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Chapter 6 Viscosity
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Introduction
How liquids flow
The concept of viscosity
Quantifying viscosity
The units of viscosity
Measuring viscosity
Viscometers
Redwood,
Ostwald, and
Other viscometers
Endnote
Tutorial sheet
©Ivor Bittle
1
Chapter 6
Viscosity
Introduction
I am in the process of describing an empirical science that is in existence and the next major
thing to describe is the way that we deal with the flow of fluids in pipes. This cannot be done
without the use of the property viscosity.
I have said that this is a personal text that lays out my view of this subject and here I am
writing for any one who uses engineering science and is a bit more curious than most. You
may not always agree with what I say but you are free to ignore it and I have found that what
follows works for me. Viscosity has always troubled me because I learnt about it through
Poiseuille and laminar flow as if this is the main application and then without more ado it was
used in every conceivable type of flow. Subsequently, however I thought about it, there
seemed to be no obvious justification for this extension of its use and yet it worked extremely
well. With the rise of the internet I looked for explanations but I have never seen any satisfying
explanation. The question that I now need to answer is, why is viscosity a property and
therefore universal in its application and what does it quantify ? In the event the question was
in the front of my mind for many days before I could put together all that I know in some
satisfactory way. Keep in mind that all we need as engineers is an explanation that lets us think
about engineering applications without making silly mistakes of principle.
I think that people do not spend enough time just looking at water on the move. Water is the
liquid that we see most often although no one else that I know ever bothers to look at it
carefully even when they have an opportunity. When I first started looking at flowing water I
could make little sense of it, mostly because my degree course was not a suitable preparation.
Sometimes I could see no sense because I did not know the shape of the surfaces that affected
the flow because they were obscured by the murky water. Year on year I learnt more, and
more observations fitted together and now I can often understand what is happening and why.
Figure 6-1
©Ivor Bittle
2
How liquids flow
Look at figure 6-1. It was taken in the River Medway in Kent in the UK. The flow in the river
was greater than normal and the water flowed from right to left past bridge piers that have
these pointed ends1. The water breaks away
from the sharply angled corner and a swirling
wake forms between the main flow and the
side of the pier. Those who seek to make
diagrams of this flow will usually just draw a
lot of squiggly lines2 to indicate no orderly
flow. But, if you have the opportunity to look
long enough, you can find some order and, in Figure 6-2
^
my view it is important to look for that order.
In this flow there is a cyclic change in the flow
pattern and what you need to know is that the foot of the pier has a large apron stretching
over, and standing proud of, the natural bed of the river to protect it from scour. This affects
the flow at the foot and these
two flow patterns get out of
phase. If you look at the wake
you cannot avoid seeing eddies
that form in the wake and get
swept downstream. There is one
in figure 6-2 just above the
pointer. These eddies are
everywhere in the natural flow
of water and air.
Figure 6-3 is from Prandtl’s
book where there are several
pictures of eddies made using
paint on glass and aluminium
powder sprinkled on the surface
of eddying water. Here he is
showing us how a twodimensional flow that has
become detached can become
re-attached as the speed of flow
increases. I am interested in
Figure 6-3
these eddies. Look at them, they
are not accidental or random,
they are an important part of the overall flow pattern. We do not know for certain whether
these eddies all have the same direction of rotation and having three or more eddies in a line
like these is quite common. Inevitably where two eddies exist side-by-side there is contra flow
between them. Eddies are not necessarily circular nor are they the same shape instant-byinstant but over time they have a well-defined shape and appear to be an essential and
predictable part of the flow pattern. Eddies do not rotate like wheels, instead the angular
velocity increases with decreasing radius and, throughout the eddies, there is shearing and this
shearing takes place continuously both within the eddies and between them. Unhappily Prandtl
has not given us sufficient of this flow pattern to be really useful but, in that, he is not alone. I
Not much has changed. Those bridge builders of long ago used an intuitive approach to their designs. It took a
long time for the pier to become rounded. Still, when someone attempts to design a leading edge for a keel or a
bow for a boat the first thought is to make it sharp. It is sad.
2I have in the past and will again in this text but I suspect that there is order in this chaos somewhere and I plan
to look for it.
1
©Ivor Bittle
3
have never seen a photograph of all the elements of a flow pattern3. It seems to me that that
such flow patterns can only come about if there are shearing forces in the water. Clearly
Nature has designed a flow pattern in which to lose mechanical energy to internal energy very
effectively.
You can see from the horizontal stone course in picture 6-1 that the surface of the river
Medway suffered a drop in level as it flowed past this bridge. This means that potential energy
has been given up and it is now contained partially in an increased velocity and partly in the
wakes from the several piers. Beyond the bridge the velocity will return to normal in an
eddying flow that is also worth considerable study and the energy given up in the drop in level
will be contained in this flow. If you try to think of ways to contain energy without an increase
in level or average velocity you will find that the options are very limited and there is only one
way for this energy to be contained and that is in eddies rotating with all sorts of different
axes.4 Go and look at your local river and see the way that it eddies after it has passed an
obstruction. The well-defined eddies will have disappeared from the surface leaving a flow
with much finer grained disturbance in it and the kinetic energy will be dispersing into the
water. The mechanism doing the dispersion is the shearing and it is reasonable to suppose
that some liquids resist this shearing more than others. This means that, say, crude oil
flowing in the natural bed of a river would present us with a new and exciting flow pattern to
study. We, in ordinary parlance, talk of thin and thick liquids and we think of them as being
runny or slow to pour. It is all about shearing. But ultimately it is not the shearing that we can
see that matters, it is the shearing that goes on at molecular level that is the basic mechanism
by which mechanical energy is dispersed into the fluid as random internal energy. That motion
is going on all the time. It is part of the character of this motion that we quantify when we
measure viscosity.
