Download PM5 RHEOLOGY

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PM5
RHEOLOGY
Elephants are useful friends
Equipped with handles at both ends.
They have a wrinkled, moth-proof hide;
Their teeth are upside down, outside.
If you think the elephant preposterous
You've probably never seen a rhinosterous.
OBJECTIVES
Aims
In this chapter you will look at how many liquids have viscosity coefficients that change with the
applied shearing stress. This is called non-newtonian behaviour. Visco-elastic effects, which
combine fluid and solid properties are also discussed.
Minimum Learning Goals
When you have finished studying this chapter you should be able to do all of the following.
1.
Explain and use the following terms: rheology, non-newtonian flow, pseudo-plasticity,
dilatancy, plasticity, thixotropy, visco-elasticity, creep, stress relaxation.
2.
Describe, in terms of shear stress, shear, rate of shear, and time, the behaviour of the following
materials: starch solution, cornflour solution, wet sand, toothpaste, quickclay, wool fibres,
pitch, wet soil, blood.
3.
Describe an experiment showing the different effect of shear stress on a newtonian fluid (e.g.
glycerine-water mixture) and a non-newtonian fluid (e.g. starch solution).
PRE-LECTURE
1.
Refer back to chapter PM1 which discusses the behaviour of purely elastic materials. Note in
particular that for these materials Hooke's Law is obeyed i.e. the stress is proportional to the
strain.
2.
Refer back to chapter PM4 which discusses viscous forces, forces which show themselves
when a fluid flows. Note in particular that for simple fluids such as water the shear stress is
proportional to the velocity gradient, a relationship which is known as Newton's law. Note as
well that this law can be expressed alternatively as the shear stress is proportional to the rate of
shear.
LECTURE
5-1 NON-NEWTONIAN FLUIDS
Rheology is the study of how bodies behave under the action of deforming forces. As the word is
usually understood it deals with all materials except the purely elastic and those which are purely
viscous and obey Newton's law. Most materials do not fall into these two extreme categories - in
fact strictly speaking none do if the relevant parameters are varied widely enough. In this lecture we
are going to look at some of the more complicated behaviour one can get.
To start we consider the fluids that do not obey Newton's law - the non-newtonian fluids. For
these, the shear stress is not proportional to the shear rate and thus their viscosities depend on shear
rate. Basically, they fall into two different classes. Firstly there are the pseudo-plastics for which
the coefficient of viscosity decreases as the shear rate increases.
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PM5: Rheology
Demonstration
A solution of starch is pseudoplastic. Its pseudoplasticity can be easily demonstrated by letting it flow
from a burette and comparing its flow with that of a suitable viscosity newtonian fluid flowing from an
identical burette. (The newtonian fluid used was glycerine and water.)
At the beginning the starch flowed faster than the glycerine; as both fluids started to slow, the starch
slowed down more than the glycerine; finally the level of glycerine solution fell below that of the starch
solution.
In explaining this experiment we note that
(i)
the shear stress is related to the pressure difference driving the liquid
(ii)
the rate of shear is related to the flow rate.
(iii) for a newtonian liquid (the glycerine-water mixture) the viscosity, that is the relation between
the shear stress and the rate of shear is constant; as the burette emptied the shear stress (and
the pressure) reduced while the rate of shear (and the flow rate) reduced correspondingly.
(iv) for the non-newtonian liquid (the starch solution) the viscosity increased as the shear stress
decreased (that is, as the head of liquid in the burette decreased). The increase in viscosity is
shown by the disproportionate decrease in the flow rate.
(v)
the amount of glycerine in the solution was chosen so that the (constant) viscosity was
originally greater but later less than that of the starch solution.
glycerine-water
(newtonian)
Viscosity
coefficient
starch solution
(pseudo-plastic)
Shear rate
Fig 5.1 Variation of viscosity coefficient with shear rate
The second type of non-newtonian fluid is the so-called dilatant fluid which has the reverse
behaviour to a pseudo-plastic fluid viz its viscosity increases as the shear rate increases.
