PM1 ELASTICITY - School of Physics · In this chapter you will see that all elastic deformations...

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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, linearextension, uniform compression, shear, pressure, fracture, stress, strain, elasticmodulus, Young'smodulus, 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.

Transcript of PM1 ELASTICITY - School of Physics · In this chapter you will see that all elastic deformations...

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PM1ELASTICITY

The lion is the king of beastsAnd husband of the lioness.Gazelles and things on which he feastsAddress him as Your LionessThere are those who admire that roar of hisIn the African jungles and veldts.But I think wherever the lion isI'd rather be somewhere else.

OBJECTIVESAimsIn this chapter you will see that all elastic deformations can be described in terms of linear, shear andbulk changes. You will be introduced to the concepts of stress, strain and material strength. Youwill apply these ideas to some real-world deformations. You will learn how to do calculationsinvolving simple situations of deformation.Minimum Learning GoalsWhen 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 simplenumerical 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 translationaland/or rotational accelerations of the body: the body is assumed to be rigid. However, no bodyis 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.

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LECTURE1-1 STRESS, STRAIN AND THE BASIC DEFORMATIONS.The study of elasticity is concerned with how bodies deform under the action of pairs of appliedforces. 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 resultingacceleration (change of motion) but there is a resulting deformation or change in the size or shapeof 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 twoare in balance the deformation will be maintained without further change. For these simple cases wemake 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 ExtensionDemonstration

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, theoriginal length as L and the extension as e. The stress and strain are then defined as follows:

A

F

F

L

e

Stress =

Strain =

F/A

e/L

Fig 1.1 Definitions of stress and strain (linear extension)

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Uniform CompressionDemonstration

If the forces are applied uniformly in all directions, we have a deformation typified by that produced by auniform hydrostatic pressure.

Write the pressure as p, the original volume as V and the change in volume as DV. The stressand strain are then defined as follows:

p

Stress =

Strain =

p

- DV/V

(the significance of the minus sign is that thevolume decreases as thepressure increases)

Fig 1.2 Definitions of stress and strain (uniform compression)ShearDemonstration

In the previous two deformations, either the length or volume of a body was changed. In the sheardeformation, only the shape of a body is changed. Shear occurs for example when oppositely directedtangential forces are applied across opposite faces of a rectangular block of material. These forces deform therectangular block into a parallelogram.

Write the force as F, the area across which the force is applied as A and the angle ofdeformation (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.

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In general, things are more complicated than this but can be resolved in terms of these basicdeformations.Demonstration

As an example of the more complicated behaviour one can get, consider a rod under the action of acompressive 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 isthin enough, one does not get a linear compression but rather the rod buckles.

1-2 ELASTIC AND NON-ELASTIC BEHAVIOURLet us now consider what happens to a body under the action of one of these types of deformingforce 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 EngineeringDepartment. A sample of mild steel was tested and the stress as a function of strain was recorded on a chartrecorder.

The complete stress-strain curve was as follows:

Fig 1.5 A stress/strain curveAs 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 isknown as the elastic regime and the point P is called the elastic limit.

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As the stress was further increased, a point Y, known as the yield point, at which the stress rapidlydropped, was reached. From J to K the material flowed like a fluid; such behaviour is called plastic f low .After a region K to L of partial elastic behaviour, plastic flow continued from L to M . Eventually (when thepoint 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 thestress did not reduce the strain to zero.

The behaviour described above for mild steel is not typical of all materials. Materials thatbehave 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 MODULIAs we have seen, when a material is stressed there are basically two different regimes: the elastic andthe non-elastic. The latter is difficult to describe in a way which is easily applicable but in the formerthe stress is proportional to the strain.

This proportionality between stress and strain is known as Hooke's law; it applies to all ofthe three basic deformations. Hence the ratio stress/strain is a constant; this constant is known asthe elastic modulus. There are three elastic moduli, one for each of the three basic deformations. Linear Extension

StressStrain =

F/Ae/L

= Ynamed Young's modulus

Uniform CompressionStressStrain =

-PDV/V = -

pVDV

= knamed bulk modulus

ShearStressStrain =

F/Aq =

FAq

= nnamed shear modulus or modulus of rigidity

<< There is one other parameter which is necessary to describe the elastic behaviour or materials. This isPoisson'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 theapplied stress

>>A Microscopic ModelThe values of the various parameters we have defined must depend on the microscopic structure ofthe material. In the unstressed state the atoms or molecules are in equilibrium positions, such that if

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they are pulled apart the forces between them are attractive and if they are pushed together the forcesare repulsive.

Where these forces as a function of distance between the atoms or molecules are known, onecould, 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 arealways too high, due to the presence, in even the purest crystals, of imperfections such asdislocations and impurity atoms.

1-4 EXPERIMENTAL MEASUREMENT OF ELASTIC MODULIThe 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 tomeasure the strain appropriate to different stresses.

An alternative method is to make use of the fact that the mechanical oscillations of bodies andthe characteristics of pressure waves propagating through them depend on the elastic moduli.Demonstration

1.

Fig 1.7 Oscillations of a coiled spring: shear modulusThe 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.

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3.

Fig 1.9 Torsional oscillations: shear modulusThe 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 samplesuitable for the test machines. Investigations of possible changes in the elasticity of bones in thebody with age and disease have been made, for example, by setting the bones into oscillation andmeasuring the oscillation frequencies.

The propagation characteristics of a pressure wave are determined by the bulk modulus ofthe 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 APPLICATIONSRepair of the Human BodyMany materials are used in the repair of the body. The prime consideration in these applications isthat the materials be strong enough. External to the body there are, for example, artificial limbs, andinternal to the body there are, for example, plates used for repairing fractures. In these latterapplications 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 materialis "composite". This is a polymer mixed with quartz and is quite strong in compression, as it must be, sincelarge compressive stresses are experienced in biting. Its compressive strength is about 2.5 ¥ 108 Pa whichcompares favourably with that of tooth enamel viz 4 ¥ 108 Pa. Since as well it looks like tooth enamel, it isa 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 atstresses much less than this if the stress is applied and removed a large number of times. Thisphenomenon is known as fatigue.Demonstration

Dentures for example can fail by fatigue.Bones etc. as Structural ElementsThe basic point in designing any element to withstand stress is to properly assess what the stressesare. 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 inthis regard are trees. These are basically columns and are in a state of compression due to their ownweight. One might think that their heights would be limited only by the requirement that thecompressive strength be not exceeded; thus no relationship between height and diameter would beexpected.

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This, however, is not the case. A column fails not by compression but by bending. Failureoccurs 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 beamsare much stronger against bending than solid beams such as this.

Fig 1.12 Solid beamsDemonstration

Many bones indeed are of tubular shape. In others their good design leads to the bone being arrangeddifferently: it is all a matter of the nature of the stresses. For example, in the top of the femur, the bone isarranged in thin sheets separated by marrow, the sheets being so arranged to give the greatest strength when thebone is experiencing those forces to which it is normally subjected.

DemonstrationNo 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 itis stressed in a way for which it was not designed.

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FibresThe elastic properties of bone and timber are different in different directions. This is so becausethese materials are fibrous. There are many fibrous materials in nature. One important class offibres are those used in making textiles: the natural fibres wool and cotton and the various syntheticfibres.

The elastic properties of these fibres is obviously important in that they determine theproperties 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.

A

B C

D

Stress

StrainO

Fig 1.13 Stress-strain curve for wool fibreThe 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 isremoved in this region the strain returns to zero. Therefore this region does not correspond to a region ofplastic flow, as for steel. It results from the long keratin molecules, of which the fibre is composed, changingfrom a coiled shape to a more extended one. With further stress, the curve again rises and finally the fibreruptures at the point D.

Arteries and the LungStrong fibrous materials, such as bone, are common in the body. There are other materials in thebody where strength is not the important thing but stretchability. The walls of the arteries fall intothis 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 aorticvalve shuts and the pressure in the arteries drops, the walls relax maintaining the blood flow. The hardening ofthe 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 ininspiration 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 withsurface tension play a large part in the operation of the lung; when the lung is filled with saline solution theseforces 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 tensionduring inspiration. (This will become clearer after the surface tension lecture - PM2 - when more details willbe given.)

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Muscle and SkinDemonstration

Other tissues in the body where stretchability is important are muscle and skin. The skin's elasticitydecreases noticeably with age.

POST-LECTURE1.6 UNITS You will have noticed that in the television lecture, on occasions, units other than SI. units have beenused.

The correct unit, as agreed by the International Conference on Weights and Measures, forstress (or strength, or any of the elastic moduli) is the pascal (Pa). That unit is used exclusively inthese notes.

1.7 TABLESThe calculated values are based on microscopic models. The lack of correspondence is the result ofdislocations and impurity atoms.