I have been looking at flow with a random element like this water in a river and perhaps in the
wind but liquids can also flow in a totally orderly way. Look at figure 6-4. It is from chapter 9
Figure 6-4
Figure 6-5 Close up of 6-4
of my book on model yachts that is on my website. It was produced on a home-built HeleShaw apparatus to try to visualise the flow round sails. (See details in book.)5 Here I am
interested in the blow-up of part of the flow pattern. It is the total orderliness that I find so
astonishing. The water was flowing between glass plates spaced about 1 millimetre apart and
the lines are of ordinary fountain pen ink. The water was flowing round a brass shape
representing a sail attached to a mast. Water in contact with the plates is stationary and the
As I have worked on this website I have become more and more frustrated by the absence of any flow patterns
that are complete in that the approach flow and the leaving flow have been recorded.
4 Sailors know this. They call it gusting, veering and backing because that is what it feels like to them.
5 I cannot now remember when I first saw a Hele-Shaw apparatus but I did have one made when I was teaching. I
learnt a lot from it but, in retrospect, not as much as I might have done. No other lecturer took any interest in it
partly I suppose because not all lecturers in engineering turn out to be good with their hands.
3
©Ivor Bittle
4
velocity distribution between the plates is parabolic. When there is a change in direction of the
lines this parabolic velocity distribution takes on a transverse component as well and the line
widens because there is the same pressure difference to produce the essential centripetal
acceleration and because the water moves at different tangential speeds the centripetal
acceleration to make them stay together is not the same for all the flow. When you actually see
the apparatus running you can see the detail of the lines widening and narrowing and those
“shaded” areas are actually sheets of ink having this parabolic shape across the flow. It is all
too evident that water can flow without any discernable mixing even though it must be
subjected to continual shearing at molecular level. I like these images that come from
Hele-Shaw, I suspect that the apparatus is more versatile and instructive than is generally
supposed but it needs experimental skill to get results from it6.
My examples above show two very different aspects of this shearing and I need to attempt to
find a mechanism that accounts for all the effects that are attributed to viscosity. I think that I
must start with Newton, who, at a time (1687) when there was no concept of a molecular
structure for solids, liquids and gases, could offer no insight to the mechanism that could
create this internal friction that he called viscosity but could cut through the maze of different
effects of viscosity and suggest a way of quantifying it.
The concept of viscosity
Newton saw that viscosity produced an effect just like solid friction in that mechanical energy
was just lost into the liquid as it might be into a solid. His concept has obvious similarities with
the way we think about solid friction as figures 6-6 and 6-7 show. In solid friction we put
F    m  g and call  the coefficient of friction. In the system shown in figure 6-7 a liquid is
imagined to fill the space between the flat and level plane and the flat plate, of surface area A ,
which is distance t above it. A horizontal force F is imagined to act on the block and to
Figure 6-6
Figure 6-7
cause it to move at a steady speed c . Newton then used this system to define a coefficient of
viscosity for the liquid. He put:c
F    A  where  is the coefficient of viscosity7.
t
In my more recent work I have become even more convinced of the value of the Hele-Shaw rig
This is not a coefficient based on a rational expression, it is just an empirical expression linking the measureable
quantities in a logical way. As a result the coefficient will have units and its numerical value will depend on the
system of units used. Be careful.
6
7
©Ivor Bittle
5
Clearly Newton has proposed a system, under which the fluid is made to move with continual
shearing, that is both well defined and easily visualised. I interpret Newton’s system as
behaving as I have shown in figures 6-7a, b and c where a straight line through the liquid layer
remains straight as the layer is continually distorted and there is a uniform rate of distortion of
c
.
t
Figure 6-8a
Figure 6-8b
Figure 6-8c
We need to have some mental model of what goes on in that layer because this is where
mechanical energy is ultimately converted to the random motion of internal energy and I want
to look towards the molecular motion of water as an example that might also apply to other
liquids.
However it is easiest for me to start with gases. The kinetic theory of gases gives a very good
picture of the structure of a gas. It postulates that a gas is composed of separate molecules,
(which may be single atoms). The molecules “fly” freely at high speed, colliding very frequently
with other molecules and with the walls of the container in which they are enclosed. The scale
of the structure of a gas is indicated by the following figures. The common gases at room
temperature and at a pressure of one atmosphere have about 2  7.1019 molecules per cubic
centimetre. Even with this concentration there is still space between the molecules for them to
move freely at high speed (about 350 m/s) through a distance of about 7 molecular
"diameters" between collisions and the number of collisions made by each molecule each
second is about 1.1011 . These are large numbers that I can accept but cannot imagine.
The molecules of the gas appear to have mass. They move with high linear speed, rotate, and
where the molecules have two or more atoms they can vibrate in the inter-atomic bonding.
Kinetic energy can be stored in the gas in these motions. (Measurements seem to suggest that
little energy is stored in vibration in a gas, nevertheless it does have this degree of freedom.)
