turbulence means increased fuel consumption in
aircraft, and many cunning and intricate devices are used to reduce turbulence.
The shape of a body will, to some extent, decide whether it will move through a fluid in
streamline or turbulent motion.
Demonstration
Shapes of marine animals, specially shaped corks.
30
PM3: Hydrodynamics
3-3 REYNOLDS NUMBER
What factors determine whether a fluid will flow in streamlined or in turbulent motion? You
could guess some of these more or less easily.
(i) Speed of flow - faster flow gets turbulent more easily.
(ii) Stickiness of fluid - thick, sticky liquids like glycerine become turbulent less easily than
thin liquids like water. [Just what physical quantity is involved here is not obvious. It is called the
kinematic viscosity and we cannot say anything about it till next lecture. The symbol for it is h/r
(see post lecture).]
(iii) A more unexpected result which turns up is that the size of the system is important.
For water flowing at the same speed through narrow pipes, the flow becomes turbulent more
easily in the tube of larger radius.
More thorough experimental investigation will collect all these results thus. 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.
Then it is found experimentally that if this number is not too large (smaller than about 2000)
the motion will be streamline; whereas if R ≥ 2000 then turbulence can set in.
There is no theoretical explanation of this value of 2000, it is just found to be the case.
3-4 THE EQUATION OF CONTINUITY
For fluids which are flowing in streamlined motion, what laws do they obey? Firstly there is
the so called equation of continuity:
for an incompressible fluid moving in streamline motion in a tube of variable cross-section,
the flow speed at any point in inversely proportional to the cross sectional area
1
Speed µ area .
The reason behind this is very easy to grasp. If you want a more rigorous statement, see the
post-lecture material.
One sees many applications of this. Four examples follow.
Demonstrations
(i) In flowing rivers, when going from deep to shallow, the flow speed increases (often becoming
turbulent).
(ii) In the circulatory system of the blood there is a branching effect.
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.
Low Speed
High Speed
Figure 3.6 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.
31
PM3: Hydrodynamics
Same Speed
High Speed
Figure 3.7 Y-junction with pipes of half the original cross-section
(Note: blood will clot if its speed falls too low.)
Most gases behave like incompressible fluids provided their flow speed is less than the speed
of sound. 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.
Demonstrations
Air conditioning systems must also be built with consideration for the branch effect.
Also the tube structure of the respiratory system is remarkably similar to that of the circulatory system.
In complicated patterns of streamline flow, the stream lines effectively define flow tubes. So
the equation of continuity says that where streamlines crowd together the flow speed must increase.
Streamlines close together:
speed high
Aerofoil
Streamlines spread out:
speed low
Fig 3.8 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.g. air) flowing through a pipe with a
constriction in it, the fluid pressure is lowest at the constriction.
In terms of the equation of continuity, the fluid pressure falls as the flow speed increases.
The reason is easy to understand. The fluid has different speeds and hence different kinetic
energies at different parts of the tube. 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.
higher speed
higher kinetic energy
lower pressure
lower speed
lower speed
lower kinetic energy
lower kinetic energy
higher pressure
higher pressure
Figure 3.9 Application of Bernoulli's Principle
The units of pressure, N.m-2, might be rewritten as J.m-3; that is, pressure is dimensionally
equivalent to work/volume.
Since the fluid is driven from regions of high pressure to those of low pressure and thus
increases its kinetic energy, we can write
kinetic energy/volume +
work/volume is constant,
l
i.e. 2 rv2 +
p = constant.
In cases where the flow is not horizontal, we should add in the gravitational potential energy/volume
l
also: 2 rv2 + p
+ rgh = constant.
This is known as Bernoulli's equation. For the very simple cases it says what we had before the fluid pressure is lowest where the flow speed is highest.
Demonstrations
The venturi effect: a fast jet of air emerging from a small nozzle will have a lower pressure than the
surrounding atmospheric pressure. You can support a weight this way:
Nozzle
Air
low
pressure
high
pressure
Figure 3.10 Example of the Venturi Effect
The chimney effect: just the venturi effect being used to suck material up. [Note in most automobiles,
petrol is sucked into the carburettor in this way.]
32
PM3: Hydrodynamics
33
Atomiser: This same effect makes atomisers and spray guns work.
Nozzle: low pressure
High pressure
(Atmospheric)
Figure 3.11 An "Atomiser"
It is most important that the free surface of the liquid should be open to the atmosphere, else the high
pressure outside the container and the low pressure inside will result in the container being crushed. {Fly
sprays always have a small air hole.]
A spinning ball or cylinder moving through a fluid experiences a sideways force.
There is a high pressure on one side (so a big force) and low pressure (small force) on the
other. The ball experiences a net sideways thrust. This is one of the ways players can get cricket or
ping pong balls to swerve.
Demonstration
Spinning cylinder.
POST-LECTURE
3-6 MORE ON REYNOLDS NUMBER
There are several points to note about the definition of the Reynolds Number.
(a) It is not a precise physical quantity. The quantities L and v are only typical values of size
and speed. It is often not possible even to say which length you are talking about. For a body
moving through a fluid it might be either length or breadth or thickness - or any other dimension
you might think of. For a fluid flowing through a channel or a tube, it turns out that it is the
diameter of the tube which enters. 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.
This imprecision in its definition reflects the fact that the basic physical law is itself rather
vague - indeed it can often only be stated as we did: "The flow of fluid in a system is more likely to
be turbulent if the system is large, than if it is small". It is not surprising then that the magic number
of 2000 is also only rough.
b) The "stickiness" index, the kinematic viscosity, is given the strange symbol h/r for the
following reason. There are many ways in which this "stickiness" or viscosity manifests itself.
Basically, how fast the fluid flows determines one measure of stickiness known as the coefficient of
viscosity (h) - see next lecture. 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. It is pointless to say
any more at this stage, except to give units.
h is measured in units of Pa.s; to give you a feeling for what numbers occur, for water h ~ 103 Pa.s.
(c) The Reynolds number is a dimensionless number as you can see from its definition:
[R] =
[m.s-1][m][kg.m-3]
[Pa.s]
This will be understood when you come to see where the Reynolds Number comes from. It is
a ratio of two quantities - essentially a scaling number.
34
PM3: Hydrodynamics
Q3.1: Work out the Reynolds Number for the following flow systems, and say in which ones you might expect
there to be a lot of energy dissipated through turbulence.
(i) A Sydney Harbour ferry
(ii) Household plumbing pipes
(iii) The circulatory system. [Take an average sort of figure for the flow speed of blood to be 0.2 m.s-1 the
diameter of the largest blood vessel, the aorta, to be ~ 10 mm; and guess that the viscosity of blood probably
is not very different from that of water.]
(iv) Spermatozoa swimming. [They are typically about 10 µm in length with speeds of about 10-5m.s-1.]
3-7 CONTINUITY
A careful derivation of the equation of continuity goes like this.
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 3.12 Figure for derivation of Equation of Continuity
The volume of fluid flowing past A in a very small time ∆t =
A1v1Dt.
So the mass which flows past A is r1 A1v1Dt . Similarly the mass of fluid flowing past B in
time ∆t is r2 A2v2∆t .
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 2 A2v2Dt .
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.
Notice that for the final statement to be true, incompressibility is important. But notice also
that if the fluid is approximately incompressible, i.e. if its density never changes by very much, then
the equation of continuity, as we quoted it, is approximately true.
The quantity appearing in this equation, Av, measures the volume of the fluid which flows past
any point of the tube divided by time. It is given the name volume rate of flow, and is usually
denoted by the symbol q. See, for example, Poiseuille's law on page 43.
Q3.2
From observations you have made, either from the TV screen or in the real world, draw in the stream lines
for a liquid flowing in streamline motion through a drain with a corner in it.
PM3: Hydrodynamics
35
Figure 3.13 A drain with a corner in it!
Use continuity to decide where the flow speeds up, and when it slows down.
You cannot apply this to water flowing around a bend in the river. A Reynolds number calculation shows that the
situation is quite different.
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. It goes along the same lines as the proof of the equation of continuity. 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. Bernoulli's equation only really applies when the motion is
strictly streamline.
Nonetheless, provided there is not too much turbulence, the law will approximately apply.
Certainly, in all of the experiments we did on screen the flow must have been pretty turbulent, yet
they all showed the characteristic effect of pressure drop.
Q 3.3 Medical textbooks often quote Bernoulli's equation simply as
p + rgh =
constant
l
implying that the kinetic energy term ( rv2 ) is not important but the gravitational potential energy term is.
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.
Q 3.4 In section 2 above, you analysed streamlined flow round a corner. Using the result of that analysis show
how the pressure changes as the liquid goes round the corner. Can you reconcile this with the kind of
simple minded diagrams drawn for the lift force on wings drawn in figure 3.3 above?
Q 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. Discuss.
36
PM4
VISCOSITY
Come crown my brow tin leaves of myrtle
I know the tortoise is a turtle.
Come carve my name in stone immortal,
I know the turtoise is a tortle.
I know to my profound despair
I bet on one to beat a hare.
I also know I'm now a pauper
Because of its tortley turtley torpor.
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. The coefficient of viscosity will be defined. 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.
1.