TENSILE STRENGTH / 108 Pa

MATERIAL THEORETICAL OBSERVED

rock salt (NaCl) 2.7 0.004 (bulk material)1.0 (single crystal)

iron 46 2.0 (bulk)14.0 (single crystal)

cellulose 11 1.0 (fibre)

For liquids and gases the shear modulus is zero; for liquids the bulk modulus is about the samevalue as for solids but it is much smaller for gases.SUBSTANCE Y/1010 Pa k/1010 Pa n/1010 Paaluminium 7.05 7.46 2.67steel 19 - 21 16.4 - 18.1 7.9 - 8.9glass (crown) 6.5 - 7.8 4.0 - 5.9 2.6 - 3.2water 0.2 0 mercury 2.1 0air (atmospheric pressure) 1.4 ¥ 10-5 0

1.8 TENSILE AND COMPRESSIVE MODULI A crystalline solid exhibits the same stress vs. strain relation whether it is under tension orcompression. On the other hand, bone and other biological materials show different behaviourunder tension and compression.

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1.10. PROBLEMSQ1.1 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

500 mm

500 mm

5.0 mmF 0.10 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 isexaggerated, for clarity, in the diagram. Calculate the magnitude of these forces.

The shear modulus of brass is 3.5 ¥ 1010 Pa.

Q1.3 By what fraction does the density of water at a depth where the pressure is 4 ¥ 105 Pa increase over thesurface density.

The bulk modulus of water is 2 ¥ 109 Pa.

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PM2SURFACE 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.

OBJECTIVESAimsIn this chapter you will look at the behaviour of liquid surfaces and the explanation of that behaviourboth in terms of forces and in terms of energy. The principle of minimum potential energy can beinvoked to explain many surface phenomenaMinimum Learning GoalsWhen 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) Perform simple numerical calculations associated with such a determination.

3 (i) Use a model of the microscopic structure of liquids to explain the phenomenon ofsurface tension in terms of potential energy.(ii) Extend this argument to explain why liquids tend to assume a shape which minimises thesurface 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 perlength 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.

(ii) Recall, explain and use the relationship h = 2Trgr 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 generalnature 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 theforce 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. Inparticular, the molecules of a substance tend to come together (pulled by the intermolecularattraction) until on the average their distances apart correspond to the cross over point betweenattraction and repulsion. This means the normal state of a substance is an average kind ofequilibrium.

(iii) Equilibrium can be discussed in terms of potential energy. The equilibrium configuration isone 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.

LECTURE2-1 PHENOMENON OF SURFACE TENSIONThe 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 ofthe 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 arecases on record where ducks have drowned in farmyard ponds into which washing water was emptied, or instreams polluted with non degradable detergents.]

2-2 MEASUREMENT AND DEFINITION OF SURFACE TENSIONThe strength of the surface membrane can be imagined to arise from a set of forces acting on eachpoint of the surface, parallel to the surface, like the skin of a drum.

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DemonstrationThe easiest way to measure these forces is with the following apparatus

BALANCE

ADJUSTABLE WEIGHT AND SAND GLASS

SLIDE

WATER

FIXEDCOUNTER WEIGHT

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 theglass and the water are vertical rather than horizontal.

SLIDE

WATER

MENISCUS

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 sameposition

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 surfaceof a drum, a bigger cut is harder to repair than a smaller one).

Length of slide used in first experiment: 38 mmLength of slide used in second experiment: 76 mmResult 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 intimatecontact (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.

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DemonstrationIn 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 tensionof 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 ENERGYTo understand why the phenomenon of surface tension arises, you must think of intermolecularattraction 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 separationis low.

Consider a fixed number of liquid molecules. If they are packed so that they have a largesurface area, their average intermolecular separation is relatively high. If they have small surfacearea, the average intermolecular separation is relatively low. Their total potential energy is lower inthe latter case.

A logical conclusion from this is that energy has to be added in order to increase the surfacearea of a liquid. The bigger the change in surface area, the more energy has to be put in. Associatedwith the surface there is a potential energy that depends on the area of the surface. This means thatan 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 totalsurface potential energy.

DemonstrationDrops of water are sphericalLoop of thread on water; detergent added inside loop; loop takes a circular shape.

Shaded area here is greater than shaded area here

CONTAINERLOOP OFTHREAD

PURE WATER

WATER ANDDETERGENT

Fig 2.3 Effect of placing a drop of detergent inside a loop of string that is floating on thesurface of water

(Surface tension of detergent and water is much lower than that of water.)

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The dimensions of energy are force ¥ length, so energyarea has the same dimensions as

forcelength .

Sometimes it is easiest to explain surface phenomena in terms of energy considerations,sometimes in terms of force considerations

Demonstration Three matches on water:

MATCHES

CONTAINER

PURE WATER

DETERGENT ADDED

becomes

Fig 2.4 Effect of placing a drop of detergent inside a triangle of matches that are floatingon the surface of water

This is basically the same as the loop of thread demonstration, but it is easier to explain whyeach match moved in terms of forces as thus for the match at the top of the diagram:

larger force(water: highersurface 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 CAPILLARITYA consequence of the phenomenon of surface tension is that many liquids will "creep up" tubes, anobservation 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.

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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:

TWO GLASS SHEETS

RUBBER BAND

PLAN ELEVATION

WIDE END

NARROW END

HYPERBOLIC !! !SHAPE

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

h = 2Trgr

The tube used in the demonstration had a bore of radius 0.50 mm and the measured rise was28!mm. For a tube of this radius, the calculated rise is

h = 2!¥ !0.06!N.m-1

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 chapterPM4.]

(ii) Chromatography.Demonstration

This is a method of chemical analysis which can be done by eye. See post lecture material for a morecareful description.

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2-5 WETTINGA question we have skimmed over is: why is there an attractive force between water and glasscausing the rise of water in a glass capillary tube? This is a question about intermolecular forceswhich only chemists can answer properly. But certainly different liquids are attracted to differentsolids in different degrees. For example, the level of mercury will fall in a glass capillary tube.

Demonstration Drops on solid surfaces.

MERCURYWATER

GLASS

MERCURYWATER

LEADFig 2.8 Water and mercury drops on glass and lead surfaces

Laboratory workers measure the intersurface forces in terms of the angle of contact definedas follows.

ANGLE OFCONTACT f

tangent line

Fig 2.9 Definition of f, the angle of contact between a liquid and a solid surfaceThe 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.

GLASS

OIL

WATER

Fig 2.10 The presence of oil results in the water forming a drop rather than spreading overthe glass surface

DemonstrationWaterproofing of material (this usually involves coating fibres with oil or polymers).

DemonstrationPreening of birds.Water birds spread oil on their feathers to make them water resistant.

DemonstrationWater resistant sands.

Some West Australian sands are virtually impervious to water as a result of fibrous materialbetween the grains making them water resistant. This leads to bad run off conditions in vast areas ofthe state.

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DetergentsThe properties of detergents arise from their complicated molecular structure. This can be illustratedschematically thus:

C

H

H

C

H

H

C

H

H

C

H

H

C

H

H

C

H

H

C

H

H

C

H

H

C

H

H

C

H

H

C

H

H

H C O-O

This end is repelled by water molecules [hydrophobic] and is attracted to oils, fats [lipiphilic]

This end is attracted to water molecules [hydrophilic]

Fig 2.11 A detergent molecule(i) 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 asurface of pure water apart

(ii) In washing up water the following sequence occurs as the water is stirred up.

watergreaseDETERGENT ADDED STIRRED

Fig 2.13 Stirring of soapy water during "washing up"

PM2: Surface Tension 20

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 intoan emulsion with water by the addition of a little detergent.

DemonstrationOil and water.

POST-LECTURE2-6 UNITS AND DIMENSIONSA 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 ofthread in the loop was 0.l!m and work out which, of the following possible shapes the loop could have, hasthe largest area.

2.5 cm

2.5 cm

1 cm

4 cm

3.3 cm

3.2 cm

3.3 cm 3.3 cm

Fig 2.14 Diagram for Q2.1Q2.2 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 aand Ú. The surface tension of the soapy water is T.

F

a

Ú

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 withsurface tension is applied.

Since the film has two surfaces, the relation between F and T is

F = 2ÚT .

Calculate the total work done in increasing the distance a by an amount b, and show it is proportional to the changein area of the soap film.

2-7 MORE ON CAPILLARITYThe law quoted in 2-4 can be derived theoretically as follows. Ask yourself first, why should waterrise 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.

Glassmolecules

Water moleculesFig 2.16 Interaction of water and glass molecules

Each water surface molecule exerts forces on those near it Since there is equilibrium the lastwater molecule must also have a force exerted on it by the glass molecule near it. Therefore, allaround the top of the water, the glass is exerting a force on the water. Because is so happens thatwater 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 theline of contact between the water and the glass is 2p times the radius of tube, so the magnitude of theupward 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 = rπr2hgand, therefore, for this situation, where the water wets the glass completely, the final height of thewater column can be written

h = 2Trgr

Q2.3 In the experiment with soil, we found that for the coarse grained soils (radius of soil grains ~ 0.3!mm) aftera 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 toapply the above relation. Apply the relation and find how much error is in fact introduced.