Consequently the gas may be regarded as having a stock of kinetic energy that is stored in a
random manner in its rectilinear motion and, in thermodynamics, this is called internal energy.
The temperature and the pressure of a gas are measurements of two different aspects of the
concentration of kinetic energy in the structure of the gas. The pressure is the result of the
very large number of collisions that occur between the molecules of the gas and the solid
surfaces containing it8. Those solid surfaces are anything but smooth at the atomic level. The
picture in figure 6-9 shows the inside surface of a piece of 15mm copper tube at a
magnification of about 1,000. Such a pipe is made by extrusion through a die. For copper, at
least, the process leaves the surface in a very rough condition as if it sticks and tears as it
emerged from the die. The white lines are the crests of hollows that appear to intertwine. In
The gases surrounding the Earth are located by the surface of the Earth and by gravity. If a gas under pressure is
suddenly free from its constraining surfaces it cannot sustain its pressure except during the short period when it is
accelerating as part of a general mass of gas.
8
©Ivor Bittle
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the middle there are two
white lines that are
0  01mm9 apart and this
corresponds to about
20,000
molecular
“diameters”.
On
a
molecular scale these
hollows are very large.
Each collision with such
a surface involves a very
small force acting for a
very short time at right
angles to the part of the
surface that it collides
with and the continuous
uniform pressure so
produced is the aggregate
of all these short-lived
forces. Of course the
Figure 6-9
^
^
pressure acts normally to
the mean surface because
all the tangential components of the impact forces cancel out. So, in some way, pressure
depends on the mass of the molecule, the mean kinetic energy of the molecules, and the
concentration of molecules in the space (which, of course, determines the frequency of
collisions with the walls). If, for a given gas, the mean kinetic energy were to be kept constant,
the pressure would simply depend on the concentration of molecules, that is, on the density.
Temperature is a concept that springs from our natural concept of hot and cold and this
appears to be essential to human survival because we are more vulnerable to temperature
changes than animals, birds and reptiles. In order to measure the temperature of a gas we bring
some thermometric device into contact with it. The atoms of the material of the device are
held together by atomic bonds, that appear to be perfectly elastic, and they vibrate in a random
manner within the limits imposed by the bonding. The molecules of the gas, when they collide
with the surface of the device, do not meet a rigid, flat, stationary surface but, as we have seen,
one that is irregular in shape, and at the molecular scale, is also in violent motion within its
atomic bonds. The molecules of the gas and the atoms of the surface continually exchange
energy. We wait until equilibrium is established between the average rate at which energy is
given to the surface of the device by the molecules of the gas and the average rate at which
energy is given to the gas by the atoms in the surface of the device. Thermodynamicists call
this thermal equilibrium and say that the gas and the solid are at the same temperature. Thus
temperature is an independent variable, to which we can relate other properties, and which is a
measure, in the case of a gas, of the mean total kinetic energy of each molecule, which
physicists observe to be the same for most gases at the same temperature.
The thermometric device will have been chosen so that some easily observed feature of it
changes during the process of reaching thermal equilibrium and reaches a steady value. The
device is calibrated to read temperature in an arbitrary system of units. The important thing to
note, is that the reading of the device, that is the temperature, is dependent on the mean
kinetic energy of any single molecule, whereas pressure is a measure of the concentration in
many molecules of random kinetic energy in the space occupied by the gas.
I set my micrometer screw gauge to have a gap of 0.01 mm and I could not see light through the gap. It is very
small.
9
©Ivor Bittle
7
We now have a picture of the gas in which each molecule is involved in many millions of
collisions every second. The image brought to mind is of a continuous exchange of kinetic
energy, both between the molecules of the gas, and between the molecules of the gas and the
atoms of the material of the container. Such a process tends to produce uniform values for the
mean velocity etc of the molecules. If there were to be some way in which the kinetic energy
of the molecules of the gas and the atoms of the isolated container could dissipate their kinetic
energy there would be a steady fall in temperature and pressure. Neither of these things
happens. We can only conclude that the energy of the molecules and atoms is not dissipated
and therefore the molecular interactions are perfectly elastic.
Now I can look at water as a liquid that behaves in a way that is typical of many other liquids.
In chapter 2, I mentioned that the molecule of water has a T shape but one might prefer to see
it as three atoms joined in a triangle. I have no mental image of an atom, except as a nucleus
with electrons whizzing round it, but I know that atoms vary in physical size so that the atom
of oxygen is larger than the atom of hydrogen and that the molecule of water will have an odd
shape because it has three atoms. Water is an agglomeration of densely packed molecules of
H 2O . I have said that, in the commonly-used, simple image of a molecular “structure”,
molecules are regarded as being perfectly elastic. This means that, for millions of millions of
molecules all moving in a seething mass, collisions and other exchanges of kinetic energy do
not lead to a change in the sum total of kinetic energy. Only an external effect can produce a
change. I need some model of the system in which this kinetic energy is stored by the
molecules. As I understand it (I may be totally wrong.) the molecules seem to experience an
intermolecular attraction when they are close together and a repulsion when they are too close
together. As both the attraction and the repulsion increase as the distance between molecules
decreases I have supposed, in order to make progress, that it might be possible to represent
these two forces by the graph in figure 6-10. For the system of two molecules to have my
claimed character these two plots must cross as I have shown them. At some point P the
attraction will equal the repulsion and when the two molecules get closer the net force will act
to push them apart and when they are further apart than P they will be attracted.
