GG101L: RHEOLOGY Handout
(Material to read instead of a Lab Book chapter – you might want to print this out
and bring it to lab)
Rheology is the study of how materials flow, and includes two main properties. The first is
viscosity (which tells you how fast something will flow), and the second is yield strength
(which tells you if something will flow). Rheology is very important in volcanology. For
example, the combined viscosity and yield strength of a magma will determine if gas bubbles
can rise through the magma chamber to escape peacefully out the top, or if they will be trapped,
allowing the pressure in the magma chamber to increase to the point of an explosive eruption.
Geologists also study the rheology of glaciers to understand what makes them flow rapidly or
slowly, the rheology of lahars to understand how they pick up and carry large boulders and
trees, the rheology of the Earth’s mantle to understand how it convects, and much more.
Although rheology is important, it is not so easy to understand, especially when people use
a lot of math to describe it. In the Rheology lab we will take a mostly qualitative but partially
quantitative approach to illustrate rheology, concentrating on viscosity.
I VISCOSITY AND CYLINDERS OF SHAMPOO AND OIL
Viscosity is the resistance to flow (it is the opposite of fluidity). Viscosity is a property of a
substance that relates a known amount of deformation (for example stirring force) to the rate
that the substance deforms (flows). In other words, how fast will something flow for any given
amount of stirring. For example, say you push with a certain force on a spoon in a bowl of
water. You will get reasonably fast flowing. However, if the bowl is full of cookie dough the
same force on the spoon produces very slow movement. The other way you could think about
it would be to see how much force on the spoon was required to give you one turn around the
bowl full of water every 2 seconds, and compare this to how much force was required to give
you the same stirring rate for the bowl of cookie dough. In either case, you would determine
that the water has a lower viscosity (it deforms quickly with only a small force), and the cookie
dough has a higher viscosity (it deforms slowly with the same amount of force or requires a
greater force to deform at the same rate as water).
There are lots of ways to measure viscosity, including attaching a torque wrench to a
paddle, using a spring of known strength to push a rod into the fluid, or measuring how fast a
fluid pours through a small hole. The method we will use is one of the oldest, and it involves
measuring how fast a sphere of known size and density sinks through a fluid. If we know the
size of the sinking sphere and its density relative to that of the fluid, we can determine the
amount of sinking force it produces. Thus, if the sphere sinks quickly then the viscosity is low
because it is offering only a little resistance. If the sphere sinks slowly the viscosity is high
because it is offering a lot of resistance.
1
This lab will also let you practice measuring lots of things, practice converting units, show
you how scientific results are never identical, and let you see how seemingly small
measurement mistakes can sometimes turn out to be kind of significant.
The formula that allows you to calculate viscosity with a sinking sphere is a modification of
Stoke's Law, named for George Gabriel Stokes (1819-1903):
viscosity = η = 2(∆
∆ρ)gr2
9v
η = viscosity (what you want; units are Pa s)
∆ρ = the difference between the density of the sphere and the density of the fluid (kg m-3)
g = the acceleration due to gravity, which is constant on Earth ( 9.8 m s-2)
r = the radius of the sphere (m)
v = the velocity that the sphere sinks (m s-1)
Note that all are in SI (Système International) units. In the lab, you will drop spheres of known
radius and density (determined by you by measuring their mass and diameter) through a fluid of
known density (determined by you by measureing its mass and volume). You will therefore
know ∆ρ and r2. By calculating how fast the spheres sink a known distance through the fluid,
you will determine the velocity, v, and at that point you can calculate the viscosity of the fluid.
II UNITS
Units: It is important to keep track of the units that go along with the numbers you are using.
The most important reason why you do this is that if the ending units don’t make sense,
probably somewhere you have made a math mistake and your answer isn’t correct. This is
captured in the saying that goes something like “Units are like sports officials – nobody pays
much attention to them until something goes wrong”.
For example, say you are in Samoa and you know that the money exchange rate is 3 Tala
per dollar (3T/$). You have $30 and you want to check into a hotel that costs 70T per night.
Can you do it?
First, you divide 30$ by 3T/$ and you get 10. You are bummed. But did you do the
calculations right? The key to using units is that you do the same operations on the units as
you do to the numbers. So, if you divided the amount of dollars you have (30) by the amount
of the conversion factor (3) then you have to divide the units of the dollars you have ($) by the
units of the conversion factor (T/$). The results of doing this is $2/T, which doesn’t make
sense. That is a pretty darn good hint that the math part is not correct either.
If instead you multiplied 30$ by 3T/$, then the number is 90. And if you do the same thing
to the units, you end up with $T/$, which cancels out to just T, which makes sense. You won’t
2
have to sleep outside under a coconut tree after all, and you’ll even have some money left for a
Vailima!