PM2: Surface Tension 22

2-8 ANGLE OF CONTACTThe angle of contact is defined to be the angle between the surface of the liquid and the solidsurface at the point of contact.

ANGLE OFCONTACT f

tangent line

Fig 2.17 Angle of contact for a liquid that does not "wet" the solid surfaceYou will observe that for a water-glass contact, as in the next diagram, the angle of contact is

much smaller;angle of contact

f

small

tangent line

Fig 2.18 Angle of contact for water-glass contactfor mercury-glass, as in the next diagram, it is almost 180°.

angle of contact

f

large

tangent line

Fig 2.19 Angle of contact for mercury-glass contactWhen 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.

PM2: Surface Tension 23

Forces associatedwith surface tension

Angle

f

Fig 2.20 Close up of part of Fig 2.16Redrawing 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 equalto 0 ̊or 180˚ is zero), which results in a usually imperceptible distortion of the wall. There is avertical component, T cos f (which for f equal to 0 ̊ or 180˚ is T), which causes the liquid in acapillary tube to rise.

So the equation for capillary rise that we wrote is not complete. The general form is

h = 2Tcosf

rgr

For clean glass-water contacts f ª 0 and cos f ª 1. So the equation was suitable for water ina 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 thesurface would be lower inside the tube?

PM2: Surface Tension 24

2-9 SCALING QUESTIONSQ2.5 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 mmand 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 linebetween 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 originalquestion succinctly.

PM2: Surface Tension 25

<< 2-10 CHROMATOGRAPHY

Chromatography is a technique for separating out the chemical constituents of mixtures. It is particularlyuseful in biological contexts. There are two commonly used forms.

Paper Chromatography : Here a few drops of the chemical mixture are put onto a piece of filter paper andallowed to dry. Next the paper is touched to a reservoir of some solvent which will dissolve the chemical substanceyou hope to detect. The solvent is sucked up into the filter paper (by capillary action), and as it flows past the driedmixture, it dissolves out the chemical constituents and carries them along. However, different chemical substancesadhere more or less strongly to the paper (i.e. the surface tension between the surface of the solution and the fibres ofthe paper differs) and so different chemical substances are carried along at different rates. So if you remove the paperfrom the solvent after a while the various chemical constituents of the original mixture will be at different positionson the filter paper.

Colour Chromatography (This is the experiment we filmed.) Here the solvent is put on top of themixture, and allowed to flow through a plug composed of grains of cellulose. Again, the adhesion between thechemical constituents of the sample (spinach leaf) and the cellulose grains is different and they all sink at differentrates. In our experiment (which we filmed in the Department of Agricultural Chemistry with the help of Dr BobCaldwell) 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 TENSIONHere are the values of surface tension of some common liquids. They are listed heremerely for the purpose of showing you what range the values of surface tension canhave.

Liquid Surface Tension/N.m-1

water (20°C) 0.073water (100°C) 0.059alcohol 0.022glycerine 0.063turpentine 0.027mercury 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

PM3HYDRODYNAMICS

Some fish are minnowsSome 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 drownedBut every fish wifeFears for her fish.What we call mermaidsand they call merfish.

OBJECTIVESAimsIn this chapter you will look at the behaviour of fluids in motion and the explanation of thatbehaviour both in terms of forces, energy and the continuity of the fluid. The distinction betweensmooth and turbulent flow is investigated.Minimum Learning GoalsWhen 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 acontinuous substance rather than individual molecules.

3. Describe how energy is dissipated in turbulent motion.4. (i) Recall the definition of Reynolds number

R = vLrh

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, theflow 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 fluidpressure decreases as the flow speed increases. (Bernoulli's Principle.)(ii) Describe and explain the following phenomena : the venturi effect, the chimney effect, theworking of an atomiser.

PM3: Hydrodynamics 27

PRE-LECTURE

Recall the following background information from earlier chapters, particularly chapters FE3, FE4and 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 eachpart of the fluid on neighbouring parts. The kinetic theory is a school of thought whichseeks to understand how this pressure arises through the collision of individual molecules ofthe fluid with the walls and with each other. We will not in fact pursue kinetic theory anyfurther - but concentrate on experimentally observable laws concerning fluid pressure and itseffects.

(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 pressurevaries 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 isin 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 intothermal energy). Typically forces such as electromagnetic or gravitational are conservative andfrictional forces are dissipative.Workers in hydrodynamics (or aerodynamics) try to classify pressures and fluid forcessimilarly. However since the origin of these effects has a more complicated microscopicexplanation, this classification is not always so straightforward. The basic criterion employedis whether or not the equation of conservation of mechanical energy is obeyed.

LECTURE3-1 THRUST AND LIFT FORCESThe study of hydrodynamics involves the study of the interaction of fluids and solid bodies. Threeapparently 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 objectsFrom 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.

Force exerted on molecule

Before collision After collision

Figure 3.1 Collision of fluid molecules with a solid surface

PM3: Hydrodynamics 28

and, by the fact that forces occur in pairs, the fluid exerts an equal and oppositely directedforce on the solid.

Force exerted on solid

Figure 3.2 Force exerted by fluid molecules on solid surface

Wind direction !WING

Deflected air stream

RESULTANTLIFT FORCE

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, bygiving a velocity to (i.e. accelerating) each molecule of the fluid.

Example : Fish tails.Motion of tail

! RESULTANTTHRUST FORCE

Pivot

Tail

Wakedirection

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, youmust push the fluid backwards.

(2) The physics of what happens is the same whether it is the fluid or the solid or both whichis moving. Application

Aeronautical engineers can predict how an aeroplane will behave in flight by observing it at rest in a windtunnel (or even in a water tank).

PM3: Hydrodynamics 29

3-2 STREAM LINES AND TURBULENCEThe 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 donot 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 obstacleSince 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 mustincrease 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 theirmotion 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 continuoussubstance rather than to concentrate on the motion of individual molecules. Particles of thiscontinuous fluid can be considered to travel along these smooth continuous paths which are giventhe name streamlines. These stream lines can of course be curved or straight, depending on theflow of the fluid.

This continuous substance can be regarded as being made up of bundles or tubes ofstreamlines. The tubes have elastic properties:

(a) A tensile strength, which means that the parts of the fluid along a particular streamline sticktogether 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 netenergy has been given to it.

But when turbulence is established, a net amount of kinetic energy is left in the fluid after thebody has passed. Application

This is very important in aeronautical engineering. Air turbulence means increased fuel consumption inaircraft, 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 instreamline or turbulent motion.

DemonstrationShapes of marine animals, specially shaped corks.

PM3: Hydrodynamics 30

3-3 REYNOLDS NUMBERWhat 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 thekinematic 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

R ≡ vLrh

where v is a typical flow speed of the fluid, L is a typical length scale and h/r the kinematicviscosity 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 CONTINUITYFor 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 areaSpeed µ

1area .

The reason behind this is very easy to grasp. If you want a more rigorous statement, see thepost-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 becomingturbulent).

(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 cross-sectional area after the branch is twice that before the branch, so the flow speed must fall to half.

High Speed

Low 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 havehalf the cross-sectional area of those before.

PM3: Hydrodynamics 31

High Speed

Same 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 theequation of continuity to describe its motion.

DemonstrationsAir 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. Sothe equation of continuity says that where streamlines crowd together the flow speed must increase.

Streamlines spread out: speed low

Streamlines close together: speed high

Aerofoil

Fig 3.8 Streamline pattern around an aerofoil

PM3: Hydrodynamics 32

3-5 BERNOULLI'S PRINCIPLEDemonstration

An interesting effect which is easy to show is that, for a fluid (e.g. air) flowing through a pipe with aconstriction 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 onthe fluid and the only forces in the tube that might do work on the fluid are the driving forcesassociated with changes in pressure from place to place.

lower speedlower kinetic energy higher pressure

lower speedlower kinetic energy higher pressure

higher speedhigher kinetic energy lower pressure

Figure 3.9 Application of Bernoulli's PrincipleThe 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 writekinetic energy/volume + work/volume is constant,

i.e. l2 rv2 + p = constant.

In cases where the flow is not horizontal, we should add in the gravitational potential energy/volume

also: l2 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.

DemonstrationsThe 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:

lowpressure

highpressure

Air

Nozzle

Figure 3.10 Example of the Venturi EffectThe 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.]

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. {Flysprays 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 orping pong balls to swerve.

Demonstration Spinning cylinder.

POST-LECTURE3-6 MORE ON REYNOLDS NUMBERThere 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 sizeand speed. It is often not possible even to say which length you are talking about. For a bodymoving through a fluid it might be either length or breadth or thickness - or any other dimensionyou might think of. For a fluid flowing through a channel or a tube, it turns out that it is thediameter of the tube which enters. It is not until you learn more about the Reynolds Number thatyou 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 rathervague - indeed it can often only be stated as we did: "The flow of fluid in a system is more likely tobe turbulent if the system is large, than if it is small". It is not surprising then that the magic numberof 2000 is also only rough.

b) The "stickiness" index, the kinematic viscosity, is given the strange symbol h/r for thefollowing 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 ofviscosity (h) - see next lecture. How easily the fluid becomes turbulent is related to this but to thedensity (r) as well - or if you like, it defines a different measure of stickiness. It is pointless to sayany 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 ~ 10-3 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 isa ratio of two quantities - essentially a scaling number.