The repulsion will obviously go on rising rapidly as the separation decreases and this must
mean that a large number of molecules forming a continuum will be effectively
incompressible. The implication of having an attraction is that the same continuum will be
capable of withstanding tension, which is, of course, the case. This tension has a significant
value compared with the forces ordinarily impressed on water in engineering applications.
The molecules have mass and if they also have
motion the molecules could oscillate between the
two positions that I have shown as l1 and l2 and
the combination of forces of attraction and
repulsion produce a motion like that of a weight
suspended from a non-linear spring. The
amplitude of the oscillation will vary with the total
kinetic energy of the two molecules. It looks to
me as though the kinetic energy can have any
value between zero and some value when the
attraction is too weak to prevent the molecules
flying apart if that is, in fact, physically possible.10
Figure 6-10
Molecules of water that are part of a continuum
of molecules do not join to form pairs. They are very closely packed so it follows that any
molecule will also be in engaging with several other molecules and have some complex system
10
This corresponds to evaporation.
©Ivor Bittle
8
of oscillation with all of them simultaneously. Given that the molecules are not symmetrical
this makes one wonder how many molecules can be in contact simultaneously. I looked at
tennis balls and they do not fit snugly round a central ball. I suppose that 8 or 9 can be in loose
contact at once. New Scientist says that water molecules form clusters of ever-changing
numbers with a probable mean size of 5 molecules and that fits with my figure.
Somehow we have to form a mental picture of innumerable closely-packed molecules
oscillating in complex ways in ever-changing groups to form an extremely active continuum of
molecules. Clearly kinetic energy will be contained at some mean level in every molecule. The
kinetic energy has no mean direction so it cannot give the mass centre of the water a direction
and so there is no way that the sum of all this kinetic energy can be extracted and stored in the
gravitational field. It is totally random energy.
There is a further piece of information that is common experience and that is that water is a
poor conductor of heat. The sum total of the random kinetic energy can be increased by
heating the water and this proves to be a slow process if it takes place by dispersion through
the molecular structure. This is consistent with this model.
Now I need to look at this model of molecular behaviour in the layer of liquid envisaged by
Newton. He applies a force to the layer of liquid. Presumably, if an external force tending to
compress the system of molecules were to be applied, it would occur through the whole
system instantly. A depression would also affect the system instantly. So it would be reasonable
to expect a shearing force to be exerted instantly throughout the molecular structure and not
depend on a slow dispersion process.
But what happens when a liquid or gas is subjected to continuous shearing ? We know that
mechanical energy is continuously imparted to the molecular structure of a liquid or gas
whenever it is subjected to continuous shearing but it is not at all obvious what mechanism is
at work to produce this result. Just looking at figure 6-9 is sufficient to be persuaded that
Newton was justified in supposing the layer in contact with a rigid surface is stationary
although it seems that the molecules of liquid or gas are just colliding with a very chaotic
surface and rebounding to behave like a stationary layer. So let me suppose that there is a
stationary layer of fluid at the boundary. There will be a layer of fluid next to it that is moving.
There will be random collisions between this layer and the stationary one but these collisions
will be different from those in a stationary fluid because every molecule coming from the
moving layer carries with it the same additional velocity due to its movement. This layer has
the organised energy that makes it into mechanical energy. After impact with the stationary
layer this organised energy is randomised and becomes part of the stock of internal energy.
The same thing will happen between layers of fluid moving at different speeds.
This seems to me to be a plausible model for the molecular structure and its behaviour and
that it is adequate to continue this study of viscosity.
Quantifying viscosity
I think that it is worth examining Newton’s model for his definition of the coefficient of
viscosity. It is wholly conceptual and quite impossible to create as a real system. The only
variable in the expression that defines the coefficient of viscosity is the velocity gradient yet
Newton, like everyone else, must have been aware that the there are plenty of liquids that flow
more easily when they are heated and have expected that his coefficient of viscosity would
decrease as the temperature rises yet there is no recognition in his model of this observation.
We must keep in mind that our attitude to this model is coloured by our knowledge of
molecular theory of gases and liquids. It seems that Newton expected that the coefficient of
c
viscosity would be mainly dependent of the velocity gradient . We cannot know.
t
©Ivor Bittle
9
Anyone thinking of using this system to actually quantify the coefficient of friction for the
liquid in the space finds out very quickly that it is not easy because any mechanical system that
is devised causes effects other than those due to viscosity. No doubt many people tried to
devise ways of realising Newton’s idea in practical hardware but the hardware that was needed
did not appear until Hagen, who was an engineer and ignored by physicists, gave a basis for it
in 1839 only for it to be rediscovered by Poiseuille in 1840. It is interesting to contemplate
what went on in the years between 1687 and 1840. We know that Robert Hooke, who was
contemporary with Newton and often entertained the fellows of the new Royal Society with
demonstrations much as the Royal Society entertains the public now, appears to have had
reasonably well-developed engineering skill at his disposal. This must mean that various
devices must have been invented that were of a mechanical nature e.g. time of discharge to
measure liquid mobility, rotating discs to measure shear, falling balls and so on. This suggests
that the failure to exploit Newton’s proposal by designing a method that could give an accurate
measurement of viscosity did not lie with the technology of the time. The one system that did
work was dependent on the use of calculus. Newton “invented” calculus but he wrote his
Principia in Latin and, even though it was translated into English in 1729 it is still very difficult
to read and understand. This must have delayed the dissemination of his ideas and, as the
methods used by both Poiseuille and Hagen depended on being able to integrate, the delay
may have been inevitable. I suppose that both Hagen and Poiseuille showed that when a liquid
flows in a tube of small diameter the pressure drop is proportional to the flow. Perhaps they
thought that this was an unlikely result because it is not true for larger pipes but they both
pursued it. They found a method of quantifying viscosity that has proved to be useful well
beyond the direct result. It made possible the non-dimensional group that we call Reynolds’
number and the use of Reynolds’ number opened the way to storage and retrieval systems for
immense amounts of valuable experimental data. It was a big step on the way to constructing
an empirical science.