The Stokes law equation has lots of units in it and they actually help us understand the eventual
SI units of viscosity, Pascal-seconds (Pa s):
∆ρ)gr2
η = 2(∆
9v
In units, the part to the right of the = is: kg x m
m3
s2
x
m2
÷ m
s
You can also write this as: (kg)(m)(m2)(s)
(m3)(s2)(m)
As we saw above, you can cancel and combine units in the same way that you can cancel and
combine numbers.
For example, you can combine (kg)(m) into what is called a Newton (N), a unit of force.
(s2)
This leaves you with (N)(m2)(s)
(m3)(m)
which reduces down to (N)(s)
(m2)
(N) is force per area, or pressure, and in SI units is called a Pascal (Pa)
(m2)
So...for viscosity you are left with a unit called a Pascal-second (Pa s), a unit of pressure times
a unit of time. Although this is the official SI unit of viscosity, it is not obviously intuitive that
it should represent the resistance of a fluid to flowing.
Let’s play with the units a little more. If you start with the starting units:
(kg)(m)(m2)(s)
(m3)(s2)(m)
and cancel everything you can, you are left with (kg)
(m)(s)
3
This is a unit of mass (in kg) divided by length (in m) and time (in s). This makes a little more
sense. Consider the same sphere (same mass) sinking for the same period of time, in other
words (kg) is constant.
(s)
Then, if the distance the sphere sinks is big, (m is big), the fraction gets small. This makes
sense - if the same sphere sinks a long distance in the same amount of time, the viscosity is
low. On the other hand, if the sphere sinks only a short distance, (m is small), then the fraction
gets big. This makes sense too - if the sphere sinks only a short distance in the same amount of
time the viscosity is high.
One way that viscosity is commonly illustrated uses a graph of the deforming pressure that
you apply to a fluid plotted against the rate that the fluid deforms (Figure 1).
Figure 1: qualitative plot of applied deformation pressure vs. strain rate for three
hypothetical Newtonian fluids.
Notice that for the same applied pressure, you get a low strain rate for the high viscosity fluid
and a high strain rate for the low viscosity fluid. Or, you can say that in order to get the same
strain rate, in a low viscosity fluid you need a low applied pressure but in a high viscosity fluid
you need a high applied pressure.
4
III YIELD STRENGTH
Yield strength is another important aspect of rheology, and it tells you how hard you have
to push on a substance in order to get it to flow. It is important to recognize that yield strength
is a property of some fluids, but not all. If you push on water, for example, no matter how
softly you push, the water will move. Thus, water has no yield strength. We call substances
that have no yield strength Newtonian. However, for many substances, tiny pushes don't
produce any movement and maybe even medium pushes don't either. For these, there is a
certain amount of force you have to exert on the substance before it will flow. That certain
amount of force is called the yield strength of that substance. Once the substance starts to
move, then the rate at which it moves is a function of its viscosity.
Yield strength is very important when studying lava flows, glaciers, salt domes, and many
other geological substances that flow. In nature, the most common source of a deforming force
is gravity. Gravity causes lava flows and glaciers to flow downhill and to spread laterally, it
causes stones to sink into lahars and glaciers, as well as many other geological motions. It also
turns out to be much easier to determine yield strength than viscosity. That is because for yield
strength, all you have to do is figure out if something flowed, and this can often be done long
after the flowing has stopped. For viscosity, you have to somehow figure out how fast
something flowed, which is difficult even if you are watching it happen, let alone if you come
along after the flowing is pau.
One of the easiest ways to visualize how yield strength works is to consider a blob of fluid
(lava, glacial ice, cake batter) on a horizontal surface (Figure 2). Gravity will cause the fluid to
try to spread out laterally. The fluid’s
viscosity will control how fast it
spreads but here we are concerned
with the yield strength, and whether it
will allow the fluid to spread at all.
The stress that is deforming the
blob of fluid depends on gravity, the
density of the lava, and the height of
the fluid. As long as the combination
of height, density, and gravity is large
enough, it can overcome any internal
strength (the yield strength) that the
lava dome has. Gravity is constant so
that doesn’t change. We can also
assume that the density of the fluid
doesn’t change either. As the blob
spreads laterally, however, its height
decreases (Figures 2 B and C), and Figure 2: a spreading blob of fluid
5
there will come a point where height, density, and gravity are no longer able to overcome the
yield strength – the fluid will have achieved an equilibrium shape. It won’t matter how long
we wait, the fluid will not be able to overcome its yield strength. This means that we can come
along long after a lava dome has solidified, for example, measure its density, height, and width,
and calculate what the yield strength must have been while it was actively flowing.
The simplest formula for calculating yield strength utilizes this balance between the
deforming forces and resisting forces:
YS = ρgh2
w
YS = yield strength (in units of pressure, N m-2)
ρ = density of the fluid (kg m-3)
h = height (or thickness) of fluid (m)
w = width of fluid at its base (m)
So you see that as a lava dome spreads laterally, h decreases while w increases. If the dome
can spread a lot, the h2/w part of the equation will be small and the yield strength will likewise
be small. On the other hand, if the dome stops spreading while h is still high and w has not
increased very much, the h2/w part of the equation will be large and the yield strength will
likewise be large.