Examples
One example of a dilatant fluid is a thin paste of cornflour.
Other examples of dilatant fluids are printing inks, vinyl resin pastes and suspensions at high solid content
such as wet beach sand which shows its dilatancy through the fact it stiffens when trodden on.
Apart from these two main categories there is one other type of non-newtonian fluid which has
the complicating feature that it does not flow, it does not behave as a fluid until a certain stress, the
yield stress, has been exceeded. Once this stress has been exceeded, the viscosity either remains
constant or decreases as the shear rate increases. These materials are called plastic.
Examples
An example of a plastic material is toothpaste. It is necessary that toothpaste will stay on the brush but it
must be easily extrudable.
Some other examples of plastic materials of which there are many, are good brushing paints and sewage
sludge.
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PM5: Rheology
The flow characteristics of the different non-newtonian fluids may be summarised thus:
plastic
Shear
stress
pseudo-plastic
newtonian
dilatant
Shear rate
Fig 5.2 Variation of shear rate with shear stress for newtonian and non-newtonian fluids
Another way in which the non-newtonian nature of a fluid can show itself is in its radial
velocity profile as it flows through a narrow tube.
For newtonian fluids this radial velocity profile is parabolic.
For non-newtonian fluids the radial velocity profile is not parabolic. It is somewhat sharper
for dilatant fluids and for pseudoplastics it is blunter. For plastic materials there is a completely flat
region in the centre where the shearing stress is less than the yield value.
Demonstration
The protoplasmic flow in the plant known as "slime mould" has a distinctly non-parabolic radial velocity
profile.
Before leaving non-newtonian fluids, one further complication needs to be mentioned viz. that
there are some fluids whose viscous behaviour depends very much on the time they have been
sheared and the time they have been at rest. There are two types of these. The thixotropic fluids
are like the pseudo-plastics in that the viscosity decreases with increasing shear rate but as well show
the property that at constant shear rate the viscosity decreases with time. Further, after being sheared
at high rates and left at rest, the fluid does not recover its higher viscosity behaviour until after a
certain characteristic time has elapsed which may be as long as several hours.
The other type is the rheopectic fluid which is akin to the dilatant fluid in the same way that
the thixotropic fluid is akin to the pseudo-plastic fluid.
It is important to realise that there is basically no difference between a thixotropic and a
pseudo-plastic material. It is just that for a pseudo-plastic material the characteristic time is so small
as to be not observed in normal circumstances.
Demonstration
Thus it is that if a thixotropic varnish which is like jelly after being left at rest for a long time is mixed
up, it becomes quite liquid and stays like that for many minutes after the mixing has ceased.
For a pseudo-plastic material, the return to its original state would be instantaneous.
Demonstration
This was shown by placing a paste of plaster of Paris on an inclined plane. It did not flow appreciably
under these circumstances but when the table was vibrated it flowed freely. On stopping the vibration,
however, the flow ceased instantaneously. If the material on the vibrating table has been "quick-clay", which is
thixotropic, the flow would have continued when the vibration ceased. This would then have been a
demonstration of what happens in certain earthquakes where buildings are destroyed because they were built on
"quick-clay".
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PM5: Rheology
5-2 VISCO-ELASTIC MATERIALS
Having discussed the effect of time on non-newtonian fluids we are led naturally to the other
major class of materials covered by the subject of rheology, viz the visco-elastic materials - the
materials which are neither purely elastic nor purely viscous, materials which show the properties of
both solids and liquids. Whether these behave as solids or liquids depends on how long the stress
is applied.
One way of distinguishing between an elastic solid and a viscous liquid is to apply a stress and
maintain this stress. A liquid will continue to deform as long as the stress is applied, but for a solid
there will be an instantaneous deformation and this will then remain constant with time. But many
materials we normally class as solids do not behave like this.
Demonstration
If we take, for example, a copper wire and stress it, we do get an instantaneous deformation but if we keep
the stress constant and observe over a period of a day or so, we find that if the stress is large enough then the
wire will continue to deform appreciably over this period - it will have flowed slowly but continuously like a
liquid.