Hagen and Poiseuille knew that, for a small diameter pipe, the pressure drop was proportional
to the flow and they must also have known that the actual pressure drop varied with
temperature. They required something that took account of both temperature and the liquid
that was flowing in the pipe. Newton’s definition of the coefficient of viscosity was there to
use and we can reconstruct their method. It is implicit in Newton’s work that the liquid in
contact with a fixed surface behaves as if it, the liquid, is stationary and this means that in flow
through a pipe the liquid in contact with the pipe has zero velocity. Obviously there must be
velocity elsewhere in the pipe and the most reasonable supposition is that across any section
the velocity at a given radius is uniform and that this velocity changes from zero to at the outer
radius to some maximum on the axis. Poiseuille imagined the flow to take place as if a series of
coaxial cylindrical layers moving in an orderly way at different speeds. This clearly reflects
Newton’s method of defining a coefficient of viscosity but does not provide uniform velocity
in the layers nor a constant area.11
11Of
course no one then knew whether this was possible and it was not until 1896 that Osborne Reynolds
demonstrated the existence of this mode of flow.
©Ivor Bittle
10
This is how the rest of the argument goes. Now we can look at figure 6-11. It shows, at some
instant, a section of the flow
of a liquid in a small tube of
radius R 12 between two
plane surfaces  l apart. The
pressure on the upstream
section is p and on the
downstream
section
is
p   p . In figure 6-11 I
have drawn a section of a
cylindrical surface of radius
r that is coaxial with the
tube. There is a net force
Figure 6-11
acting on this element equal
to:-
p  r 2   p   p  r 2 =  p  r 2
If the flow pattern is taken to be symmetrical about the axis, we can replace the
c
of
t
dc
, use Newton’s relationship, and put the force exerted on the liquid
dr
in the small tube equal to:dc
  2   r  l 
dr
Then :dc
dc
will be
 p   r 2     2    r   l  , where the minus sign allows for the fact that
dr
dr
negative.
Newton’s definition by
Then we can rearrange and put:dc 
dp rdr
which can be integrated to give :
dl 2  
c
 p r2

 A where A is a constant.
 l 4
Using Newton’s simplifying decision that the fluid behaves as if the solid boundary is the same
 p R2
as a stationary layer of liquid we get c  0 where r  R and then A 
and :
 l 4
p 1
 R 2  r 2  , which is the equation to a parabola.
c

 l 4
This is not yet in the form of a rational expression because c is not a measurable quantity. We
need an expression in terms of the mean velocity c and:R
V
1
c

2    r  dr  c 

2
R
 R2
0
R
2r  p 1

  R 2  r 2   d
= 2 
 l 4
0 R
=
12
 p R2
 p D2
or, changing to diameter, =


 l 8
 l 32
Engineers do not generally work in radii but it is convenient here.
©Ivor Bittle
11
If this is now integrated along the pipe we get:32    l  c
p1  p2 
and this has been named after
D2
Poiseuille and sometimes also called the Hagen-Poiseuille expression.
It is not in the form that is best for engineering. We need:-
V
( p1  p2 )    D 4
128  l
This is an expression that contains only measurable quantities. When it was first produced no
one knew whether liquid could flow with this parabolic velocity distribution. It could only be
successful if the coefficient of viscosity was unaffected by the rate of shearing which clearly
varies in flow with a parabolic velocity distribution. It was remarkably successful for liquids
like water and oil. Look in any science data book and you will find values of  quoted at 20C
for a string of liquids. In effect it was found that what Newton called a coefficient of viscosity
was, for a wide range of liquids and gases, in fact, behaved as if it is a property just like
density13 for those liquids and gases. Presumably these liquids share the fact that they are all
agglomerations of one type of molecule and not of curious mixtures of liquids, solids and
semi-solids and, as a result, the coefficient does not depend on the rate of shearing. We do not
find paint, milk or confectionary cream on the list or raspberry jam.
We have found that all the runny substances can be divided into two classes depending on
whether the viscosity can be regarded as a property. We call them Newtonian and NonNewtonian fluids.
The units of viscosity
Viscosity has units and these can be deduced from Newton's original definition in any given
system of units. Unfortunately for engineers, systems of units have cycles of acceptability like
fashion and as a result useful data is stored in at least three consistent systems and in many
inconsistent systems as well.