PM3: Hydrodynamics 34

Q3.1 : Work out the Reynolds Number for the following flow systems, and say in which ones you might expectthere 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 thediameter of the largest blood vessel, the aorta, to be ~ 10 mm; and guess that the viscosity of blood probablyis 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 CONTINUITYA careful derivation of the equation of continuity goes like this.

Consider a fluid flowing through an irregular tube like this

Speed =

Area =

v

1

1

A

Speed = v2

Area = 2

AA

B

Figure 3.12 Figure for derivation of Equation of ContinuityThe volume of fluid flowing past A in a very small time ∆t = A1v1Dt.So the mass which flows past A is r1A1v1Dt . Similarly the mass of fluid flowing past B in

time ∆t is r2A2v2∆t .

Now, when the flow is steady all the material which goes past A must go past B in the sametime (or else it will continually piling up somewhere) so

r1A1v1Dt . = r2A2v2Dt .

r1A1v1 = r2A2v2

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, thenthe 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 pastany point of the tube divided by time. It is given the name volume rate of flow, and is usuallydenoted 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 linesfor 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 thesituation is quite different.

3-8 BERNOULLI'S EQUATIONIf you really want a more careful derivation of Bernoulli's equation, you can look it up in anotherbook. 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 noenergy dissipation through turbulence. Bernoulli's equation only really applies when the motion isstrictly 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, yetthey all showed the characteristic effect of pressure drop.Q 3.3 Medical textbooks often quote Bernoulli's equation simply as

p + rgh = constant

implying that the kinetic energy term ( l2 rv2) is not important but the gravitational potential energy term is.

Use the average speed for blood flow quoted above and a typical human blood pressure of 104Pa to explain whythis is so.

Q 3.4 In section 2 above, you analysed streamlined flow round a corner. Using the result of that analysis showhow the pressure changes as the liquid goes round the corner. Can you reconcile this with the kind ofsimple 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 ofyour mouth. Discuss.

36

PM4 VISCOSITY

Come crown my brow tin leaves of myrtleI 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 despairI bet on one to beat a hare.I also know I'm now a pauperBecause of its tortley turtley torpor.

OBJECTIVES

AimsIn this chapter you will look at the effect of the application of shear stresses to fluids and theassociated phenomenon of fluid viscosity. The coefficient of viscosity will be defined. Poiseuille'sequation, which describes the flow rate of viscous liquids through pipes is presented, discussed andapplied to a number of situations

Minimum Learning GoalsWhen 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 ofviscosity).

3. (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.

4. (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 sheardeformation.

5. (i) Explain qualitatively the dependence of the rate of streamline flow of liquid in a pipe onpressure 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 relateto Poiseuille's equation.

6. Present the analogy between current in an electric circuit and fluid flow in a pipe system andexplain 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 isimportant 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 systemwas likely to be turbulent or streamline. The same number will turn up again to decide whetheror not the flow is viscous. The basic reason for the existence of this number and why it takesthe form it does is perhaps one of the most important questions in the whole study of fluidflow.

(ii) The definition of the shear deformation and Hooke's Law as it applies to bodies which behaveelastically under shear stresses. As we have pointed out it is the behaviour of a substanceunder 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 returnto its original shape afterwards. A liquid has a very, very small shear modulus. You can slideone bit of a liquid past another bit, and there will be no noticeable tendency for the two toregain their original shape when you shop pushing. Nonetheless the sliding of one bit doeshave 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 mathematicaltechniques of working with it (like Ohm's Law and Kirchhoff's Theorem). The flow of waterthrough pipes is an important part of this lecture - and obviously much the same kind ofmathematical 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 withdistance (x), being big at some point and small at another, we say a pressure gradient exists.

A measure of this is the derivative dpdx ; or, more crudely, the ratio:

difference!in!pressure!at!2!pointsdistance!between!those!2!points .

In a fluid we might expect the flow speed to change from point to point, and we could describethis variation by measuring the velocity gradient.

LECTURE4-1 VISCOSITYA feature which distinguishes one liquid from another is their "thickness" or the ease with whichthey pour.

DemonstrationObserve the flow of water, glycerine, oil, treacle, lava, pitch.<< This last experiment is on show in the Physics Department, University of Queensland. The

experimental record is:1920 Pitch poured in funnel1938 (Dec.) First drop fell1947 (Feb.) Second drop fell1954 (Aug.) Third drop fell1962 (May)` Fourth drop fell1970 (Aug.) Fifth drop fell. >>

The physical property which distinguishes these liquids from one another is something to dowith how well the liquid molecules adhere to one another; and this molecular adhesion leads to ahost of rather complicated effects.

PM4: Viscosity 38

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 connectionwith one another is very complicated. However, we have to start somewhere. We must select onephysical effect to measure, and try to understand the others in terms of it. We choose to concentrateon 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 thewall does not move. However, away from the wall the flow speed is not zero. So a velocity gradientexists.

Fluid Flow

High speed

Low speed

Stationary Wall

Fig 4.1 Velocity gradient in a stream of fluid moving past a stationary wallWe 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 VISCOSITYA simple experiment set-up capable of demonstrating the law of viscosity involves a small metalplate suspended in a tank of liquid. Before the experiment starts the weight of an attached pan isadjusted so that the plate is neutrally buoyant - i.e. it does not tend to sink in the liquid or to rise.

Metal plate(side view)

Applied Force

Liquid A

B C

D

E

F

G

H

Fig 4.2 Experiment to measure coefficient of viscosity: static situation

PM4: Viscosity 39

An extra force is now applied to cause the plate to move through the liquid. When the plate ismoving with speed v through the liquid, there will be a velocity gradient between AB and FE; alsobetween DC and HG. The complete apparatus is

Scale

Pointer

Glycerine

Plate (of known size)

Weights (to provide known force)

Fig 4.3 Experiment to measure coefficient of viscosity: complete apparatusDemonstration

(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 85 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 areaMass on pan/g Time to go 100 mm/s

2 13.55 57 3

We interpret these two sets of results as indicating that the speed of the plate increases with theshearing 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

v/d

distance betweenwall and plate =

Fig 4.4 Increased velocity gradient when the plate is closer to the wallMass on pan Time (close to wall)

7 g 4.2 s

We interpret this result as saying that a given shearing stress sets up a velocity gradient in thefluid.

PM4: Viscosity 40

More careful experimentation, or more detailed theoretical analysis, will clarify theseconclusions into Newton's law of viscosity which says: When a shearing stress acts within a fluidmoving in a streamlined motion, it sets up in the liquid a velocity gradient which is proportional tothe stress.

s = h dvdx

(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) youcan 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 apparatuswe used above.

Weights

Liquidunder test

AxleInner cylinder (rotates)

Outer cylinder (fixed)

ELEVATION

PLAN

Liquidvelocity gradient set up here

Fig 4.5 A torsional viscometerOf course the viscosity of a liquid can change.Demonstration

The viscosity depends on temperature, usually increasing as the temperature decreases (which is whyautomobiles need different oils in hot countries than in cold countries and indeed why the engine runs morefreely as it heats up).

However, this kind of detail you can catch up on later, when you come to talk about viscouseffects in your own discipline.

PM4: Viscosity 41

4-3 ALTERNATIVE STATEMENT OF NEWTON'S LAWSince there are many manifestations of viscosity, there are many different statements of the basiclaw. We have given Newton's statement, relating velocity gradient to shear stress (pressure).

Another statement, which research workers use, specifically points up the difference betweensolids and liquids.

SolidsWhen a shearing stress is applied to a solid it suffers a shear (i.e. a shear deformation)

A

B

C

y

x

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.

LiquidsWhen 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 thestress 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 = length!AClength!AB }

Viscous liquids obey Newton's law which saysshear stress µ 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:

velocity gradient = speed

transverse!length

=

!dxdty

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.

PM4: Viscosity 42

4-4 POISEUILLE'S LAWWe 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 onvelocity gradients, it must be true that liquid will flow faster through a wide pipe than a narrow one.

DemonstrationPolystyrene 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 flowingthrough 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'slaw of viscosity would give the volume rate of flow, qv , of fluid of viscosity h, through a pipe ofradius r and length l, when driven by a pressure difference ∆p as

qv =∆p.r4!p

!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 apipe 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 isfixed by the requirements of the population of Sydney. When Sydney doubles in size, the Water Board willhave to use twice as many pipes or replace the present pipes by ones of (2)1/4 times the radius. [This is anoversimplification, see post lecture.]