This phenomenon is known as creep.
[Since it flows one can of course ascribe a viscosity to such a solid. Conventionally a material
is taken to be a solid if its viscosity is >1014 Pa.s. At such a viscosity a 25 mm edge cube would
support a man for a year and sink only 2.5 mm.]
Another way to distinguish between a solid and a liquid is to produce a strain and maintain that
strain. For a solid a certain stress is required to produce the strain, and to maintain the strain the
stress must continue at this value. For a liquid, however, the stress instantaneously goes to zero once
the strain is produced.
Demonstration
But if we take a wool fibre for example and carry out this experiment, we find that the stress does not
remain constant nor drop instantaneously to zero. Rather, the stress to maintain a given strain gradually drops
as shown in the following diagram:
Elastic
behaviour
Viscous
behaviour
Stress
Strain
Time
From here the strain is kept constant;
the axis now represents time
Fig 5.3 Stress-strain curve behaviour for a wool fibre. The curve shows elastic (solid) and
viscous (liquid) effects
This phenomenon is known as stress relaxation. The time for the stress to drop by e-1, (e is
the exponential function) is known as the stress relaxation time, t. Conventionally, a material is
called a solid if t > 104 s, and a liquid if t < 10-4 s. The materials in between are the visco-elastic
materials.
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PM5: Rheology
There are many visco-elastic materials for example pitch, wool fibres, nylon, silk, vulcanised
rubber and bakers' dough.
Demonstrations
A good example is the material known as "silly" putty. If this is stressed in short times, it behaves
elastically - thus it will bounce. If, however, a stress is applied over a long time, it will flow - it behaves as a
viscous liquid.
Another example is egg white. This will flow but it also shows elastic properties in that it recoils if the
flowing stream is broken or cut.
We can see that the visco-elastic materials behave as solids when they are stressed in short
times but as liquids when the stress is applied over long periods.
<<
An interesting property of visco-elastic materials is the phenomenon known as the Weissenberg
effect.
Demonstration
This is the effect that if a rod is rotated in a large mass of material, the material will climb the shaft. This is
observed when cake mix is mixed with a rotating beater.
>>
5-3 SOILS AND CLAYS
A group of materials whose rheological properties are very important are the soils and clays which
compose a large part of the surface of the earth. These materials are quite complicated. They are
two-phase materials consisting of solid and liquid (usually water) and their properties depend very
much on the concentration of water. The clays are particularly complex in that their properties
depend markedly on the presence of electrolytes.
Let us first consider these materials in the natural state, in the ground, where they are of
importance to people like engineers who are interested in their weight-bearing properties.
At low concentrations of water, these materials behave as solids, and if one does have a flow
problem it can be considered as the flow of the water through the soil rather than as the flow of a
composite material.
Demonstration
This is shown by maintaining a constant head of water over a bed of sand in a sheet pile structure and
injecting dye into the top of the sand on the high pressure side.
Water
Dye
lines
Sand
Fig 5.4 Flow of water through wet sand
The dye follows well-defined lines which can be predicted mathematically. The dye lines show the velocity
is fastest where the cross-section is narrowest, and thus illustrate the equation of continuity.
As the water concentration in a soil or clay increases its properties change markedly. Clay for
example can become either plastic or pseudo-plastic or indeed thixotropic and at all but the highest
concentrations of water, is visco-elastic.
Demonstration
The reduction in the load-carrying capacity of a sand as the water content increases was shown.
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PM5: Rheology
Model
building
Level
gradually
raised
Water
Sand
Sand
Fig 5.5 Effect of increasing water content in wet sand
Clays are not only important in the ground. They have found widespread application
particularly in the ceramic industry. A method of ceramic manufacture known as "slip-casting"
makes use of the thixotropy of certain clay-water mixtures and also of the pronounced effect of
electrolytes on the viscosity of these.