If we start with the current fashion for S.I. we can put each term in kilograms, metres and
seconds :c
F t
F    A  can be rearranged to give  
which, when written in basic units, gives 
t
c A
the units of
kg  metres / sec metres
.
metres  metres / sec
kg
which rearranges to give  the units of
in the S.I system. It can also be expressed
metre  sec
Newton  sec
in
. Neither unit has a name.
metre 2
In the c.g.s. system the units are gram/cm.sec and 1 gram/cm.sec is called a poise and
frequently viscosities are quoted in centipoise to give easily remembered numbers like 1
centipoise as the viscosity of water. The two Imperial systems give units of pound/foot.sec
and slug/foot.sec neither of which have names.
What we have ended up with is a coefficient of viscosity found from Newton's definition by experiments
carried out with fluids flowing in glass tubes of small bore. We set out to find the fluid equivalent of the
coefficient of friction between solids. Whether what we have is a property will always be open to doubt. But, for
the purposes of engineering it is good enough to use it as a property. Those who use the Navier-Stokes equations
may not be so lucky.
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Measuring viscosity
The Hagen-Poiseuille expression is used as the basis for measuring the viscosity of liquids and
gases. If a straight pipe of uniform diameter can be set up horizontally, and a liquid or gas
made to flow through it, the pressure drop and the rate of flow can be measured and, from
this, the value of the coefficient of viscosity can be deduced. Any attempt to do this will show
that the coefficient of viscosity varies with temperature and a further constraint must be
imposed on the method of measuring the coefficient of viscosity; the measurement must be
made at uniform temperature. As laminar flow necessarily involves internal heating, the flow
can only be at constant temperature if it is slow with a small pressure drop and undertaken in
an environment that is temperature controlled. Then, for liquids and gases for which  is
independent of the velocity gradient it is possible to measure . We, as engineers, must accept
that physicists are quite capable of doing this and of producing accurate values of viscosity of
all the common liquids and gases. We should note that laminar flow at constant temperature is
a very special case and normally, especially in oil hydraulics, the heating is significant and, if we
choose to use the Poiseuille expression to deal with such a flow, we should expect it to
become inaccurate in such cases and be prepared to create a computer model that allows for
the heating.
The method outlined above is regarded as a primary method of measuring viscosity and,
without doubt, it is a method for use in a laboratory. One must contemplate the position of
engineers in respect of viscosity.
Viscometers
The proper place for making measurements of viscosity that are to be used in evaluating nondimensional groups is in a physics laboratory using primary methods. This would not be done
in the normal day-to-day business of engineering.
The main fluids that are moved about by engineers are probably water, oil, beer and milk in
that order and water and oil dominate. The movement of oil is the one that causes the
engineer to become involved with viscosity. The oil that is extracted from the Earth’s crust
varies in its composition. It is made up of all sorts of inflammable compounds of hydrogen
and carbon and, when they are separated, some are heavy and thick and some are light and
runny and everything between. For use in motor-cars, whether petrol or diesel, crude oil is
refined and the light fractions separated to become petrol or light diesel14. What is left after
distillation is inflammable and therefore a fuel. It will still contain a mix of different
compounds some of which are not very desirable in flue gas. It is also thick stodgy stuff that
needs careful management if it is to be burnt in power stations or ships. It is also typical of tar
oils.
The main problems for engineers can be illustrated by that of burning heavy oil in power
stations. There the oil has to be kept hot enough to flow into pumps to shift it into and out of
tankers whether they are road tankers or sea tankers and also for the rest of its journey to the
boilers. Once delivered to storage tanks on site it must be kept warm usually by steam heating
and then pumped through burners in the boilers. These burners atomise the oil, that is, they
produce a continuous spray of finely divided oil that must evaporate before it can burn. The
flame is so hot that, it can only be viewed through heavily smoked glass like that for welding
goggles, but even so it can be stratified by the different temperatures at which the several
fractions evaporate. Burning this oil so that the combustion products are least harmful and the
combustion most efficient needs careful control of the flame by controlling the inlet
conditions to the burner. Atomisation depends on the high pressure at the burner, the mobility
of the oil and I suppose other properties of the oil like film strength and surface tension. It is
Generally slow speed engines can burn heavy diesel oil and ships use very thick oil and lorries something in
between.
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©Ivor Bittle
13
not just about viscosity and even if it were to be there is still the problem of its composition
of many fractions. I think that the flame is mainly monitored by just looking at it.
From the above it is clear that engineers want to know about the state of the oil as it goes
through its various stages before burning and, whilst it might be interesting to have an accurate
value for the variation of the viscosity of the oil with temperature, some hands-on method of
finding a working value of viscosity is much more desirable at the final point of injection at the
burner. There are various viscometers available and I can make a assessment of their suitability
for engineering use.
The Ostwald viscometer
I must start with the Ostwald viscometer because it is the one that might just escape from the
laboratory. The following (In blue.) is what I wrote for a laboratory exercise for students in
1984. I make no apologies about including it because it tells us how empirical sciences grow. I
have no time to bring it up to date and, for my purpose, it would be pointless.
British Standard 188:1977
This standard is the latest (In 1984.) in a series that have been published in 1923, 1929, 1937,
1957, and now 1977. and the original reason for the publication was to encourage the use of
CGS units of viscosity rather than the arbitrary units such as Redwood seconds and Saybolt
seconds.
The standard introduced by the BSI is shown in figure 6-11 with its standardised dimensions.
In 1957 there was a range of 8 viscometers to measure the viscosities of most liquids. However
between 1957 and 1977 manufacturers of
viscometers produced their own ranges with
detailed differences in design and, in 1977 BSI
removed the length constraints and approved
commercial designs.