PM4: Viscosity 43

(ii) Respiratory system: The flow of gas here is also Poiseuillean. The resistance to flow is determinedprimarily by the narrow tubes leading to the alveoli. Any general constriction of the pipes, as occurs inbronchospasm 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 harderto 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; andif a fluid obeys one it obeys the other. There are a large number of important liquids which do notobey these laws, and they are called non-newtonian liquids. One of the simplest ways torecognise a newtonian liquid is to examine its velocity profile, which should be parabolic for aliquid which obeys Newton's (or Poiseuille's ) law.

Demonstrationflow of syrup.

4-5 REYNOLDS NUMBERPoiseuille's Law shows that viscosity is responsible for loss of pressure - and hence is an energydissipating phenomenon. We are not talking about energy loss due to turbulence, but about energyloss which occurs even when the flow is streamlined. It comes about through friction between thestreamlines moving past one another.

The criterion whether or not much energy is lost in this way, is therefore whether or not thereis much of a velocity gradient throughout the whole of the liquid. Since most of this gradient occursnear a boundary (in the so called Boundary layer, it is the ratio of the size of the system to the sizeof the boundary layer which is important.

System relatively largeenergy mostly conserved

System relatively smallmuch 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 numbersituations: in thick liquids (h large), or in small slow flow systems. On the other hand viscouseffects will not be important in thin liquids or in large, fast flow systems. (However in these lattersystems remember that turbulence is always possible, and energy can be lost through that means.)

PM4: Viscosity 44

In general then, flow patterns will be different in systems with low and with high Reynoldsnumbers. In particular, in very low Reynolds number systems, the flow is perfectly reversible sinceno turbulent effects can occur anywhere.

DemonstrationThe 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-LECTURE4-6 MORE ON POISEUILLE'S LAWPoiseuille's law can be derived from Newton's law; but to go through the complete derivation, even inpost-lecture material, would be completely opposed to the philosophy of this course. If you feel youcannot do without it, there are plenty of books you can look up. But in broad outline the derivationfollows 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 flowis streamlined, then this shearing resistance acts all over the surface of imaginary tubes offluids concentric with the tube trough which the fluid is flowing; and thus the retarding forcewill 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; andwhen 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 iswhere Newton's law comes in. There is most fluid-on-fluid slipping toward the outside of thetube (purely by geometry) so the velocity gradient is larges there, and is zero in the centre. Infact the velocity profile is a parabola.

Velocitycomponent

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 thevolume 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.

PM4: Viscosity 45

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 recallthat in the experiment done on screen glycerine flowed through the wide pipe more slowly thanwould 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 choiceof 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 columnsof soils. It was observed that the water rose faster in the column with the coarser grains. Can you say nowwhy this is so?

4-7 ELECTRICAL ANALOGUEOhm's Law says : V = R I

Poiseuille's Law says : Dp = 8p

l!hr4 ¥qv

The comparison is obvious, and hence it is most convenient to talk about flow through anykind of tube in terms of resistance defined thus:

R ≡ 8l!hp!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 quitepossible 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,

A B C

750 mm

pressure 5.0 Pa

pressure 1.0 Pa

250 mm

Fig 4.10 Data for Q4.4If you measured the fluid pressure at point B, what value would you get?

PM4: Viscosity 46

Q4.5 Consider these two different streamlined flow systems,

AB C

A BC

(a)

(b)

Fig 4.11 Data for Q4.5The 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 bothcases 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 circulatorysystem, why does most pressure drop occur over the capillaries?"

4-8 MORE ON REYNOLDS NUMBERIn 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 theviscous 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

=

12!r!v2

Dp ~

12!r!v2

h!dv/dx (by Newton's law)

Now, in a system where the boundary layer is comparable in size with the scale length of thesystem (L), we can approximate the velocity gradient in this expression by

dvdx ~

vL

and therefore the above ratio is = r!vL!2h

which, apart from the factor 2, is just the definition of the Reynolds number

PM4: Viscosity 47

Q4.6 If you were designing a circulatory system for the human body, where a prime requirement is that as littleenergy as possible should be dissipated, in order not to require the heart to pump any harder than absolutelynecessary, 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 VISCOSITYAs pointed out in last lecture, and as can be checked from the equation in 4-2, the units of viscosityare those of (pressure) ¥ (length)/(speed) or Pa.s. Some books of tables quote numbers in the oldcgs unit - the poise. You can easily convert by remembering that

1 Pa.s = 10 poisePurely to show you what range the values of viscosity can have, here are some common fluids:

FLUID VISCOSITY / Pa.swater (20°C) 1.0 ¥ 10-3

water (100°C) 0.3 ¥ 10-3

alcohol 1.2 ¥ 10-3

glycerine 1.5mercury 1.8 ¥ 10-3

air 1.8 ¥ 10-5

48

PM5 RHEOLOGY

Elephants are useful friendsEquipped with handles at both ends.They have a wrinkled, moth-proof hide;Their teeth are upside down, outside.If you think the elephant preposterousYou've probably never seen a rhinosterous.

OBJECTIVESAimsIn this chapter you will look at how many liquids have viscosity coefficients that change with theapplied shearing stress. This is called non-newtonian behaviour. Visco-elastic effects, whichcombine fluid and solid properties are also discussed.Minimum Learning GoalsWhen 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 inparticular that for these materials Hooke's Law is obeyed i.e. the stress is proportional to thestrain.

2. Refer back to chapter PM4 which discusses viscous forces, forces which show themselveswhen a fluid flows. Note in particular that for simple fluids such as water the shear stress isproportional to the velocity gradient, a relationship which is known as Newton's law. Note aswell that this law can be expressed alternatively as the shear stress is proportional to the rate ofshear.

LECTURE5-1 NON-NEWTONIAN FLUIDSRheology is the study of how bodies behave under the action of deforming forces. As the word isusually understood it deals with all materials except the purely elastic and those which are purelyviscous and obey Newton's law. Most materials do not fall into these two extreme categories - infact strictly speaking none do if the relevant parameters are varied widely enough. In this lecture weare 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. Forthese, the shear stress is not proportional to the shear rate and thus their viscosities depend on shearrate. Basically, they fall into two different classes. Firstly there are the pseudo-plastics for whichthe coefficient of viscosity decreases as the shear rate increases.

PM5: Rheology 49

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 anidentical 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 starchslowed down more than the glycerine; finally the level of glycerine solution fell below that of the starchsolution.

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 (andthe 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 stressdecreased (that is, as the head of liquid in the burette decreased). The increase in viscosity isshown by the disproportionate decrease in the flow rate.

(v) the amount of glycerine in the solution was chosen so that the (constant) viscosity wasoriginally greater but later less than that of the starch solution.

glycerine-water (newtonian)

starch solution(pseudo-plastic)

Shear rate

Viscositycoefficient

Fig 5.1 Variation of viscosity coefficient with shear rateThe 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 hasthe complicating feature that it does not flow, it does not behave as a fluid until a certain stress, theyield stress, has been exceeded. Once this stress has been exceeded, the viscosity either remainsconstant or decreases as the shear rate increases. These materials are called plastic.

ExamplesAn 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.

PM5: Rheology 50

The flow characteristics of the different non-newtonian fluids may be summarised thus:

Shear stress

Shear rate

plastic

newtonian

pseudo-plastic

dilatant

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 radialvelocity 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 flatregion in the centre where the shearing stress is less than the yield value.

DemonstrationThe 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. thatthere are some fluids whose viscous behaviour depends very much on the time they have beensheared and the time they have been at rest. There are two types of these. The thixotropic fluidsare like the pseudo-plastics in that the viscosity decreases with increasing shear rate but as well showthe property that at constant shear rate the viscosity decreases with time. Further, after being shearedat high rates and left at rest, the fluid does not recover its higher viscosity behaviour until after acertain 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 thatthe thixotropic fluid is akin to the pseudo-plastic fluid.

It is important to realise that there is basically no difference between a thixotropic and apseudo-plastic material. It is just that for a pseudo-plastic material the characteristic time is so smallas to be not observed in normal circumstances.

DemonstrationThus 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 appreciablyunder 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 isthixotropic, the flow would have continued when the vibration ceased. This would then have been ademonstration of what happens in certain earthquakes where buildings are destroyed because they were built on"quick-clay".

PM5: Rheology 51

5-2 VISCO-ELASTIC MATERIALSHaving 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 - thematerials which are neither purely elastic nor purely viscous, materials which show the properties ofboth solids and liquids. Whether these behave as solids or liquids depends on how long the stressis applied.

One way of distinguishing between an elastic solid and a viscous liquid is to apply a stress andmaintain this stress. A liquid will continue to deform as long as the stress is applied, but for a solidthere will be an instantaneous deformation and this will then remain constant with time. But manymaterials we normally class as solids do not behave like this.

DemonstrationIf 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 thewire will continue to deform appreciably over this period - it will have flowed slowly but continuously like aliquid.

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 wouldsupport 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 thatstrain. For a solid a certain stress is required to produce the strain, and to maintain the strain thestress must continue at this value. For a liquid, however, the stress instantaneously goes to zero oncethe strain is produced.