Explain and use the following terms: shear stress, velocity gradient, viscosity, newtonian
liquid.
2.
(i) Describe an experiment that shows qualitatively a relationship between shear stress and
velocity gradient.
(ii) Define the coefficient of viscosity in terms of this relationship (Newton's Law of
viscosity).
3.
4.
(i)
Identify the unit of viscosity as 1 Pa.s.
(ii)
Recall that the coefficient of viscosity for water is about 1 ¥ 10- 3 !Pa.s.
(i)
Describe the different response of liquids and solids to an applied shear stress.
(ii) State Newton's law of viscosity in terms of shear stress and the rate of shear
deformation.
5.
(i) Explain qualitatively the dependence of the rate of streamline flow of liquid in a pipe on
pressure difference, pipe length, pipe radius and the coefficient of viscosity of the liquid.
(Poiseuille's equation.)
(ii)
Use Poiseuille's equation, when quoted, to do simple calculations.
(iii) Describe three phenomena, including both water pipes and the human body, which relate
to Poiseuille's equation.
6.
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.
7.
(i)
Explain how energy is dissipated by viscosity.
(ii) Use the Reynolds number to determine whether or not viscous dissipation of energy is
important in simple systems.
PM4: Viscosity
37
PRE-LECTURE
Keep in mind two particular points that have been made so far in these Properties of Matter lectures.
(i) The definition of the Reynolds number, and its importance essentially as a scaling number. 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 same number will turn up again to decide whether
or not the flow is viscous. 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.
(ii) The definition of the shear deformation and Hooke's Law as it applies to bodies which behave
elastically under shear stresses. As we have pointed out it is the behaviour of a substance
under shear which essentially distinguishes between a solid and a liquid . A solid (usually)
has a large shear modulus, i.e. if you try to deform it (in shear) it will deform, but then return
to its original shape afterwards. A liquid has a very, very small shear modulus. You can slide
one bit of a liquid past another bit, and there will be no noticeable tendency for the two to
regain their original shape when you shop pushing. Nonetheless the sliding of one bit does
have an influence on the other, and this is what viscosity is all about.
(iii) Also you should recall discussion of electrical resistance and the various mathematical
techniques of working with it (like Ohm's Law and Kirchhoff's Theorem). 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.C. circuits.
(iv) Recall also the meaning of the word gradient. If some quantity (say pressure) varies with
distance (x), being big at some point and small at another, we say a pressure gradient exists.
dp
A measure of this is the derivative dx ; or, more crudely, the ratio:
difference!in!pressure!at!2!points
distance!between!those!2!points .
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.
LECTURE
4-1 VISCOSITY
A feature which distinguishes one liquid from another is their "thickness" or the ease with which
they pour.
Demonstration
Observe the flow of water, glycerine, oil, treacle, lava, pitch.
<< This last experiment is on show in the Physics Department, University of Queensland.
experimental record is:
1920
Pitch poured in funnel
1938 (Dec.) First drop fell
1947 (Feb.) Second drop fell
1954 (Aug.) Third drop fell
1962 (May)` Fourth drop fell
1970 (Aug.) Fifth drop fell.
>>
The
The physical property which distinguishes these liquids from one another is something to do
with how well the liquid molecules adhere to one another; and this molecular adhesion leads to a
host of rather complicated effects.
38
PM4: Viscosity
Demonstration
(i) The rate at which solids fall through liquids (this has already been discussed in chapter FE4).
(ii) The spin-down effect, tea leaves in the bottom of a stirred cup migrate to the centre (not the outside as
you might expect).
(iii)
Smoke rings.
(iv)
Vortex rings in liquids.
These are all traceable (in the end) to molecular adhesion, but their explanation and connection
with one another is very complicated. However, we have to start somewhere. We must select one
physical effect to measure, and try to understand the others in terms of it. We choose to concentrate
on the existence of a velocity gradient.
When a fluid (e.g. air) flows past a stationary wall (e.g. table top), the fluid right close to the
wall does not move. However, away from the wall the flow speed is not zero. So a velocity gradient
exists.
High speed
Fluid
Flow
Low speed
Stationary Wall
Fig 4.1 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. We will use this fact to define viscosity.
Demonstration
Observe the velocity gradient in a tank of treacle.
4-2 THE COEFFICIENT OF VISCOSITY
A simple experiment set-up capable of demonstrating the law of viscosity involves a small metal
plate suspended in a tank of liquid. Before the experiment starts the weight of an attached pan is
adjusted so that the plate is neutrally buoyant - i.e. it does not tend to sink in the liquid or to rise.
Applied
Force
Liquid
F
A
D
H
Metal plate
(side view)
E
B
C
G
Fig 4.2 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. When the plate is
moving with speed v through the liquid, there will be a velocity gradient between AB and FE; also
between DC and HG. The complete apparatus is
Pointer
Scale
Weights (to provide
known force)
Glycerine
Plate (of known size)
Fig 4.3 Experiment to measure coefficient of viscosity: complete apparatus
Demonstration
(i) The speed with which the plate rises, increases as the force pulling it increases:
Mass on pan/g
Time to go 100 mm/s
2
8
5
2.6
(ii) For the same force, the speed of the plate decreases as the area of the plate increases.
For a plate of twice the surface area
Mass on pan/g
Time to go 100 mm/s
2
13.5
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.
velocity gradient = v/d
!
distance between
wall and plate = d
v
Fig 4.4 Increased velocity gradient when the plate is closer to the wall
Mass on pan
7g
Time (close to wall)
4.2 s
We interpret this result as saying that a given shearing stress sets up a velocity gradient in the
fluid.
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.
dv
s = h dx
(stress) = (constant) ¥ (velocity gradient)
The constant h is called the coefficient of viscosity, and is different for different liquids.
Water is obviously a lot less viscous than glycerine.
Though this equipment can roughly show Newton's Law to be plausible, it cannot be used for
accurate measurement of viscosity. Some common measuring devices are:
Demonstrations
(i) Commercial oil companies use simpler viscometers. These essentially measure how fast the oil pours.
The "grade" of an oil is the number of seconds it takes to pour a measured amount through a certain tap.
(ii) If not much liquid is available, or cannot be removed (e.g. protoplasm in a cell, or sap in a plant) you
can observe how fast bubbles rise or particles sink in the liquid.
(iii) Laboratories usually use a torsional viscometer, which is really a very refined version of the apparatus
we used above.
ELEVATION
Weights
Outer cylinder
(fixed)
Liquid
under test
Axle
Inner cylinder
(rotates)
PLAN
Liquid
velocity gradient
set up here
Fig 4.5 A torsional viscometer
Of course the viscosity of a liquid can change.
Demonstration
The viscosity depends on temperature, usually increasing as the temperature decreases (which is why
automobiles need different oils in hot countries than in cold countries and indeed why the engine runs more
freely as it heats up).
However, this kind of detail you can catch up on later, when you come to talk about viscous
effects in your own discipline.
41
PM4: Viscosity
4-3 ALTERNATIVE STATEMENT OF NEWTON'S LAW
Since there are many manifestations of viscosity, there are many different statements of the basic
law. We have given Newton's statement, relating velocity gradient to shear stress (pressure).
Another statement, which research workers use, specifically points up the difference between
solids and liquids.
Solids
When a shearing stress is applied to a solid it suffers a shear (i.e. a shear deformation)
x
A
C
y
B
Fig 4.6 Shearing of a solid (side view)
A solid deforms instantaneously and then stops deforming. When the shearing stress is
removed, if the solid is elastic the deformation recovers.
Liquids
When a shearing stress is applied to a liquid it suffers a shear deformation also, sometimes
slowly sometimes fast. However, so long as the shear is applied it continues to shear. When the
stress is removed, the shearing stops, but does not recover.
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.
However the velocity gradient is the same thing as time rate of change of shear deformation..
This can be seen as follows, with reference to figure 4.6:
speed
velocity gradient
= transverse!length
!dx
dt
=
y
rate of shear
=
d/dt (x/y)
=
1/y (dx/dt)
provided y is constant, as it is.
So, Newton's law can be restated
shear stress µ rate of shear deformation.
42
PM4: Viscosity
4-4 POISEUILLE'S LAW
We want to find out what effect viscosity has when fluids flow in more relevant situations.
The most important one, which is the only one we will consider is flow through a long pipe.
We could go through a mathematical analysis and apply Newton's law (of viscosity) to this
problem, but we will not. One thing however is obvious. Because viscosity puts restrictions on
velocity gradients, it must be true that liquid will flow faster through a wide pipe than a narrow one.
Demonstration
Polystyrene chips on surface of a treacle tank.
Start
Later
Fig 4.7 Velocity profile for moving treacle
[This technique of showing the velocity profile of the flow will become important later.]
In trying to find out what other factors control how fast fluids can flow through pipes, the following
factors are easy to isolate:
(i) the pressure difference between the ends of the pipe. The bigger the pressure difference, the faster will be
the flow;
(ii) the length of the pipe. More liquid will flow through a shorter than a longer pipe in the same time.
(iii) the radius of the pipe. More liquid will flow through a wide than a narrow pipe in the same time.
This dependence is very marked. Even in our rough demonstration we get 8 times as much glycerine flowing
through a pipe twice the radius of the other. (Theory says we should have got 16 times as much.)