In 1977 BSI changed to SI units of viscosity and
also introduced a new definition for coefficient of
viscosity and introduced ideas of shear stress and
rate of shear to take the concept of viscosity as a
property a further step away from Newton’s
concept.
The standard lays down test procedures for
measurement of viscosity and these are more
suited to a physics laboratory than to a power
station.
The standard says “ the principle of the test is the
measurement of the time taken for a reproducible
volume of liquid to flow through a capillary
viscometer under an accurately reproducible head
and at a closely controlled temperature. The
kinematic viscosity ( 

=µ/ρ) is then calculated

from the measured flow time and the calibration
factor of the viscometer.”
©Ivor Bittle
Figure 6-11
14
In use a viscometer is filled with the test sample to just above level G. It is then mounted in a
bath of water that is gently stirred and heated under the control of an accurate thermostat.
When the temperature is steady at some desired value the level of the sample is adjusted to be
at G by extracting some of the sample. The sample is then sucked up tube 2 until the level is
above E. Then the time is taken for the level to fall from E to F.
Then the kinematic viscosity is found from an expression of the form:B
Kinematic viscosity  Ct 
where t is the elapsed time and B and C are
t
calibration constants.
The Ostwald viscometer uses a time-of-discharge method where the liquid flows under a
known pressure difference through a small bore tube. It is a secondary viscometer and must be
calibrated before use. It requires a glass water bath with heating and a stirrer and is half way
from a physics laboratory to an engineering testing house.
Look on the internet and you will see that BSI have not displaced Redwood seconds and
Saybolt seconds as practical measurements of viscosity for use by engineers. In my experience
the BSI instrument is bound to fail. Their viscometer is much more a device for use in a
laboratory and once it goes into a lab bureaucratic paperwork and procedural delays make it
useless for the engineer who could well have a whole tank full of oil to burn efficiently in a
boiler now, not in a month’s time. Redwood viscometers and Saybolt viscometers have been
reworked to make them more “user-friendly” and are still very much in use.
It is hard to see where BSI was going when it sought to alter the definition of viscosity. But the
mindset of BSI is that of the physicist and not that of the engineer just as the NPL was when
SI was introduced. They seem not to be interested in how engineers, or anyone else for that
matter, go about their work.
The Redwood viscometer In the late nineteenth century three secondary viscometers came
to prominence, the Redwood viscometer in Great Britain, the Saybolt viscometer in America
and the Engler viscometer in Europe. They all depended on the same method of performing a
“time-of-discharge” test on a sample where the time depended on performing a "time-ofdischarge" test on a sample. The time depended on laminar flow through a short pipe. The
sample of oil to be tested is held in a pot that is filled to a set point and, when the temperature
of the sample is at some desired value, the sample is allowed to drain from the pot through the
short pipe until a set volume has drained away. The kinematic viscosity, if that is what is
measured, is calculated from the elapsed time for the draining.
The Redwood viscometer was made15 by Sir Boverton Redwood in about 1880. I looked for an
authentic drawing of a Redwood viscometer but the important details seem to have changed in
with successive re-drafting for textbooks. Lewitt’s book of 1943 shows in figure 6-12 the
design of the viscometer as it was somewhere between 1923 and 1943 and I am inclined to
think that this is correct. I give a copy of Lewitt’s diagram in figure 6-6 and I have drawn the
essential central pot to a larger scale in figure 6-13. The brass pot that has a height of about
90mm or 3.5 and an internal diameter of 46.5mm, that is, the inside diameter of 2 tubing
having a 14 swg wall thickness. The bottom of the pot is clearly a turned insert. The allimportant feature is the agate insert. It has a bore of 1/16 or 1.62 mm and is 1/2 or 12.5
mm long. This is an Imperial size but it was a volume of 50 cc that was allowed to drain out.16
It is not clear whether he designed it.
Engineers still use a mix of units. I found a model engine that used ball races of metric and imperial sizes on
the crankshaft. It is nor a problem.
15
16
©Ivor Bittle
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The unlikely detail is the creation of a spherical seat at entry to the pipe so that it might be
sealed with a ball on the end of a wire. I think that we would avoid this shape of entry to the
pipe nowadays. The pot is fitted with a hook gauge to give an accurate position for the upper
surface of the sample before carrying out the test. The level drops by 30mm during the test to
drain away 50cc of the sample. This is measured with the small graduated flask under the hole.
The pot is surrounded by a water bath that is electrically heated by coils round the bath. The
water is manually stirred with the paddles round the pot. Thermometers in the sample and in
Figure 6-12
Figure 6-13
the water permit the adjustment of temperature. The test is started by lifting the ball off its seat
but I do not know where it is placed during the test, perhaps it just rests in the groove in the
base plate. Wherever it goes its position is part of the test procedure. The Redwood viscometer
is still in regular use as are the others and I need to offer reasons for this to be the case.