DemonstrationBut 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 dropsas shown in the following diagram:

Stress

Strain Time

From here the strain is kept constant;the axis now represents time

Elasticbehaviour Viscous

behaviour

Fig 5.3 Stress-strain curve behaviour for a wool fibre. The curve shows elastic (solid) andviscous (liquid) effects

This phenomenon is known as stress relaxation. The time for the stress to drop by e-1, (e isthe exponential function) is known as the stress relaxation time, t. Conventionally, a material iscalled a solid if t > 104 s, and a liquid if t < 10-4 s. The materials in between are the visco-elasticmaterials.

PM5: Rheology 52

There are many visco-elastic materials for example pitch, wool fibres, nylon, silk, vulcanisedrubber and bakers' dough.

DemonstrationsA 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 aviscous liquid.

Another example is egg white. This will flow but it also shows elastic properties in that it recoils if theflowing stream is broken or cut.

We can see that the visco-elastic materials behave as solids when they are stressed in shorttimes but as liquids when the stress is applied over long periods.

<< An interesting property of visco-elastic materials is the phenomenon known as the Weissenbergeffect.

DemonstrationThis 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 CLAYSA group of materials whose rheological properties are very important are the soils and clays whichcompose a large part of the surface of the earth. These materials are quite complicated. They aretwo-phase materials consisting of solid and liquid (usually water) and their properties depend verymuch on the concentration of water. The clays are particularly complex in that their propertiesdepend markedly on the presence of electrolytes.

Let us first consider these materials in the natural state, in the ground, where they are ofimportance 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 flowproblem it can be considered as the flow of the water through the soil rather than as the flow of acomposite material.

DemonstrationThis 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

Sand

Dyelines

Fig 5.4 Flow of water through wet sandThe 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 forexample can become either plastic or pseudo-plastic or indeed thixotropic and at all but the highestconcentrations of water, is visco-elastic.

DemonstrationThe reduction in the load-carrying capacity of a sand as the water content increases was shown.

PM5: Rheology 53

Levelgraduallyraised

Model building

Water

SandSand

Fig 5.5 Effect of increasing water content in wet sand

Clays are not only important in the ground. They have found widespread applicationparticularly 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 ofelectrolytes on the viscosity of these.

DemonstrationsA 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, byadding 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 thixotropicclay-water mixture which, when the casting wall is sufficiently great, can be rendered liquid by agitation andpoured off.

5-4 BLOODAnother 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, whichwork with very small samples.

Measurement show that blood is thixotropic. At high shear rates, normal blood has aviscosity of between 5 and 6 times that of water; but at low shear rates it may be several hundredtimes that of water. The viscosity for people with certain diseases such as myocardial infarction andthrombosis is much higher, particularly at low shear rates.Demonstration

The data presented on the screen, is taken from many patients at Sydney Hospital:

PM5: Rheology 54

Rate of shear / s -1

10

1

0.1

0.01

0.01 0.1 1 10 100 0.01 0.1 1 10 100

Viscosity / Pa.s NORMAL MEN INFARCTION ANDTHROMBOSIS

Fig 5.6 Blood flow dataNotice 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 lowerviscosity.

This however is not the whole story - the viscosity of the interior of the red cells plays a majorpart. If the red cells were rigid particles, when their concentration reached 65%, blood would havethe 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 redcells are not rigid but in fact fluid. Thus it is that blood can flow in the small capillaries - the cellsdeform as they flow. Any condition which leads to more rigid red cells leads to a much greaterblood viscosity.

DemonstrationAn 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 aroller type. They pump the blood so that the red cells are not damaged by too high shear rates but yet at a ratesufficiently great so that aggregation does not occur.

One final aspect to mention is the effect of drugs on the blood flow. These in general affectthe flow just by dilating or contracting the blood vessels.

DemonstrationsThus 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 bypassing a light beam through it.

It is thought, however, that as well, prolonged ingestion of nicotine has a long term effectleading to an increase in aggregation of cells and hence a higher viscosity and reduced blood flow.

PM5: Rheology 55

POST-LECTURE5-5 NOMENCLATUREThough most materials can be classified into broad categories such as those mentioned above itshould 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. Furtherthere 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 PROBLEMQ5.1 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

PM6FRICTION

At midnight in the museum hallThe fossils gathered for a ball.There were no drums or saxophonesbut just the clatter of their bones,A rolling, rattling, carefree circusOf mammoth polkas and mazurkas.Pterodactyls and brontosaurusesSang ghostly prehistoric choruses.Amid the mastodonic wassailI caught the eye of one small fossil.Cheer up, sad world, he said, and winked -it's kind of fun to be extinct.

OBJECTIVESAimsIn this chapter you will study the phenomena of friction, determine the laws of friction and considerthe explanation of these laws in terms of a microscopic model. The effects of naturally occludingsurface layers on of lubricants on friction are discussed.Minimum Learning GoalsWhen 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, boundarylubrication, 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 isflow 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 ofNewton's law of viscosity (Chapter PM4).

PM6: Friction 57

LECTURE6-1 OUR EVERYDAY EXPERIENCE OF FRICTION

DemonstrationTo slide heavy objects requires considerable effort. The heavier the object the greater the effort.

Fig 6.1 Sliding an objectDemonstration

It is a lot easier to move a heavy object by rolling it than by sliding it.

Fig 6.2 Rolling an objectWe 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 skatingand skiing.

Friction plagues us in many contexts.Demonstration

Considerable power is lost in overcoming friction in engines. Even more important than the power lossis the wear which results from friction.

Lubricants can be used to reduce friction and wear. Early applications were the use of animal fats tolubricate 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 FRICTIONThe laws of sliding friction were first formulated by Leonardo da Vinci and re-discovered in 1699by Admontons. They are empirical laws which give the dependence of the friction force on therelevant parameters.

DemonstrationOne 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 resistingthe motion, can be deduced. (See post lecture material.)

PM6: Friction 58

Fig 6.3 Sliding down an inclineThe laws were, however, determined using the following apparatus:

Fig 6.4 An experiment to determine the laws of frictionA 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 onthe 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 F r 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 theforce required to just get the object moving. The other is called the kinetic coefficient andcorresponds 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.

PM6: Friction 59

Friction force with 3 blocks = 20 N (steel on aluminium)

The laws stated are crude laws, the sort obtained with crude apparatus. Even with thisapparatus, complications are evident - the tendency of the object to stick again after it has started toslide. With more refined apparatus, these complications can be examined. As well, the dependenceof 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 forceThis 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 decreasewhen the velocity became high.6-3 EXPLANATION OF THE LAWS OF SLIDING FRICTION FOR METALSTo 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 aremicroscopically rough. This is shown by:

DemonstrationsPhotographs 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 sectionSince 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 contactmust be small - the surfaces must only touch at a few points.

DemonstrationThis 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.

PM6: Friction 60

Contact points

Fig 6.7 The "contact" between two plates, viewed at a microscopic levelDemonstration

The increase in area of contact with load was also shown by measuring the voltage drop across two surfacesin contact in the following circuit. The voltage drop depends on the resistance of the path; this resistancedecreases as the (real) area of contact increases.

Digitalvoltmeter

Surfacesin "contact"

A

Ammeter

V

Fig 6.8 Experimental arrangement to measure the potential drop between the surfacesIf 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 contactpoints are large enough to make the metals deform plastically.

The final important piece of experimental information is that strong adhesion occurs at thepoints of contact. The points of contact are in effect cold-welded forming a continuous solid. If thematerials are to be slid over each other, these junctions have to be sheared.

DemonstrationsThat 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 slidon another (which can occur in screwing or hammering) is of relevance in bone surgery. Sincetransfer can occur, it is very important to use tools of the same material as the metal plates etc. usedto 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 offriction can be given. If we assume the friction force is just that required to shear all the junctionsthen since the real area of contact increases with the load, then so must the friction force. This is thesecond 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 areasof 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 acontribution 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 BEHAVIOURThe 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 onlybecause surface layers exist that metals can be slid on each other. If the surfaces are cleaned in avacuum and the metals slid on each other in the vacuum, it is found the surfaces bond together sothat it is not only impossible to slide one on the other but it is impossible to pull them apart.

DemonstrationThe 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 - thecoefficient 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 athigher loads. The coefficient of friction then changes from that for surface layer sliding on surfacelayer 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 materialssuch as rubber this results from the materials deforming elastically rather than plastically at thepoints of contact. For other materials such as plastics this behaviour arises because they are visco-elastic.

DemonstrationAn 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 inartificial hip joints.

Finally, a few words about rolling friction. One form of this is the traction type as in a carwheel 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 thanany coefficient for sliding friction. The mechanism for frictional energy loss is quite different fromthat for sliding friction.

PM6: Friction 62

6-5 HEATING EFFECTSFriction 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 for10-4 !seconds or less are produced.