(iv) the coefficient of viscosity of the liquid. Water flows much more easily than glycerine.
Had we gone through a mathematical analysis of the situation, we could show that Newton's
law of viscosity would give the volume rate of flow, qv , of fluid of viscosity h, through a pipe of
radius r and length l, when driven by a pressure difference ∆p as
∆p.r4!p
qv = !lh!8
This is known as Poiseuille's law.
The similarity between this equation and Ohm's law is very marked; most workers talk about
"resistance" of a pipe. See post lecture material.
Situations in which Poiseuille's law has important effects:
Examples
(i) Irrigation pipes. 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.
(ii) Pipes from Warragamba Dam. Here ∆p and l are fixed (by geography), and the volume rate of flow is
fixed by the requirements of the population of Sydney. When Sydney doubles in size, the Water Board will
have to use twice as many pipes or replace the present pipes by ones of (2)1/4 times the radius. [This is an
oversimplification, see post lecture.]
PM4: Viscosity
43
(ii) Respiratory system: The flow of gas here is also Poiseuillean. The resistance to flow is determined
primarily by the narrow tubes leading to the alveoli. Any general constriction of the pipes, as occurs in
bronchospasm for instance, increases the resistance to flow and makes breathing much more difficult.
(iv) Circulatory system: Two points are worth making.
(a) There is a decrease in pressure across each section of the tubes. Blood pressure is highest when it
leaves the heart (through the aorta) and lowest when it returns (through the inferior vena cava).
Most pressure loss occurs over the capillaries. 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). So the heart has to work harder
to compensate. And at times of stress, when an increased flow rate is required, there can be a breakdown.
(v) Urinary tract. You work out the relevant physics.
Anyhow, just remember that Poiseuille's law is essentially the same thing as Newton's law; and
if a fluid obeys one it obeys the other. There are a large number of important liquids which do not
obey these laws, and they are called non-newtonian liquids. One of the simplest ways to
recognise a newtonian liquid is to examine its velocity profile, which should be parabolic for a
liquid which obeys Newton's (or Poiseuille's ) law.
Demonstration
flow of syrup.
4-5 REYNOLDS NUMBER
Poiseuille's Law shows that viscosity is responsible for loss of pressure - and hence is an energy
dissipating phenomenon. We are not talking about energy loss due to turbulence, but about energy
loss which occurs even when the flow is streamlined. It comes about through friction between the
streamlines moving past one another.
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. Since most of this gradient occurs
near a boundary (in the so called Boundary layer, it is the ratio of the size of the system to the size
of the boundary layer which is important.
System relatively large
System relatively small
energy mostly conserved
much energy dissipated
Fig 4.8 Effect of boundary layer in velocity profile
As will be shown in the post-lecture, the ratio of total energy of flow to energy dissipated, is
very closely related to the Reynolds number which we introduced in last lecture.
So now we can appreciate another reason why the Reynolds number is important. Viscous
effects can never be neglected (i.e. the energy dissipated is appreciable) in low Reynolds number
situations: in thick liquids (h large), or in small slow flow systems. On the other hand viscous
effects will not be important in thin liquids or in large, fast flow systems. (However in these latter
systems remember that turbulence is always possible, and energy can be lost through that means.)
44
PM4: Viscosity
In general then, flow patterns will be different in systems with low and with high Reynolds
numbers. In particular, in very low Reynolds number systems, the flow is perfectly reversible since
no turbulent effects can occur anywhere.
Demonstration
The method of swimming is quite different for fishes (R ~ 10,000) and spermatozoa (R ~ 0.0001).
Modes of boat propulsion which work in thin liquids (water) will not work in thick liquids (glycerine).
It is possible to stir glycerine up, and then unstir it completely. You cannot do this with water.
POST-LECTURE
4-6 MORE ON POISEUILLE'S LAW
Poiseuille's law can be derived from Newton's law; but to go through the complete derivation, even in
post-lecture material, would be completely opposed to the philosophy of this course. If you feel you
cannot do without it, there are plenty of books you can look up. But in broad outline the derivation
follows these lines:
(i) The force trying to push the fluid through the tube is of course due to the pressure difference
∆p. The retarding force comes from viscous drag, which acts to prevent shearing. If the flow
is streamlined, then this shearing resistance acts all over the surface of imaginary tubes of
fluids concentric with the tube trough which the fluid is flowing; and thus the retarding force
will depend on the surface area of the tubes of fluid - and hence on the length of the tube.
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.
(ii) Trying to calculate how fast the fluid must flow to produce the necessary resisting force is
where Newton's law comes in. 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. In
fact the velocity profile is a parabola.
Velocity
component
Distance from axis of tube
Fig 4.9 Parabolic velocity profile
You can say therefore that the average flow speed is likely to depend on r2. And so the
volume rate of flow (equal to the average speed • area of the tube) is likely to depend on r4.
(iii) The factor p/8 you must take on faith, or work it out for yourself.
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. If turbulence occurs
than you must be very careful about using Poiseuille's law to calculate flow rates. (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.)
Q4.1
Why should turbulence mean that the volume rate of flow is less than in streamlined flow?
Q4.2
When a builder designs the drainage system for the roof of a house, what factors should influence the choice
of the size of the downpipe?
Would he be correct in basing his calculations on Poiseuille's Law
Q4.3
The experiment was done in chapter PM2 in which water rose, by capillary attraction, through two columns
of soils. It was observed that the water rose faster in the column with the coarser grains. 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 l!h
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. m-4. s-1.
It is even possible to do this when the flow is turbulent. It only means that Poiseuille's
equation is not valid, and you cannot use this explicit formula for the resistance. But it is still quite
possible to define a 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.
Q4.4
Consider water flowing along a l.0 m long pipe at a steady rate,
250 mm
pressure
5.0 Pa
A
750 mm
B
C
Fig 4.10 Data for Q4.4
If you measured the fluid pressure at point B, what value would you get?
pressure
1.0 Pa
46
PM4: Viscosity
Q4.5
Consider these two different streamlined flow systems,
B
A
C
(a)
B
A
C
(b)
Fig 4.11 Data for Q4.5
The lengths of the two pipes in the section BC are equal to the lengths AB in both cases. The radii of all three pipes
in case (a) are the same; but in case (b) the radii of the two pipes in BC are half that of pipe AB. In both
cases also, the pressure at A is 4 Pa, and at C is 1 Pa.
What is the pressure at B in case (a) and case (b)?
Can you guess from these two answers, the answer to the question posed in the lecture notes: "in the circulatory
system, why does most pressure drop occur over the capillaries?"
4-8 MORE ON REYNOLDS NUMBER
In the lecture, we talked about the ratio:
Total!energy!of!flow!(per!unit!volume)
Energy!dissipated!(per!unit!volume)
We can easily evaluate this ratio, since the energy dissipated is just the work done by the
viscous forces divided by unit volume. And, by the same arguments used in 4-5 of chapter PM3,
this must be equal to the pressure drop. Hence this ratio
1
1
2
2
2!r!v
2!r!v
=
Dp ~
h!dv/dx (by Newton's law)
Now, in a system where the boundary layer is comparable in size with the scale length of the
system (L), we can approximate the velocity gradient in this expression by
dv
v
~
dx
L
r!vL
and therefore the above ratio is
=
!2h
which, apart from the factor 2, is just the definition of the Reynolds number
47
PM4: Viscosity
Q4.6
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.
4-9 VALUES OF VISCOSITY
As pointed out in last lecture, and as can be checked from the equation in 4-2, the units of viscosity
are those of (pressure) ¥ (length)/(speed) or Pa.s. Some books of tables quote numbers in the old
cgs unit - the poise. You can easily convert by remembering that
1 Pa.s = 10 poise
Purely to show you what range the values of viscosity can have, here are some common fluids:
FLUID
VISCOSITY / Pa.s
water (20°C)
1.0 ¥ 10-3
water (100°C)
0.3 ¥ 10-3
alcohol
1.2 ¥ 10-3
glycerine
1.5
mercury
1.8 ¥ 10-3
air
1.8 ¥ 10-5
48
PM5
RHEOLOGY
Elephants are useful friends
Equipped with handles at both ends.
They have a wrinkled, moth-proof hide;
Their teeth are upside down, outside.
If you think the elephant preposterous
You've probably never seen a rhinosterous.
OBJECTIVES
Aims
In this chapter you will look at how many liquids have viscosity coefficients that change with the
applied shearing stress. This is called non-newtonian behaviour. Visco-elastic effects, which
combine fluid and solid properties are also discussed.
Minimum Learning Goals
When you have finished studying this chapter you should be able to do all of the following.
1.
Explain and use the following terms: rheology, non-newtonian flow, pseudo-plasticity,
dilatancy, plasticity, thixotropy, visco-elasticity, creep, stress relaxation.
2.
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.
3.
Describe an experiment showing the different effect of shear stress on a newtonian fluid (e.g.
glycerine-water mixture) and a non-newtonian fluid (e.g. starch solution).
PRE-LECTURE
1.
Refer back to chapter PM1 which discusses the behaviour of purely elastic materials. Note in
particular that for these materials Hooke's Law is obeyed i.e. the stress is proportional to the
strain.