There can be no dispute about the fact that if you carry out the same test with the same oil in
identical equipment you will get the same result.17 This means that, if you want to buy fuel for
a power station and a supplier offers the 10,000 tons you need with a Redwood number, when
tested at a set temperature, of say 1,200 seconds and you know that your equipment can burn
fuel with this Redwood number, there is no physical reason not to buy if the price is agreeable
to you both. The seller knows that when the fuel arrives you can test a sample long before very
much of it has entered your storage tanks. Here we have the legal and practical basis for a
transaction. Nowhere in this is viscosity even mentioned, only the Redwood seconds because,
in the end, the final adjustment that ensures that the fuel is burnt as well as possible is all about
looking at the flame just as a plumber does to adjust his gas torch. The Redwood viscometer
that makes this transaction possible is a simple piece of equipment that is very suited to the
task for which it was designed and this is why it, and the other two viscometers, will go on
17
Science depends on it.
©Ivor Bittle
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being used. There are two versions of this viscometer and the second is designed for heavy
oils.
The Redwood viscometer is capable of being reproduced very accurately and however much
one might question the use of a ball with its spherical seat that produces such an unlikely entry
to the short pipe it cannot be changed now and there is no undisputed case to do so.
So what does the Redwood viscometer actually measure?
If the energy equation is applied to the viscometer at some instant during a test when the
difference in level between the free surface and the exit from the short pipe is h we get:c2
p
c2
p
c2
h  2  t. 1  1  z1  2  2  z2  the loss between 1 and 2.
2g
 g 2g
 g 2g
As p1 and p2 are both atmospheric pressure, c1 is zero and z1  z2 = h , this reduces to
c22
 the loss between 1 and 2.
2g
The loss is the energy lost in laminar flow in the short pipe. There is little option but to use the
Pouseuille expression and put:32    l  c
The loss between 1 and 2 
where d and l are the diameter and
g d2
length of the short pipe, c  c2 and  and  are the viscosity and density of the sample.
Then:c22 32    l  c2
h

2g
g d2
This expression tells us that the Redwood viscometer will only work if the kinetic energy is
very small when compared with the loss in the pipe. So, if the expression is reduced to
32    l  c
we can examine it.
h
g d2
gh  d 2 1
1
It can be rearranged to give
c
 or k  h  where k is a constant that can be
32  l 


calculated from the dimensions of the viscometer and   .

2
d
  t will flow from the viscometer
Now, in a short time  t a volume of the sample  c 
4
d2
4
 t 
and cause a drop in level of  h  c 
where d and D are the diameters of the
4
 D2
pipe and the pot.
Substituting for c and re-arranging we get:D2
h
 t  2  k  
which can be integrated between 0 and
d
h
T and the start and finish values of h. The result of this is that the time for 50 cc of sample to
drain from the pot is given by T = a constant  .
h

which crops

up so often that it is given a special name, the kinematic viscosity and denoted  . It now has
units of metres2/second where it used to have units of centistokes with units of centimetres2
/second. To change from centistokes to SI units divide by 10,000.
So the Redwood viscometer does not give us a value for the viscosity but for
©Ivor Bittle
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So the Redwood viscometer has the potential to measure kinematic viscosity but, of course, it
all depends on whether we were justified in using the Pouseuille expression to relate the time
of discharge and the kinematic
viscosity. We cannot know from
argument but it is evident that the
pipe is too short for the shape at
entry to have no effect so we must
turn to the actual performance of the
viscometer.
In the graph in figure 6-15 I have
plotted the performance of a
Redwood number 1 viscometer
when tested using fluids of known
viscosity at 21C. I think that most
engineers would be delighted to have
any basic data with this sort of
accuracy. The graphs for the
viscometer at 60C and 93C show
Figure 6-15
little more error.
I was trained to denigrate these secondary viscometers as being very poor compared with U
tube viscometers. I regret passing this view on to others when lecturing. This text gives me a
chance to offer a more reasoned view.
Other viscometers
Stokes showed that the force exerted on a sphere moving with laminar flow of fluid round it
F  6     R  c
can be given by
where R is the radius of the sphere and c its
velocity.
If a ball, e.g. a steel bearing ball, falls freely through a liquid that is expansive compared with
the size of the ball it will reach its terminal velocity and if that velocity is measured the value of
the viscosity can be deduced from it. This is the principle of the falling ball viscometer.
There used to be a falling ball viscometer that was really portable. It was in two parts, a steel
ball about 1 in diameter and a steel cup attached to a handle. The cup had the same radius as
the ball but it had three high spots that created a small clearance between the ball and cup
when one was fitted inside the other. In use the ball was immersed in a sample of oil and the
cup, on its handle, was placed over the ball under the oil and left for the ball and cup to come
into contact when sufficient oil had been squeezed out. The clearance was then full of oil. The
viscometer was then lifted and the time taken for the ball to fall out of the cup was measured
to give a useful estimate of the viscosity.
There is a case for continuously monitoring the viscosity of fuel oil just before it enters the
burners of a boiler. It could then become part of an automatic control system. Then a cylinder
rotating inside a sleeve would give a suitable element to form part of a transducer. It could be a
rotating disc.
Endnote
The fact that viscosity is measured from laminar flow in a pipe does not mean that it has no
significance in other modes of flow. The fact is that the use of laminar flow is just a means to
quantify viscosity that is now regarded as a property of the molecular structure of the liquid or
©Ivor Bittle
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gas. Of course the viscosity will be important in many practical applications in engineering like
bearings and oil hydraulics but it also has an important role in creating storage and retrieval
systems in our empirical science and computer fluid dynamics would be impossible without it.
Dealing with non-Newtonian fluids is a whole additional field of study but it seems that it is
tackled mainly by experiment.
Scroll down for exercises
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