DemonstrationThese 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 meltingpoint polishing agent is necessary for efficient polishing.

It is the heating of the surfaces which causes the coefficient of friction to decrease at highvelocities. The high temperature enhances the plastic flow, and if it is high enough a layer ofessentially liquid metal is produced which acts as a lubricant.

DemonstrationThe 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 isknown as lubrication.

DemonstrationA block will slide much easier on a table if a layer of oil is spread on it.

Hydrodynamic lubricationThe 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 of0.001 and completely eliminates wear.

DemonstrationThis 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 (hydrodynamiclubrication)

In this type of lubrication, the frictional energy loss is due only to the viscous forces in thelubricant (see post-lecture material). The viscosity cannot be reduced indefinitely, however, since theseparation between the surfaces decreases as the viscosity decreases and eventually the surfacescome into contact.

PM6: Friction 63

Boundary lubricationWhen 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 isreduced to localised patches a few molecules thick. Lubrication under these conditions is known asboundary 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 itselffirmly to the clean metal surfaces formed as the cold-welded junctions are sheared. A layer is thenformed 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 itseems that the lubrication is of another type known as elastohydrodynamic lubrication in whichthe surfaces deform appreciably, the elastic deformations being comparable with the lubricating filmthickness so that it is maintained. In the synovial joints the cartilage elasticity is such as to allow thisprocess. The rheological properties of the synovial fluid are also important; in particular thesynovial fluid is thixotropic. This theory of synovial joint lubrication is supported by the fact thatsynovial fluid from rheumatoid arthritis cases is newtonian, and that wear in the joint occurs whenhard calcifying material is deposited on the cartilage. (This decreases the cartilage elasticity.)

PM6: Friction 64

POST LECTURE6-7 PROBLEMSQ 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 interms 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 FRICTIONThe second law of friction can be explained in terms of the shearing of cold-welded junctions. If sis the shear strength of the junctions and A the total real area of contact, then the frictional force F isgiven 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 expressionN = Ap

where p is the yield pressure, the stress at which plastic flow occurs.

Thus F = sp N

and so m = sp .

If s and p are taken as those of the softer material, this expression predicts reasonable valuesfor m. Really good agreement is not obtained because of the influence of surface layers.

PM6: Friction 65

6-9 MORE PROBLEMSQ6.3 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 ofuniform thickness d exists between the shaft and the bearing and that the length of shaft supported by thebearing is Ú.

Lubricant

r

d

Rotating shaft

Stationary bearing

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 ofthe 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 theshaft, 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 viscosityhas to be decreased. You should also note that the theory given here is that pertaining to the torsionalviscometer which was described in chapter PM4.)

66

PM7SOUND

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.

OBJECTIVESAimsIn this chapter you will study the phenomena of sound. You will find that the speed of sounddepends on the stiffness and the density of the medium it propagates through. The property specificacoustic impedance, of a medium is defined and its role in determining the reflection of sound fromboundaries between two media is discussed. The physics involved in the ear and in hearing isdiscussed.Minimum Learning GoalsWhen 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 theamplitudes of incident and the reflected waves and in terms of the specific acoustic impedancesof 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 whichthe problems arising from such a mismatch may be avoided.

5 Describe an experiment to analyze the frequency components present in a musical note orother sound.

6 (i) Draw a schematic diagram of the ear, identifying the following : outer ear, auditorycanal, eardrum (tympanic membrane), ossicles, oval window, cochlea.(ii) Draw a schematic diagram of the cochlea identifying the basilar membrane and the nervecells.(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 substanceswhich are deformed by volume compression stresses.

(ii) In Electrical Circuit Theory there is a theorem called the power matching theorem. Itconcerns 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 ismuch 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 lecturefollowing this television lecture.

In many, many cases involving the transport of energy from one place to another, the same kindof reasoning will be found to apply. There will be a source and a receiver (load); and there will beproperty of each which will effectively determine its ability to accept and transmit energy; thiswill usually be given the name impedance. Then, unless the impedance of the source is roughlythe 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 factthat this kind of oscillation is essentially an elastic phenomenon.

You will recall from that discussion that the frequency of a simple harmonic oscillation wasdetermined by (a) the mass of the object and (b) its spring constant (or alternately its Young'sModulus). Sound consists of elastic vibrations also (pressure oscillations in fact) and you wouldtherefore expect that a mathematical analysis would yield a similar result: viz. that the parametersdescribing the propagation of sound waves through a medium would also depend on twoquantities: 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 acomplicated oscillation into a sum of simpler sinusoidal oscillations. Each of these sinusoidalcomponents has an amplitude and a phase.The inverse problem, that of starting with simple oscillations and combining them into a morecomplicated shape is called fourier synthesis.

LECTURE

It is assumed that you already know a fair bit about sound, particularly how it is generated. We willconcentrate on two aspects only: its propagation and its analysis. We do this with specific referenceto the EAR and electrical hearing devices.

PM7: Sound 68

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 anothermembrane, the oval window (E). Beyond that point the information in the sound is converted intoelectrical signals to be sent to the brain.To understand the structure of the outer ear, we must talk in general terms about the propagation ofpressure waves through an elastic medium.

DemonstrationA 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 massmove 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 willthey 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 ofsound to increase as the density decreases, and as the bulk modulus increases.The formula is

(k is the bulk modulus, r is the density.)speed of sound: c = k / r

PM7: Sound 69

Consider now the boundary between two media. What determines how a wave propagates fromone 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 absolutelyidentical. [Why?]If any one ball is heavier or lighter than the others, then the disturbance will result in somereflection as well as propagation of kinetic energy.

DemonstrationA 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 areidentical. Reflection will occur at any point where there is as CHANGE of either mass or springstrength.

DemonstrationHowever, 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 ifyou 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

to serve as an index to tell us whether sound energy can efficiently be transferred from one mediumto another.

Clearly we could redefine this quantity (as is more usual) in terms of the velocity of sound, c:z = rc

z ≡ kr

PM7: Sound 70

The specific acoustic impedance is a property of the medium. A particular acoustic devicewill be described by a quantity called the acoustic impedance, which depends on both the shapeand size of the device.

The name impedance comes from the analogy with electrical circuit theory (as described inthe pre-lecture material). It may be helpful in understanding this subject to realize that acousticimpedance plays a role analogous in some respects to resistance; and in the same way specificacoustic impedance is analogous to resistivity.

Consider again the ear. Since both the density and bulk modulus of skin are much greaterthan 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 energyof 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 thegeometry of the device; and the narrowing of the auditory canal plays a most important role. Theimpedance of a layer of air near the drum is considerably greater than a layer of air near the outerear. Crudely speaking, a narrow column of air is more difficult to get moving than completely freeair, because of viscous effects at the side of the tube. Its impedance is increased by the narrownessof 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 theproblem 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 badlymismatched, and most of any sound wave is reflected away.

DemonstrationClinical measurements of drum impedance feed into the ear a sound of known intensity, and measurethe intensity reflected from the drum.

This gives the impedance of the drum (relative to that of air [see post-lecture]. Obviously, forexample, if no sound is reflected, the impedance of the drum is exactly the same as that of air.

PM7: Sound 71

Since you know the elasticity and density of skin and bone, it should be possible to work out theacoustic 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 waydrum impedance varies with pressure and to compare with a known normal ear. By this means, it ispossible to pick up certain specific defects in the drum or the ossicular chair - for example aperforated drum or a calcified ossicular chain.

To sum up: there is an enormous impedance mismatch between the eardrum and the outside air, soyou only detect a very small fraction of the energy in any sound wave. From this point of view, theear is in an inefficient hearing device. Electrical hearing devices for example get over this problemby 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 couldbe 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 ofits geometrical shape, different parts of this membrane resonate with oscillations of differentfrequencies. Nerve cells connecting this membrane to the auditory nerve, tell the brain which part ofthe basilar membrane is vibrating - and thus what simple harmonic oscillations are present in theoriginal sound wave.

Fig 7.6 Helmholtz Spectrum Analyzer

DemonstrationA musical note of the correct frequency will cause one of these cavities to resonate. The increasedamplitude of vibrations thus set up, are made to cause a gas jet to flicker up and down. Which cavity isresonating can then be detected in a rotating mirror system (this apparatus was built circa 1890).

PM7: Sound 72

DemonstrationThis 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 theoutput 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 timeand "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 variousmusical instruments their distinctive sounds. It is just a question of what harmonics (or overtones) arepresent, and in what strength (amplitude).*In order to get a feeling for the relationship between a sound and its spectrum, just listen to thesounds as they are produced and see if you can correlate the most prominent features of the spectrumyou see with what you hear.[Note: It seems to be the basic philosophy of much modern music that the older instruments havebecome stale and uninteresting. And indeed the spectrum of one wind instrument for example, is verylike that of another. So musicians today are trying to get sounds out of all sorts of unlikelyinstruments - 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 ofelectrical signals which the brain uses as a sensory input. And the information going to the brain isof 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 oscilloscopepattern of a sound made by the teacher. In the way this technique is usually used, the pattern the childhas 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 changethe 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 iscarried out with complete spectrum analyzers rather than simple oscilloscopes, that it will prove mosteffective.]