2.
Refer back to chapter PM4 which discusses viscous forces, forces which show themselves
when a fluid flows. 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. Note as
well that this law can be expressed alternatively as the shear stress is proportional to the rate of
shear.
LECTURE
5-1 NON-NEWTONIAN FLUIDS
Rheology is the study of how bodies behave under the action of deforming forces. As the word is
usually understood it deals with all materials except the purely elastic and those which are purely
viscous and obey Newton's law. Most materials do not fall into these two extreme categories - in
fact strictly speaking none do if the relevant parameters are varied widely enough. In this lecture we
are going to look at some of the more complicated behaviour one can get.
To start we consider the fluids that do not obey Newton's law - the non-newtonian fluids. For
these, the shear stress is not proportional to the shear rate and thus their viscosities depend on shear
rate. Basically, they fall into two different classes. Firstly there are the pseudo-plastics for which
the coefficient of viscosity decreases as the shear rate increases.
49
PM5: Rheology
Demonstration
A solution of starch is pseudoplastic. Its pseudoplasticity can be easily demonstrated by letting it flow
from a burette and comparing its flow with that of a suitable viscosity newtonian fluid flowing from an
identical burette. (The newtonian fluid used was glycerine and water.)
At the beginning the starch flowed faster than the glycerine; as both fluids started to slow, the starch
slowed down more than the glycerine; finally the level of glycerine solution fell below that of the starch
solution.
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.
(iii) for a newtonian liquid (the glycerine-water mixture) the viscosity, that is the relation between
the shear stress and the rate of shear is constant; as the burette emptied the shear stress (and
the pressure) reduced while the rate of shear (and the flow rate) reduced correspondingly.
(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). The increase in viscosity is
shown by the disproportionate decrease in the flow rate.
(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.
glycerine-water
(newtonian)
Viscosity
coefficient
starch solution
(pseudo-plastic)
Shear rate
Fig 5.1 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.
Examples
One example of a dilatant fluid is a thin paste of cornflour.
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.
Apart from these two main categories there is one other type of non-newtonian fluid which has
the complicating feature that it does not flow, it does not behave as a fluid until a certain stress, the
yield stress, has been exceeded. Once this stress has been exceeded, the viscosity either remains
constant or decreases as the shear rate increases. These materials are called plastic.
Examples
An example of a plastic material is toothpaste. It is necessary that toothpaste will stay on the brush but it
must be easily extrudable.
Some other examples of plastic materials of which there are many, are good brushing paints and sewage
sludge.
50
PM5: Rheology
The flow characteristics of the different non-newtonian fluids may be summarised thus:
plastic
Shear
stress
pseudo-plastic
newtonian
dilatant
Shear rate
Fig 5.2 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 newtonian fluids this radial velocity profile is parabolic.
For non-newtonian fluids the radial velocity profile is not parabolic. It is somewhat sharper
for dilatant fluids and for pseudoplastics it is blunter. For plastic materials there is a completely flat
region in the centre where the shearing stress is less than the yield value.
Demonstration
The protoplasmic flow in the plant known as "slime mould" has a distinctly non-parabolic radial velocity
profile.
Before leaving non-newtonian fluids, one further complication needs to be mentioned viz. that
there are some fluids whose viscous behaviour depends very much on the time they have been
sheared and the time they have been at rest. There are two types of these. The thixotropic fluids
are like the pseudo-plastics in that the viscosity decreases with increasing shear rate but as well show
the property that at constant shear rate the viscosity decreases with time. 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.
The other type is the rheopectic fluid which is akin to the dilatant fluid in the same way that
the thixotropic fluid is akin to the pseudo-plastic fluid.
It is important to realise that there is basically no difference between a thixotropic and a
pseudo-plastic material. It is just that for a pseudo-plastic material the characteristic time is so small
as to be not observed in normal circumstances.
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.
For a pseudo-plastic material, the return to its original state would be instantaneous.
Demonstration
This was shown by placing a paste of plaster of Paris on an inclined plane. It did not flow appreciably
under these circumstances but when the table was vibrated it flowed freely. On stopping the vibration,
however, the flow ceased instantaneously. If the material on the vibrating table has been "quick-clay", which is
thixotropic, the flow would have continued when the vibration ceased. This would then have been a
demonstration of what happens in certain earthquakes where buildings are destroyed because they were built on
"quick-clay".
51
PM5: Rheology
5-2 VISCO-ELASTIC MATERIALS
Having discussed the effect of time on non-newtonian fluids we are led naturally to the other
major class of materials covered by the subject of rheology, viz the visco-elastic materials - the
materials which are neither purely elastic nor purely viscous, materials which show the properties of
both solids and liquids. Whether these behave as solids or liquids depends on how long the stress
is applied.
One way of distinguishing between an elastic solid and a viscous liquid is to apply a stress and
maintain this stress. 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. But many
materials we normally class as solids do not behave like this.
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.
This phenomenon is known as creep.
[Since it flows one can of course ascribe a viscosity to such a solid. Conventionally a material
is taken to be a solid if its viscosity is >1014 Pa.s. At such a viscosity a 25 mm edge cube would
support a man for a year and sink only 2.5 mm.]
Another way to distinguish between a solid and a liquid is to produce a strain and maintain that
strain. For a solid a certain stress is required to produce the strain, and to maintain the strain the
stress must continue at this value. For a liquid, however, the stress instantaneously goes to zero once
the strain is produced.
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. Rather, the stress to maintain a given strain gradually drops
as shown in the following diagram:
Elastic
behaviour
Viscous
behaviour
Stress
Strain
Time
From here the strain is kept constant;
the axis now represents time
Fig 5.3 Stress-strain curve behaviour for a wool fibre. The curve shows elastic (solid) and
viscous (liquid) effects
This phenomenon is known as stress relaxation. The time for the stress to drop by e-1, (e is
the exponential function) is known as the stress relaxation time, t. Conventionally, a material is
called a solid if t > 104 s, and a liquid if t < 10-4 s. The materials in between are the visco-elastic
materials.
52
PM5: Rheology
There are many visco-elastic materials for example pitch, wool fibres, nylon, silk, vulcanised
rubber and bakers' dough.
Demonstrations
A good example is the material known as "silly" putty. If this is stressed in short times, it behaves
elastically - thus it will bounce. If, however, a stress is applied over a long time, it will flow - it behaves as a
viscous liquid.
Another example is egg white. This will flow but it also shows elastic properties in that it recoils if the
flowing stream is broken or cut.
We can see that the visco-elastic materials behave as solids when they are stressed in short
times but as liquids when the stress is applied over long periods.
<<
An interesting property of visco-elastic materials is the phenomenon known as the Weissenberg
effect.
Demonstration
This is the effect that if a rod is rotated in a large mass of material, the material will climb the shaft. This is
observed when cake mix is mixed with a rotating beater.
>>
5-3 SOILS AND CLAYS
A group of materials whose rheological properties are very important are the soils and clays which
compose a large part of the surface of the earth. These materials are quite complicated. They are
two-phase materials consisting of solid and liquid (usually water) and their properties depend very
much on the concentration of water. The clays are particularly complex in that their properties
depend markedly on the presence of electrolytes.
Let us first consider these materials in the natural state, in the ground, where they are of
importance to people like engineers who are interested in their weight-bearing properties.
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.
Demonstration
This is shown by maintaining a constant head of water over a bed of sand in a sheet pile structure and
injecting dye into the top of the sand on the high pressure side.
Water
Dye
lines
Sand
Fig 5.4 Flow of water through wet sand
The dye follows well-defined lines which can be predicted mathematically. The dye lines show the velocity
is fastest where the cross-section is narrowest, and thus illustrate the equation of continuity.
As the water concentration in a soil or clay increases its properties change markedly. 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.
Demonstration
The reduction in the load-carrying capacity of a sand as the water content increases was shown.
53
PM5: Rheology
Model
building
Level
gradually
raised
Water
Sand
Sand
Fig 5.5 Effect of increasing water content in wet sand
Clays are not only important in the ground. They have found widespread application
particularly in the ceramic industry. A method of ceramic manufacture known as "slip-casting"
makes use of the thixotropy of certain clay-water mixtures and also of the pronounced effect of
electrolytes on the viscosity of these.
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. A low viscosity mixture with very high clay content can be made, however, by
adding a small volume of electrolyte (sodium silicate and soda ash). This process is known as deflocculation
(the aggregation of the clays particles is greatly affected by the electrolyte).
If a deflocculated clay-water suspension is placed in a plaster mould, the water is absorbed into the mould.
As the water leaves, the wall of the casting gradually builds up. In the centre of the casting is a thixotropic
clay-water mixture which, when the casting wall is sufficiently great, can be rendered liquid by agitation and
poured off.
5-4 BLOOD
Another material of obvious importance is blood. This has most interesting rheological
properties.
Blood is a complex fluid, consisting of a plasma in which are suspended a variety of cells, the
predominant ones being the red cells.
Demonstration
To measure its viscous properties, special rotating viscometers have to be constructed and used, which
work with very small samples.