Demonstrations
A clay-water mixture suitable for pouring into a mould or piping throughout a factory has to have quite a
high concentration of water. A low viscosity mixture with very high clay content can be made, however, by
adding a small volume of electrolyte (sodium silicate and soda ash). This process is known as deflocculation
(the aggregation of the clays particles is greatly affected by the electrolyte).
If a deflocculated clay-water suspension is placed in a plaster mould, the water is absorbed into the mould.
As the water leaves, the wall of the casting gradually builds up. In the centre of the casting is a thixotropic
clay-water mixture which, when the casting wall is sufficiently great, can be rendered liquid by agitation and
poured off.
5-4 BLOOD
Another material of obvious importance is blood. This has most interesting rheological
properties.
Blood is a complex fluid, consisting of a plasma in which are suspended a variety of cells, the
predominant ones being the red cells.
Demonstration
To measure its viscous properties, special rotating viscometers have to be constructed and used, which
work with very small samples.
Measurement show that blood is thixotropic. At high shear rates, normal blood has a
viscosity of between 5 and 6 times that of water; but at low shear rates it may be several hundred
times that of water. The viscosity for people with certain diseases such as myocardial infarction and
thrombosis is much higher, particularly at low shear rates.
Demonstration
The data presented on the screen, is taken from many patients at Sydney Hospital:
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PM5: Rheology
10
Viscosity / Pa.s
INFARCTION AND
THROMBOSIS
NORMAL MEN
1
0.1
0.01
0.01
0.1
1
10
100
0.01
0.1
1
10
100
Rate of shear / s-1
Fig 5.6 Blood flow data
Notice the similarity between these figures, and figure 5.1.
Blood's viscous behaviour is partly due to aggregation of the red cells. These aggregates can,
and do, form when the shear rate is low, but at higher shear rates they break up, giving a lower
viscosity.
This however is not the whole story - the viscosity of the interior of the red cells plays a major
part. If the red cells were rigid particles, when their concentration reached 65%, blood would have
the consistency of concrete. This however does not happen.
Blood is still very fluid even at 99% red cell concentration. The reason for this is that the red
cells are not rigid but in fact fluid. Thus it is that blood can flow in the small capillaries - the cells
deform as they flow. Any condition which leads to more rigid red cells leads to a much greater
blood viscosity.
Demonstration
An important consequence of the rheological nature of blood is that when it is artificially pumped, as it is
in certain types of heart surgery where the heart is by-passed, special pumps have to be used. These are of a
roller type. They pump the blood so that the red cells are not damaged by too high shear rates but yet at a rate
sufficiently great so that aggregation does not occur.
One final aspect to mention is the effect of drugs on the blood flow. These in general affect
the flow just by dilating or contracting the blood vessels.
Demonstrations
Thus it is that when a cigarette is smoked, there is a short term reduction in blood flow due to the nicotine
constricting the blood vessels. This can be seen by measuring the blood flow in the vessels of the ear lobe by
passing a light beam through it.
It is thought, however, that as well, prolonged ingestion of nicotine has a long term effect
leading to an increase in aggregation of cells and hence a higher viscosity and reduced blood flow.
PM5: Rheology
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POST-LECTURE
5-5 NOMENCLATURE
Though most materials can be classified into broad categories such as those mentioned above it
should be emphasised that some materials are very complex and don't neatly fit into such categories.
Thus it is that some non-newtonian fluids also show pronounced visco-elastic behaviour. Further
there are no sharp dividing lines between the categories, particularly as regards their time behaviour.
Thus it is that the terms thixotropic and pseudoplastic are often used almost interchangeably.
5-6 PROBLEM
Q5.1
List any materials of everyday experience, other than those given here, which you think are visco-elastic or
non-newtonian. Categorise them and say why you think they fall into these categories.
5-7 REFERENCES
"The emergence of rheology"
Markovitz, Phy. Today , p 23, April 1968.
"Quick clay"
Kerr, Sci. Amer., p 132, November 1963.
"The flow of matter"
Reiner, Sci. Amer., p 122, December 1959.
"Non-newtonian viscosity and some aspects of lubrication"
Stanley, Phy. Educ., p 193, 1972.