Demonstration(ii) The inverse of a Fourier analyzer is a Fourier synthesizer - and electronic organs act as suchwhen they build up a complicated oscillation from the fundamental and a few harmonics.

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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 thesetwo pressure-time patterns are quite different, they have the same fourier spectrum - the variousharmonics 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 thegeneric name synthesizers. The most commonly known perhaps is the Moog.DemonstrationThe new breed of musical instruments based on this simple electronic organ principle are given thegeneric 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 animpedance mismatch, like for example at the ear or a microphone, is very complicated, because it isthe mismatch of impedances of the device itself, and the small part of the air right next to it whichmust be analyzed. However when we consider the passage of a sound wave from one relatively largequantity of one medium to another, things are a little simpler. Then we need to consider only thebulk 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 Aris reflected from the boundary; then the sound power reflection coefficient ar, defined as the ratioof reflected sound energy to incident flow or sound energy, is given by

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 thetwo 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]

a r = Ar2

Ai2 = z2 - z1

z2 + z1

Ê

Ë Á

ˆ

¯ ˜

2

PM7: Sound 74

7-4 IMPEDANCE MATCHING

When there is an impedance mismatch between two media, it is possible to take steps toincrease the transfer of energy between the two. One method, for specific devices, was mentioned inthe lecture. For large quantities of the media, this can be done by intervening a third mediumbetween the two. Then so long as the specific impedance of this third medium is intermediatebetween that of the first two, it is found that the transmission of energy is greatly increased.

Perfect transmission of energy occurs, in theory, when

(where z3 is the specific impedance of the intervening medium) and where the thickness ofthe medium is a quarter wavelength.

You may recall from your optics lecture (L4) that a very similar condition - with refractiveindex rather than specific acoustic impedance - describes the ideal way to reduce reflections at air toglass boundaries in optical systems.

7-5 FREQUENCY RESPONSE OF THE EAR

Because of physical limitations, the ear will not respond to all frequencies. The mainlimitation comes from the geometry of the cochlea. If you remember what you have learnt aboutresonance, then a solid body can resonate with a sound wave if its size is roughly similar to thewavelength of the sound (in that material). Hence if you consider the basilar membrane to lookschematically 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 thehighest frequency will correspond to the width of where exactly the last nerve cell is located at thesmall end.

However, there are other factors which limit the frequency range, especially at the highfrequency and the most important is the elasticity of the eardrum. This determines its ability tofollow a very high frequency vibration. It is found, and you would expect it to be so, that as peopleage, 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 upperlimit of hearing can drop by 80 Hz every six moths.

Also the state of the joints in the ossicular chain clearly influence frequency response, sincethese too must vibrate at the same frequency as the drum.

Also the state of the joints in the ossicular chain clearly influence frequency response, sincethese 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 sizeof 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.

z3 = z1 z2

75

PM8ULTRASONICS

Puccini was Latin, and Wagner Teutonic,And birds are incurably philharmonicSuburban yards and rural vistasAre filled with avian Andrews Sisters.The skylark sings a roundelay,The crow sings The Road to Mandalay,The nightingale sings a lullabyAnd the seagull sings a gullaby.That's what shepherds listen to in ArcadiaBefore somebody invented the radia..

OBJECTIVESAimsIn this chapter you will study ultrasonics. Much of the lecture is concerned with the applications ofultrasonics using techniques of echoscopy and the Doppler technique.

Minimum Learning GoalsWhen 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 andultrasonic 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 andof its importance in the transmission of sound from one medium to another.

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LECTURE

8-1 INTRODUCTION

Sound experienced by human falls in the frequency range 0 - 20 kHz. Sound above this frequency isknown as ultrasound. It can be detected by some animals.

DemonstrationA 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, justbecause 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 playingthe tapes back at a slower speed (this reduces the frequency of the recorded sound) it was found thatthey make sounds in the 40 - 55 kHz regime. They use this ultrasound for navigational purposes andalso for locating their prey. The ultrasound is produced in short duration screeches of about l0 - l5milliseconds and that part of it which has bounced off something back into the direction of the bat isheard by it. The elapsed time gives the bat information on how far the object reflecting the pulse ofultrasound is from it.

8-2 GENERATION AND DETECTION OF ULTRASOUND

DemonstrationSound 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 atultrasound frequencies. One device for doing this is the Galton whistle but ultrasond can be producedmuch more conveniently and efficiently by making use of piezo-electric materials such as bariumtitanate.

These have the property that when a voltage is applied in a certain direction, the dimension of thematerial in that direction increases, and if the sense of the voltage is revesed then the dimensiondecreases. By applying a high frequency alternating voltage, the material is caused to vibrate at a highfrequency and so ultrasound is produced.

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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.

DemonstrationUltrasound falling on one of them causes it to vibrate and hence a voltage is produced across it, avoltage 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 intosmall 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 studycells by splitting them open, to produce small lesions in the brain, to treat a disease of the inner earknown as Meniere's disease, to remove hard deposits on the valves of the heart and so on. It canalso be used for more prosaic applications such as drilling and cutting.

DemonstrationUltrasound 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 causesnegative 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 fromthat when the source is stationary. This is known as the DOPPLER EFFECT. When the source ismoving towards the observer, the frequency is increased and when the source is moving away thefrequency is decreased. The effect also applies to a wave reflected from a moving object.

DemonstrationThe 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 workingproperly just by using a tuning fork).

A common example of the Doppler effect is the sound of an ambulance siren as the ambulanceapproaches and passes. The "red shift" i.e. the shift to lower frequencies, of light from othergalaxies is interpreted to mean that the galaxies are moving away from each other and hence theconcept of the expanding universe.

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DemonstrationThe 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 thedetected sound can be measured with a digital frequency meter..

Detailed measurements with such apparatus show that the change in frequency is proportionalto the velocity of the source. Since the change in frequencyu is proportional to the velocity, ameasurement of this change can be used to determine the velocity of the source. Or, alternatively, ifthe wave is reflected from a moving surface the change in frequency of the reflected wave relative tothe incident one gives the velocity of the moving surface. In this latter form there are many uses ofDoppler techniques with ultrasound in medicine. For example it is possible to detect the foetalheartbeat as early as the l0th week, and by measuring the motion of blood vessel walls it is possibleto 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. Apaste is applied between the skin and probe to improve impedance matching and so lessen power lossby reflection. The ultrasound produced by the probe is reflected from the flowing blood and thendetected by the probe. A probe such as this shows quite different sounds for arteries and veins. Forarteries, the sound is characteristic of the pulsatile blood flow in arteries. In veins, it is more like awind-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 researchprojects on animals.

DemonstrationSmall 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 apackage on the animal. In this way it is possible to look at patho-physiological conditions inconscious animals in realistic situations. This technique has been used for example on dogs whichhave been made hypertensive. It has also been used on small monkeys to study the effect of severeoxygen lack on the circulation. In a proposed experiment it is to be used on baboons to study theeffect 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 thepuse to return after being reflected from a distant object is measured then this time is a measure ofthe 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 ofcourse the technique used by bats for navigation. It is used it as sonar for depth sounding, detectionof submarines and shoals of fish.

In medicine, the technique is used and is then known as echoscopy. If ultrasound travelling inone medium encounters another, in general some will be transmitted into the other medium as well asbeing reflected.

How much energy is reflected depends on the sound power reflection coefficient, which in turndepends on the specific acoustic impedances of the two media If these are almost the same, littleenergy is reflected; if they are widely different, much energy is reflected.

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There are not large differences bvetween the specific acoustic impedances of the various softtissues in the body (see post lecture material). Thus the pulse-echo technique can be used to obtaintwo-dimensional cross-sectional views through various organs in the body. As the ultrasoundpenetrates into the organ, some is reflected at a soft-tissue boundary but most is transmitted to sufferfurther reflections from successive boundaries. The time delays and amplitudes of the signals fromthe 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-raypicture. The latter is a three dimensional view compressed into the two dimensions of the x-rayplate. 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 retinaldetachment and is extremely valuable in detecting tumours behind the eye. It should be noted that theacoustic impedances of the transducer and the eye are matched in this machine by having thetransducer 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 outersurface of the eye.

Demonstration

This technique is also used in obstetrics to scan the pregnant uterus. The matching of acousticimpedances should again be noticed. The technique is of majeor importance in this field for unlike xrays, ultrasound appears to be completely safe. The technique gives information on the size of thefoetus, how many there are, if it is growing at a reasonable rate, whether it has miscarried, whetherthere are any gross abnormalities etc.The technique can also be used for examining the non-pregnantabdomen 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 1.53blood 1534 1.04 1.59fat 1440 0.97 1.40brain 1510 1.03 1.55liver 1590 1.03 1.64muscle 1590 1.03 1.64bone 3360 2.00 6.62air 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]