Measurement show that blood is thixotropic. At high shear rates, normal blood has a
viscosity of between 5 and 6 times that of water; but at low shear rates it may be several hundred
times that of water. The viscosity for people with certain diseases such as myocardial infarction and
thrombosis is much higher, particularly at low shear rates.
Demonstration
The data presented on the screen, is taken from many patients at Sydney Hospital:
54
PM5: Rheology
10
Viscosity / Pa.s
INFARCTION AND
THROMBOSIS
NORMAL MEN
1
0.1
0.01
0.01
0.1
1
10
100
0.01
0.1
1
10
100
Rate of shear / s-1
Fig 5.6 Blood flow data
Notice the similarity between these figures, and figure 5.1.
Blood's viscous behaviour is partly due to aggregation of the red cells. These aggregates can,
and do, form when the shear rate is low, but at higher shear rates they break up, giving a lower
viscosity.
This however is not the whole story - the viscosity of the interior of the red cells plays a major
part. If the red cells were rigid particles, when their concentration reached 65%, blood would have
the consistency of concrete. This however does not happen.
Blood is still very fluid even at 99% red cell concentration. The reason for this is that the red
cells are not rigid but in fact fluid. Thus it is that blood can flow in the small capillaries - the cells
deform as they flow. Any condition which leads to more rigid red cells leads to a much greater
blood viscosity.
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. These are of a
roller type. 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.
One final aspect to mention is the effect of drugs on the blood flow. These in general affect
the flow just by dilating or contracting the blood vessels.
Demonstrations
Thus it is that when a cigarette is smoked, there is a short term reduction in blood flow due to the nicotine
constricting 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.
It is thought, however, that as well, prolonged ingestion of nicotine has a long term effect
leading to an increase in aggregation of cells and hence a higher viscosity and reduced blood flow.
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.
Thus it is that some non-newtonian fluids also show pronounced visco-elastic behaviour. Further
there are no sharp dividing lines between the categories, particularly as regards their time behaviour.
Thus it is that the terms thixotropic and pseudoplastic are often used almost interchangeably.
5-6 PROBLEM
Q5.1
List any materials of everyday experience, other than those given here, which you think are visco-elastic or
non-newtonian. Categorise them and say why you think they fall into these categories.
5-7 REFERENCES
"The emergence of rheology"
Markovitz, Phy. Today , p 23, April 1968.
"Quick clay"
Kerr, Sci. Amer., p 132, November 1963.
"The flow of matter"
Reiner, Sci. Amer., p 122, December 1959.
"Non-newtonian viscosity and some aspects of lubrication"
Stanley, Phy. Educ., p 193, 1972.
56
PM6
FRICTION
At midnight in the museum hall
The fossils gathered for a ball.
There were no drums or saxophones
but just the clatter of their bones,
A rolling, rattling, carefree circus
Of mammoth polkas and mazurkas.
Pterodactyls and brontosauruses
Sang ghostly prehistoric choruses.
Amid the mastodonic wassail
I caught the eye of one small fossil.
Cheer up, sad world, he said, and winked it's kind of fun to be extinct.
OBJECTIVES
Aims
In this chapter you will study the phenomena of friction, determine the laws of friction and consider
the explanation of these laws in terms of a microscopic model. The effects of naturally occluding
surface layers on of lubricants on friction are discussed.
Minimum Learning Goals
When you have finished studying this chapter you should be able to do all of the following.
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.
2
(i)
Recall the experimental laws of sliding friction.
(ii)
Do simple calculations based on the second of these laws.
(iii) Describe an experiment to verify these laws.
3
(i)
Using a microscope model, explain the laws of friction.
(ii) Describe how electrical measurements and radioactivity measurements can (separately)
be used to confirm parts of this explanation.
4
Describe and explain how surface layers alter the value of the coefficient of friction.
5
Describe the process of polishing and explain how it is based on the fact that frictional forces
are non-conservative.
6
State the differences among three types of lubrication (hydrodynamic, boundary and
elasto-hydrodynamic) and give one example of each type.
PRE-LECTURE
1.
Remind yourself of the concept of a shear stress (Chapter PM1).
2.
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).
3.
Remind yourself of the basic concepts of the flow of a viscous liquid and in particular of
Newton's law of viscosity (Chapter PM4).
PM6: Friction
57
LECTURE
6-1 OUR EVERYDAY EXPERIENCE OF FRICTION
Demonstration
To slide heavy objects requires considerable effort. The heavier the object the greater the effort.
Fig 6.1 Sliding an object
Demonstration
It is a lot easier to move a heavy object by rolling it than by sliding it.
Fig 6.2 Rolling an object
We can express these facts by saying that there is a friction force resisting the motion. The
force for rolling friction is less than for sliding friction.
Demonstrations
The force of sliding friction depends markedly on the surfaces involved. Friction is very small in skating
and skiing.
Friction plagues us in many contexts.
Demonstration
Considerable power is lost in overcoming friction in engines. Even more important than the power loss
is the wear which results from friction.
Lubricants can be used to reduce friction and wear. Early applications were the use of animal fats to
lubricate chariot wheels.
Though friction plagues us a lot, it is of considerable advantage in many contexts.
Demonstrations
Friction of a rope on something or on itself as in a knot enables large loads to be easily controlled.
Friction enables the braking of moving vehicles.
Traction of cars, bikes, trains etc. depends on friction.
Traction in walking depends on friction.
6-2 LAWS OF SLIDING FRICTION
The laws of sliding friction were first formulated by Leonardo da Vinci and re-discovered in 1699
by Admontons. They are empirical laws which give the dependence of the friction force on the
relevant parameters.
Demonstration
One apparatus which can be used to find these laws is a tilting board on which an object is placed. The
board is tilted until the object slides. From the measured angle the friction force, the tangential force resisting
the motion, can be deduced. (See post lecture material.)
PM6: Friction
Fig 6.3 Sliding down an incline
The laws were, however, determined using the following apparatus:
Fig 6.4 An experiment to determine the laws of friction
A steel block was placed on a horizontal sheet of steel and connected via a string which passed over a
pulley to a weight-carrier which hung vertically. Weights were added to the carrier until the block was just on
the point of sliding. The weight was recorded, this being the friction force.
Friction force with 1 block = 4.5 N
(steel on steel)
The block was then turned over onto another face of smaller area and the measurement repeated. The
weight was the same as before.
Thus we have:
First Law of Friction.
The friction force Fr is independent of the area of contact.
The experiment was then repeated with 2 and 3 blocks of the same weight with the following results:
Friction force with 2 blocks = 10 N
(steel on steel)
Friction force with 3 blocks = 15 N
(steel on steel)
Thus we have:
Second Law of Friction.
The friction force F is proportional to the normal component of the contact force N.
Thus F Ç N and so F = mN, where m is a constant known as the coefficient of friction. Often
two values of this coefficient are given. One is called the static coefficient and corresponds to the
force required to just get the object moving. The other is called the kinetic coefficient and
corresponds to the force required to keep the object sliding at a constant velocity.
The coefficient of friction depends on the surfaces involved.
Demonstration
This was shown by repeating the measurement for 3 blocks on a sheet of aluminium.
58
PM6: Friction
Friction force with 3 blocks = 20 N
59
(steel on aluminium)
The laws stated are crude laws, the sort obtained with crude apparatus. Even with this
apparatus, complications are evident - the tendency of the object to stick again after it has started to
slide. With more refined apparatus, these complications can be examined. As well, the dependence
of friction on velocity can be investigated.
Demonstration
In a simple apparatus of this type, an object connected to a horizontal spring balance rests on a table
which can be rotated underneath it. The spring balance measures the friction force.
Fig 6.5 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.
6-3 EXPLANATION OF THE LAWS OF SLIDING FRICTION FOR METALS
To explain the laws of friction, it is necessary to introduce additional experimental information.
Firstly, a variety of techniques show that even when the surfaces look smooth, they are
microscopically rough. This is shown by:
Demonstrations
Photographs taken with electron microscopes.
The oblique sectional technique. If cuts at small angles to a surface are made, surface irregularities are
magnified.
Fig 6.6 An oblique section
Since the surfaces are rough, it is tempting to think that the friction must be due to the
intermeshing of the surfaces. But the sliding of such surfaces over each other is non-dissipative.
That this cannot be the explanation is also shown by the fact that after a certain degree of polishing,
further polishing results in an increase of the friction.
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.
Demonstration
This was shown by placing two irregular lead plates in contact with each other. It was further shown that
the real area of contact increased as the load increased.
60
PM6: Friction
Contact points
Fig 6.7 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. The voltage drop depends on the resistance of the path; this resistance
decreases as the (real) area of contact increases.
Surfaces
in "contact"
Digital
voltmeter
V
Ammeter
A
Fig 6.8 Experimental arrangement to measure the potential drop between the surfaces
If precise experiments of this nature are made, it is found that the real area of contact increases
proportionately with the load. 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 final important piece of experimental information is that strong adhesion occurs at the
points of contact. The points of contact are in effect cold-welded forming a continuous solid. If the
materials are to be slid over each other, these junctions have to be sheared.
Demonstrations
That strong junctions are formed can be shown by oblique sections of the friction tracks formed when one
material is slid on another. These show that material is transferred from one surface to the other.