IV MORE UNITS
The units of the above yield strength equation are:
(kg)(m)(m2)
(m3)(sec2)(m)
again, you can take out the (kg)(m)
(sec2)
which is a Newton (N)
all the leftover m's reduce down to 1
(m2)
so, the whole thing becomes
N
(m2)
,
which is our old friend the Pascal (Pa), a unit of pressure, or stress.
6
This unit is a little bit more intuitive than the Pa s is for viscosity. This is because it makes
sense that pressure is what causes a fluid to move. If the pressure is too low (below the yield
strength) the fluid will not flow, no matter how long you wait. If the pressure is above the
yield strength, the fluid will flow, and how fast it flows depends on its viscosity.
Yield strength is also often illustrated graphically (Figure 3)
Figure 3: qualitative plot of applied pressure vs. deformation rate for two hypothetical nonNewtonian fluids
This graph is similar to Figure 2 except for one key feature. For fluids that have a yield
strength (non-Newtonian fluids), the graph of applied pressure vs. strain rate doesn’t go
through the origin. For Fluid B, for example, any applied pressure that is less than the yield
strength produces no deformation, and therefore a deformation rate of zero. For Fluid A, an
even higher applied pressure needs to be occur before there is deformation. Thus, the yield
strengths are defined by the values of the y-intercepts. Both these fluids also have viscosities
(called non-Newtonian viscosities), which, as before, are determined by the slopes of the lines.
7
IV What does any of this have to do with Geology???
Figure 4 shows an outcrop of the 1868 Mauna Lava
flow. The outcrop was divided into equal-sized boxes
marked with chalk, and in each box the number of
olivine crystals ~4 mm across was counted. The counts
of olivines have been chalked next to the boxes.
Figure 5 shows what we think is going on.
Specifically, when a flow comes to a halt, there is an
even distribution of olivines from top to bottom (Fig.
5a). The flow cools from the top down and from the
bottom up, and the very top and very bottom cool
quickly enough to preserve the initial olivine
concentration (Fig. 5b).
However, the middle of the flow takes a while to
cool, and as long as they can “escape” the cooling front
that is working its way down from the top, they will
sink until they land on the cooling front that is working
its way up from the bottom (Figs. 5c-d). The final part
of Figure 5 shows the abundance of olivines with depth,
characterized by a “C-number”, which is a measure of
olivine concentration or depletion relative to the starting
Figure 4. 1868 Lava flow
concentration. We need to do two things: 1) figure out
how far they sank by assuming that at least some of them made it from just below the top to the depth
where the greatest concentration is; and 2) figure out how much time was available for them to do this
sinking by calculating how long it takes the lava to
cool from the bottom up to the depth where the
greatest concentration is. Once we know sinking
distance and sinking time available, we can
calculate a sinking velocity, and just like with the
shampoo, calculate the viscosity of the lava using
Stokes’ law.
Figure 5. Olivine crystal sinking diagram
8
Figure 6 is based on data that
were collected from temperature
probes lowered into a lava lake
that formed in Alae crater during
an eruption of Kīlauea in 1963.
What they did was stick a
thermocouple probe through the
lake surface as the lake cooled, and
determined at what depth each
temperature was with respect to
time. What we care about,
however, is the temperature
variation from the base up, because
it is the lava cooling from the
bottom that produces a “floor”
onto which the olivines sank. The
geologists back in 1963 didn’t
measure from the bottom up.
However, thermal theory tells us
that a flow will cool from the
bottom up at a rate ~75% of that at
Figure 6. depth vs. time to cool to 1130º C
which it cools from the top down.
This allows us to produce Figure 6.
It is a graph of height, in meters, up from the base of the lava lake (on the x axis) vs. the time, in
seconds, that it took that level in the lake to cool to 1130º C. Even though 1130º C is still pretty hot, at
this temperature the lava will have gained a yield strength that will prevent 4 mm “diameter” olivines
from sinking. Diameter is in quotes because olivines aren’t really shaped like spheres.
So…how do you use this graph? Let’s say you had a flow that was 1 m thick in total, and you
determined that the concentration of olivines was at 60 cm (0.6 m) below the surface of a flow. This
means that the concentration of olivines is 40 cm (0.4 m) above the base of the flow. You know,
therefore, that at least some of the olivines managed to sink 0.6 m in the time that it took the lava 0.4 m
from the base to cool to 1130º C. Figure 6 tells you what this time was. Find 0.4 on the x-axis and go
up until you hit the curve. Then go left, and you hit the y-axis at about 257,000 seconds. That’s how
long it took the lava at 0.4 m above the base to cool to 1130º C, and therefore how long the olivines had
to collect there. You therefore know the distance they sank (0.6 m) and you know the time it took them
to do the sinking (257,000 s), so you can calculate a sinking velocity of 2.3 x 10-6 m s-1, which is pretty
darn slow!
9