This transfer can also be shown by sliding a radio-active metal on a non-radioactive metal. After sliding,
radioactivity is detected on the 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. Since
transfer can occur, it is very important to use tools of the same material as the metal plates etc. used
to repair the bones. Otherwise, contact EMFs are set up which result in corrosion.]
Bringing the various pieces of experimental information together, an explanation of the laws of
friction can be given. If we assume the friction force is just that required to shear all the junctions
then since the real area of contact increases with the load, then so must the friction force. This is the
second law (see post-lecture material for further details).
Further, since the real area of contact depends only on the load and not on the apparent areas
of contact, the friction force is independent of the apparent areas of contact. This is the first law.
Actually the friction force is not only that required to shear the junctions. There is also a
contribution associated with the "ploughing" of the hills of one surface through the other surface.
This is small unless one surface is very much harder than the other.
PM6: Friction
61
6-4 OTHER FRICTIONAL BEHAVIOUR
The above picture of sliding friction for metals is incomplete. Invariably, surface layers exist
on the metals and these play a major part in determining the frictional behaviour. Indeed, it is only
because surface layers exist that metals can be slid on each other. If the surfaces are cleaned in a
vacuum and the metals slid on each other in the vacuum, it is found the surfaces bond together so
that it is not only impossible to slide one on the other but it is impossible to pull them apart.
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.
The presence of surface layers can result in a breakdown of the F = mN relationship - the
coefficient of friction µ can vary with the load N.
This can happen if the layers are such that they remain intact at low loads but break down at
higher loads. The coefficient of friction then changes from that for surface layer sliding on surface
layer to that of metal on metal or metal on surface layer.
Layers of soft metal are placed on the surfaces of bearings to reduce friction.
The sliding of non-metals on each is explainable in much the same way as it is for metals.
Generally, however, the coefficient of friction is much more load-dependent. For some materials
such as rubber this results from the materials deforming elastically rather than plastically at the
points of contact. For other materials such as plastics this behaviour arises because they are viscoelastic.
Demonstration
An important non-metal as regards its frictional behaviour is teflon. This has a very low coefficient of
friction of 0.05 - 0.1 arising from the nature of its molecular structure which is 'streamlined".
Teflon is an important bearing material being used for example as one of the surfaces in
artificial hip joints.
Finally, a few words about rolling friction. One form of this is the traction type as in a car
wheel on the road where frictional grip is essential. The other is "free" rolling.
Demonstration
"Free" rolling is typified by a ball-race.
In "free" rolling, the coefficient of friction is very low, less than 0.001, which is much less than
any coefficient for sliding friction. The mechanism for frictional energy loss is quite different from
that for sliding friction.
PM6: Friction
6-5 HEATING EFFECTS
Friction is a non-conservative force. When objects slide on each other, kinetic energy is
converted to heat resulting in increase in the temperature of the surfaces.
Demonstrations
An abrasive saw cutting a pipe produces sparks.
Fire can be produced by the high speed rubbing of one piece of wood on another.
Refined experiments show that very localised temperature increases of up to 2000 K for
10- 4 !seconds or less are produced.
Demonstration
These local hot spots are basic to the polishing process. When a metal such as a denture casting is
polished, local thermal softening of the metal leads to flow and filling up of gaps. Obviously, a high melting
point polishing agent is necessary for efficient polishing.
It is the heating of the surfaces which causes the coefficient of friction to decrease at high
velocities. 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.
Demonstration
The heating of the surfaces is basic to skiing. A lubricating layer of water is produced. As the ambient
temperature decreases, it is harder to produce and maintain this layer and the skis stick.
6-6 LUBRICATION
The reduction of friction between two surfaces by placing another material between them is
known as lubrication.
Demonstration
A block will slide much easier on a table if a layer of oil is spread on it.
Hydrodynamic lubrication
The type of lubrication in which the surfaces are completely separated by a thin film of fluid is
known as hydrodynamic lubrication. It results in very low coefficients of friction, of the order of
0.001 and completely eliminates wear.
Demonstration
This type of lubrication is used in journal bearings. A complete film is formed if the load is not too high
and the speed of the rotating shaft is great enough.
Lubricant
Fig 6.9 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). The viscosity cannot be reduced indefinitely, however, since the
separation between the surfaces decreases as the viscosity decreases and eventually the surfaces
come into contact.
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. If the journal speed is further decreased, the lubricant is
reduced to localised patches a few molecules thick. Lubrication under these conditions is known as
boundary lubrication.
represent lubricant
molecules
Fig 6.10 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 good boundary lubricant is one which will attach itself
firmly to the clean metal surfaces formed as the cold-welded junctions are sheared. A layer is then
formed which acts as a lubricating film, and if it can be easily sheared than the friction is low.
Typically coefficients of friction of the order of 0.1 are obtained and the wear is slight.
Elastohydrodynamic lubrication
Of considerable interest is the lubrication of synovial joints in animals, such as the hip joint.
This can be explained neither in terms of hydrodynamic nor boundary layer lubrication. Rather it
seems that the lubrication is of another type known as elastohydrodynamic lubrication in which
the surfaces deform appreciably, the elastic deformations being comparable with the lubricating film
thickness so that it is maintained. In the synovial joints the cartilage elasticity is such as to allow this
process. The rheological properties of the synovial fluid are also important; in particular the
synovial fluid is thixotropic. 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. (This decreases the cartilage elasticity.)
PM6: Friction
POST LECTURE
6-7 PROBLEMS
Q 6.1
In the inclined plane apparatus used for measuring the coefficient of friction ,(shown in the diagram below)
the block starts to move down the plane when the angle of the plane is q.
Fig 6.11 Diagram for Q6.1
Derive an expression for the coefficient of friction in terms of this angle q.
Q6.2.
Friction is a dissipative force. 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.
Explain why this sort of process is non-dissipative.
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. If s
is the shear strength of the junctions and A the total real area of contact, then the frictional force F is
given by
F = A s.
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.
s
Thus
F = p N
s
and so
m = p.
If s and p are taken as those of the softer material, this expression predicts reasonable values
for m. Really good agreement is not obtained because of the influence of surface layers.
64
65
PM6: Friction
6-9 MORE PROBLEMS
Q6.3
Since friction is a dissipative force, the model which explains it in terms of the shearing of cold-welded
junctions must involve energy dissipation. Explain how this energy dissipation occurs.
Q6.4
A shaft of radius r is rotating with angular velocity w in a bearing. Assume that a thin film of lubricant of
uniform thickness d exists between the shaft and the bearing and that the length of shaft supported by the
bearing is Ú.
Stationary
bearing
Rotating
shaft
r
Lubricant
d
Fig 6.12 Diagram for Q6.4
(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.
(c)
Hence obtain an expression for the friction force at the surface of the shaft, the force at the surface of the
shaft, the force which is resisting its rotation.
(This result shows, as mentioned in the lecture, that to reduce the resistance to motion, the coefficient of viscosity
has to be decreased. You should also note that the theory given here is that pertaining to the torsional
viscometer which was described in chapter PM4.)
66
PM7
SOUND
Some claim that pianists are human,
And quote the case of Mr Truman.
St Saens, upon the other hand,
Considered them a scurvy band.
Ape-like they are, he said, and simian,
Instead of normal men and wimian.
OBJECTIVES
Aims
In this chapter you will study the phenomena of sound. You will find that the speed of sound
depends on the stiffness and the density of the medium it propagates through. The property specific
acoustic impedance, of a medium is defined and its role in determining the reflection of sound from
boundaries between two media is discussed. The physics involved in the ear and in hearing is
discussed.
Minimum Learning Goals
When you have finished studying this chapter you should be able to do all of the following.
1
Explain and use the following terms: acoustic impedance, specific acoustic impedance,
impedance matching, Fourier analysis, Fourier synthesis.
2
Recall how the acoustic impedance of a medium depends on the bulk modulus and the density
of the medium.
3
(i) Recall how the specific acoustic impedance of a medium depends on the bulk modulus
and the density of the medium.
(ii) Give a definition of sound power reflection coefficient between two media in terms of the
amplitudes of incident and the reflected waves and in terms of the specific acoustic impedances
of the two media.
(iii) Describe an experiment to verify these laws.
4
Describe what is meant by an acoustic impedance mismatch; and describe a method by which
the problems arising from such a mismatch may be avoided.
5
Describe an experiment to analyze the frequency components present in a musical note or
other sound.
6
(i) Draw a schematic diagram of the ear, identifying the following : outer ear, auditory
canal, eardrum (tympanic membrane), ossicles, oval window, cochlea.
(ii) Draw a schematic diagram of the cochlea identifying the basilar membrane and the nerve
cells.
(iii) State the general form of the information processed in the cochlea and sent to the brain.
(iv) Describe how ageing limits the frequency response of the ear.
PM7: Sound
67
PRE-LECTURE
Recall the following information from various lectures scattered throughout this course.
(i) The definition of the Bulk Modulus, and Hooke's Law as it applies to substances
which are deformed by volume compression stresses.
(ii) In Electrical Circuit Theory there is a theorem called the power matching theorem. It
concerns the problem of getting energy from a source (battery) to somewhere it can be used (load).
In order to get efficient power transfer, it is found that it is necessary for the impedance
(resistance) of the load to match the internal resistance of the battery. 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. This will be discussed in greater detail in the live lecture
following this television lecture.
In many, many cases involving the transport of energy from one place to another, the same kind
of reasoning will be found to apply. There will be a source and a receiver (load); and there will be
property of each which will effectively determine its ability to accept and transmit energy; this
will usually be given the name impedance. 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.
(iii) A mathematical discussion of simple harmonic oscillations was given in lecture FE7.
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.
You will recall from that discussion that the frequency of a simple harmonic oscillation was
determined by (a) the mass of the object and (b) its spring constant (or alternately its Young's
Modulus). 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. that the parameters
describing the propagation of sound waves through a medium would also depend on two
quantities: the density or the medium, r and its Bulk Modulus k.
(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. Each of these sinusoidal
components has an amplitude and a phase.
The inverse problem, that of starting with simple oscillations and combining them into a more
complicated shape is called fourier synthesis.
LECTURE
It is assumed that you already know a fair bit about sound, particularly how it is generated. We will
concentrate on two aspects only: its propagation and its analysis. We do this with specific reference
to the EAR and electrical hearing devices.
PM7: Sound
7-1 ACOUSTIC IMPEDANCE
The ear may be sketched schematically thus:
Fig 7.1 Cross-section of the ear
Sound waves impinge on the outer ear (A) and are conducted through the narrowing column of air
(B) to the drum (C). 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). Beyond that point the information in the sound is converted into
electrical signals to be sent to the brain.
To understand the structure of the outer ear, we must talk in general terms about the propagation of
pressure waves through an elastic medium.
Demonstration
A mechanical model of an elastic medium might be:
Fig 7.2 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. Clearly the heavier the objects, the more slowly will each mass
move after being pushed by its neighbour, and the more slowly will the "wave" propagate.
ii) the strength of the springs. Clearly the stronger the springs, the more quickly will
they expand after being compressed, and so the more quickly will the "wave" propagate.
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.The formula is
speed of sound:
(k is the bulk modulus, r is the density.)
c= k/r
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PM7: Sound
69
Consider now the boundary between two media. What determines how a wave propagates from
one to the other?
*An extremely simple mechanical model is a row of billiard balls:
Fig 7.3 A disturbance “propagating” through a row of billiard balls.
The disturbance propagates with no net motion of the medium only if all balls are absolutely
identical. [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.
Demonstration
A better mechanical model is again cars and springs.
A disturbance will propagate with no net motion in the medium, only if all cars and all springs are
identical. Reflection will occur at any point where there is as CHANGE of either mass or spring
strength.
Demonstration
However, an increase in mass can, in part, be compensated for by a decrease in spring strength.
Careful experimentation will show that you can maximize propagation and minimize reflection if
you keep the product of mass and spring strength constant along the medium.
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.
Clearly we could redefine this quantity (as is more usual) in terms of the velocity of sound, c:
z = rc
PM7: Sound
The specific acoustic impedance is a property of the medium. 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.
The name impedance comes from the analogy with electrical circuit theory (as described in
the pre-lecture material). 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.
Consider again the ear. 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.
Hence there is an enormous mismatch of specific impedances, and only a tiny fraction of the energy
of the sound wave can get from the air into the eardrum. The rest is simply reflected back.
This is a bit of a simplification. 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. The
impedance of a layer of air near the drum is considerably greater than a layer of air near the outer
ear. 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. Its impedance is increased by the narrowness
of the tube.
Fig 7.4 Impedance and width of tube
(The use of a horn shape to get impedance matching comes into loudspeaker design, where the
problem is to get energy from inside out into the air.)
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.
Demonstration
Clinical measurements of drum impedance feed into the ear a sound of known intensity, and measure
the intensity reflected from the drum.
This gives the impedance of the drum (relative to that of air [see post-lecture]. Obviously, for
example, if no sound is reflected, the impedance of the drum is exactly the same as that of air.
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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.
This is rather an impossible calculation; but standard audiological procedure is to measure the way
drum impedance varies with pressure and to compare with a known normal ear. 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.
To sum up: there is an enormous impedance mismatch between the eardrum and the outside air, so
you only detect a very small fraction of the energy in any sound wave. From this point of view, the
ear is in an inefficient hearing device. Electrical hearing devices for example get over this problem
by incorporating an AMPLIFIER as an essential component.
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 7.5 A “straightened out” cochlea
Pressure variations in the oval window are transmitted by fluid to the basilar membrane. Because of
its geometrical shape, different parts of this membrane resonate with oscillations of different
frequencies. Nerve cells connecting this membrane to the auditory nerve, tell the brain which part of
the basilar membrane is vibrating - and thus what simple harmonic oscillations are present in the
original sound wave.
Fig 7.6 Helmholtz Spectrum Analyzer
Demonstration
A musical note of the correct frequency will cause one of these cavities to resonate. The increased
amplitude of vibrations thus set up, are made to cause a gas jet to flicker up and down. Which cavity is
resonating can then be detected in a rotating mirror system (this apparatus was built circa 1890).
PM7: Sound
Demonstration
This apparatus is in principle capable of identifying all the harmonies present in any musical note.
However, it is too cumbersome to use seriously. Modern spectrum analysis is all done electronically.
Sound is fed into a microphone and the result displayed on an oscilloscope screen. You interpret the
output thus: for a reasonably pure note you might get something like
Fig 7.7 Spectrum of sound as seen on a spectrum analyzer screen
[Note: in order to analyze the spectrum accurately the apparatus must listen to the note for some time
and "count" how many oscillations occur in that time. That is why you see such a slow rate of scan.
The slower the rate the more accurate the analysis.]
*With the aid of a spectrum analyzer you can determine what gives the human voice or various
musical instruments their distinctive sounds. It is just a question of what harmonics (or overtones) are
present, and in what strength (amplitude).
*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.
[Note: It seems to be the basic philosophy of much modern music that the older instruments have
become stale and uninteresting. And indeed the spectrum of one wind instrument for example, is very
like that of another. So musicians today are trying to get sounds out of all sorts of unlikely
instruments - to produce completely new spectra for the ear to listen to.]
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. And the information going to the brain is
of the general form:
(i) what simple harmonic frequencies are present, and
(ii) what their amplitudes are.
That this is a true representation of the ear can be confirmed by two observations.
Demonstration
(i)
In some deaf schools, children are taught to speak by getting them to match the oscilloscope
pattern of a sound made by the teacher. 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. However,
you will notice that the teacher has obviously found from experience that it is most effective to change
the scale every now and then, thus directing the child's attention to specific harmonics.
[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.]
Demonstration
(ii) The inverse of a Fourier analyzer is a Fourier synthesizer - and electronic organs act as such
when they build up a complicated oscillation from the fundamental and a few harmonics.
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73
PM7: Sound
For example:
Fig 7.8 Fourier synthesis
If you change only the phase of one of these
Fig 7.9 Another example of Fourier analysis with a minor change from Fig 7.8
It is a demonstrable fact that the ear cannot distinguish between these two sounds. And though these
two pressure-time patterns are quite different, they have the same fourier spectrum - the various
harmonics present have the same amplitudes. It is only their phases which are different.
*The new breed of musical instruments based on this simple electronic organ principle are given the
generic name synthesizers. The most commonly known perhaps is the Moog.
Demonstration
The new breed of musical instruments based on this simple electronic organ principle are given the
generic name synthesizers. The best known perhaps is the Moog.
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. However when we consider the passage of a sound wave from one relatively large
quantity of one medium to another, things are a little simpler. Then we need to consider only the
bulk property of each medium - the specific acoustic impedances.
If a wave of amplitude Ai is incident on a boundary between two media; and a wave of amplitude Ar
is reflected from the boundary; then the sound power reflection coefficient ar, defined as the ratio
of reflected sound energy to incident flow or sound energy, is given by
ar =
Ar 2
Ai2
=
Ê z2 - z1 ˆ
Á
˜
Ë z2 + z1 ¯
2
where z1 and z2 are the specific acoustic impedances of the media before and after the boundary.
It can be appreciated immediately that complete transmission (i.e. ar = 0) will only occur when the
two media have exactly the same specific acoustic impedances.
Q7.1: 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. One method, for specific devices, was mentioned in
the lecture. For large quantities of the media, this can be done by intervening a third medium
between the two. Then so long as the specific impedance of this third medium is intermediate
between that of the first two, it is found that the transmission of energy is greatly increased.
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.
You may recall from your optics lecture (L4) that a very similar condition - with refractive
index rather than specific acoustic impedance - describes the ideal way to reduce reflections at air to
glass boundaries in optical systems.
7-5 FREQUENCY RESPONSE OF THE EAR
Because of physical limitations, the ear will not respond to all frequencies. The main
limitation comes from the geometry of the cochlea. 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). Hence if you consider the basilar membrane to look
schematically like this
Fig 7.10 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.
However, there are other factors which limit the frequency range, especially at the high
frequency and the most important is the elasticity of the eardrum. This determines its ability to
follow a very high frequency vibration. It is found, and you would expect it to be so, that as people
age, the elasticity of their skin decreases and so therefore does the highest frequency they can hear.
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.
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.
Q7.2: 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.
van Begerjk et al “Waves and the Ear”, Heinemann Science Study Series.
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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.
The skylark sings a roundelay,
The crow sings The Road to Mandalay,
The nightingale sings a lullaby
And the seagull sings a gullaby.
That's what shepherds listen to in Arcadia
Before somebody invented the radia..
OBJECTIVES
Aims
In this chapter you will study ultrasonics. Much of the lecture is concerned with the applications of
ultrasonics using techniques of echoscopy and the Doppler technique.
Minimum Learning Goals
When you have finished studying this chapter you should be able to do all of the following.
1.
State the frequency range of ultrasound.
2.
Do simple calculations relating the wavelength, frequency and speed of sound waves and
ultrasonic waves.
3.
Describe an ultrasonic transducer and explain how it is used to generate or detect ultrasound.
4.
(i) Explain how the short wavelength of ultrasound makes it possible to focus a lot of
energy into a small space.
(ii) State two applications of this property.
(ii) State an illustration of the Doppler effect with electromagnetic waves.
(iii) State and describe two applications of the Doppler effect with ultrasonic waves.
5.
(i) Describe what is meant by the Doppler effect.
(ii) State one illustration of the Doppler effect with electromagnetic waves.
(iii) State and describe two applications of the Doppler effect with ultrasonic waves.
6.
Describe and distinguish the techniques of radar, sonar and echoscopy.
7.
Describe and explain one example of the use of echoscopy.
PRE-LECTURE
Refer back to lecture PM7 to remind yourself of the concept of specific acoustic impedance and
of its importance in the transmission of sound from one medium to another.
PM8: Ultrasonics
LECTURE
8-1 INTRODUCTION
Sound experienced by human falls in the frequency range 0 - 20 kHz. Sound above this frequency is
known as ultrasound. It can be detected by some animals.
Demonstration
A dog can hear low frequency ultrasound such as that produced by a Galton whistle (the principle of
this whistle is discussed in the T.V. lecture).
Ultrasound has many important applications some of which will be discussed in this lecture.
These arise partly because sound at these high frequencies has short wavelengths and partly, just
because it is sound, it is a pressure wave and hence will travel in materials.
Though the use of ultrasound by man is of relatively recent origin, BATS have always used it.
For a long time it was thought bats made no noise but by recording them on magnetic tape and playing
the tapes back at a slower speed (this reduces the frequency of the recorded sound) it was found that
they make sounds in the 40 - 55 kHz regime. They use this ultrasound for navigational purposes and
also for locating their prey. The ultrasound is produced in short duration screeches of about l0 - l5
milliseconds and that part of it which has bounced off something back into the direction of the bat is
heard by it. The elapsed time gives the bat information on how far the object reflecting the pulse of
ultrasound is from it.
8-2 GENERATION AND DETECTION OF ULTRASOUND
Demonstration
Sound is produced when an object vibrates
Fig 8.1 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. One device for doing this is the Galton whistle but ultrasond can be produced
much more conveniently and efficiently by making use of piezo-electric materials such as barium
titanate.
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. By applying a high frequency alternating voltage, the material is caused to vibrate at a high
frequency and so ultrasound is produced.
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PM6: Friction
77
Fig 8.2 Production of ultrasound using a transducer
A device to produce ultrasound based on this principle is known as a transducer.
These transducers can detect ultrasound by using them in reverse.
Demonstration
Ultrasound falling on one of them causes it to vibrate and hence a voltage is produced across it, a
voltage which can be detected using, for example, an oscilloscope.
Since ultrasound is at high frequency, its wavelength is short and so it can be focussed into
small regions. When used at high power, this fact gives ultrasound a variety of important uses.
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. It can
also be used for more prosaic applications such as drilling and cutting.
Demonstration
Ultrasound at high power is also used extensively for clearning. The ultrasound is produced in a
bath of liquid in which the object to be clearned is placed. The high intensity ultrasound causes
negative pressures in the liquid and as a result bubbles called "cavitation" bubbles are produced.
Agitation produced by these cleans the object. If the object is kept too long in the cleaner,
particularly if it is thin, it can be extensively damaged.
8-3 DOPPLER TECHNIQUES
The frequency of a wave emitted by a moving source is different to a stationary observer from
that when the source is stationary. This is known as the DOPPLER EFFECT. When the source is
moving towards the observer, the frequency is increased and when the source is moving away the
frequency is decreased. The effect also applies to a wave reflected from a moving object.
Demonstration
The police make use of the Doppler effect in their radar speed traps. (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).
A common example of the Doppler effect is the sound of an ambulance siren as the ambulance
approaches and passes. The "red shift" i.e. 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.
PM8: Ultrasonics
Demonstration
The effect can be demonstrated by having an ultrasound generator mounted on a car which can
run on rails either towards or away from a stationary ultrasonic detector. The frequency of the
detected sound can be measured with a digital frequency meter..
Detailed measurements with such apparatus show that the change in frequency is proportional
to the velocity of the source. 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. Or, alternatively, if
the wave is reflected from a moving surface the change in frequency of the reflected wave relative to
the incident one gives the velocity of the moving surface. In this latter form there are many uses of
Doppler techniques with ultrasound in medicine. For example it is possible to detect the foetal
heartbeat as early as the l0th week, and by measuring the motion of blood vessel walls it is possible
to learn about their elasticity.
Demonstration
In clinical medicine, the technique is used to detect blood flow in arteries and veins by means of
an external probe. This is placed against the skin and more or less angled along the blood vessel. A
paste is applied between the skin and probe to improve impedance matching and so lessen power loss
by reflection. The ultrasound produced by the probe is reflected from the flowing blood and then
detected by the probe. A probe such as this shows quite different sounds for arteries and veins. For
arteries, the sound is characteristic of the pulsatile blood flow in arteries. In veins, it is more like a
wind-storm which cycles with the respiration. The probe can detect blockages in arteries and veins.
(N.B. The medical term "patent" which is used in describing this technique means "unblocked".)
Doppler techniques have been used in a different way to measure the blood flow in research
projects on animals.
Demonstration
Small probes are placed around arteries during an operation. They heal in place with the leads
coming out of the skin. In an experiment, the leads are connected to a telemetering device carried in a
package on the animal. In this way it is possible to look at patho-physiological conditions in
conscious animals in realistic situations. This technique has been used for example on dogs which
have been made hypertensive. It has also been used on small monkeys to study the effect of severe
oxygen lack on the circulation. In a proposed experiment it is to be used on baboons to study the
effect of diet on coronary disease.
8-4 ECHOSCOPY
If a pulse of waves of known velocity is setn out from a transmitter and the time taken for the
puse to return after being reflected from a distant object is measured then this time is a measure of
the distance of the object. This technique is called the "pulse-echo" technique and with electro,magnetic waves is of course radar. This technique can also be used with ultrasound. It is of
course the technique used by bats for navigation. It is used it as sonar for depth sounding, detection
of submarines and shoals of fish.
In medicine, the technique is used and is then known as echoscopy. If ultrasound travelling in
one medium encounters another, in general some will be transmitted into the other medium as well as
being reflected.
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.
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PM6: Friction
There are not large differences bvetween the specific acoustic impedances of the various soft
tissues in the body (see post lecture material). Thus the pulse-echo technique can be used to obtain
two-dimensional cross-sectional views through various organs in the body. As the ultrasound
penetrates into the organ, some is reflected at a soft-tissue boundary but most is transmitted to suffer
further reflections from successive boundaries. The time delays and amplitudes of the signals from
the different boundaries make up the two dimensional picture called an echogram.
In looking at an echogram it is important that it be not thought of in the same way as an x-ray
picture. The latter is a three dimensional view compressed into the two dimensions of the x-ray
plate. The echogram is, however, a true two dimensional cross-section through an object.
Demonstration
One use of this technique is in a machine designed to scan the eye. It can pick up retinal
detachment and is extremely valuable in detecting tumours behind 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.
If the ultrasonic waves were sent through the air to the eye, most would be reflected at the outer
surface of the eye.
Demonstration
This technique is also used in obstetrics to scan the pregnant uterus. The matching of acoustic
impedances should again be noticed. The technique is of majeor importance in this field for unlike x
rays, ultrasound appears to be completely safe. The technique gives information on the size of the
foetus, how many there are, if it is growing at a reasonable rate, whether it has miscarried, whether
there are any gross abnormalities etc.The technique can also be used for examining the non-pregnant
abdomen for picking up tumours.
POST-LECTURE
8-4 ACOUSTIC PROPERTIES OF VARIOUS MATERIALS
The following table gives the acoustic properties of various materials.
Material
Velocity/m.s
Density/103 kg.m– 3
Specific acoustic impedance/106 kg.s– 1.m– 2
water
1530
1.00
1.53
blood
1534
1.04
1.59
fat
1440
0.97
1.40
brain
1510
1.03
1.55
liver
1590
1.03
1.64
muscle
1590
1.03
1.64
bone
3360
2.00
6.62
air
340
0.00012
10–4
Q8.1 Why is the frequency of ultrasound which bats use so high?
[Ans 6]
Q8.2 Show the importance of acoustic impedance matching in echoscopy by comparing the
fractional power reflected for a water-flesh interface with that for an air-flesh interface. [Ans. 13]