### Video Transcript

In this video, we’re talking about
applications of viscosity. Viscosity is a property of fluids
that affects how those fluids move as well as how other objects can move through
those fluids.

To get a sense for what viscosity
means practically, imagine that we have here a mixing bowl and a spoon and currently
all the bowl has in it is some water. From experience, we know that if we
were to stir this water up, it would be fairly easy to do. The water offers very little
resistance to the movement of the spoon through it. When we see this, we can say it’s a
result of the water having a low viscosity. And we can define this term by
saying that it refers to the magnitude of internal friction of a fluid.

So, if we picture, say, all the
layers of water in our mixing bowl, we know that as the spoon stirs the layers
because this is a low viscosity fluid, there’s very little friction between the
layers. And that actually means that the
layers of this liquid tend to mix easily with one another. That’s because friction between the
layers of a fluid actually helps to reinforce the boundaries of those layers. So, when there’s not much internal
friction, as is the case for a low-viscosity fluid like water, the boundaries
between fluid layers are weak and, therefore, the layers mix easily.

So, in the case of pure water,
stirring up the water doesn’t take much effort, and it also results in significant
mixing of the layers of this fluid. This means, by the way, that we
could describe the flow of this fluid under these conditions as turbulent. So, we have this fairly thin, that
is, low-viscosity fluid. But what if we add something to
make it a bit thicker? Say we pour some flour into the
water.

As our flour mixes in with the
water, the viscosity of this mixture will go up. And we’ll probably be able to feel
this difference as we continue stirring. The fluid will feel a little bit
thicker, and it will take a little bit more effort to move the spoon through it. And here’s another thing we can
say.

If we were to continue stirring
with the same exact force on the spoon before and after adding the flour, then once
the flour has been added, the different layers of our fluid — now, a water-flour
mixture — would do a better job at maintaining their separate boundaries than the
water did by itself before. That’s because the internal
friction of our fluid is increased with the added flour, which means the boundaries
between these layers of fluid are now stronger. They still do mix, but not as much
as they did before.

And if we were to add even more
flour to the mix, this trend would continue in the same direction. The fluid would get harder and
harder to stir. But if we did continue stirring it
with the same force as we had all along, then the layers of fluid would tend to keep
more and more separate. They would flow more and more
smoothly, that is.

By now, our fluid in the bowl is
starting to get pretty thick. And when this happens, there’s a
certain effect we may start to notice. Say that we were moving the spoon
through our fluid at a very slow speed very gradually in circles. Our fluid, because of its
viscosity, would resist that movement to some extent. Then, imagine that we rapidly
increase the speed with which we’re stirring. So, the spoon is moving much more
quickly round and round in circles through this mixture.

Now, here’s a question; in which
case, with the slow stirring or the fast stirring, will we need to apply more force
to the spoon? Well, we know from experience that
it’s the fast stirring that takes more force, and not just because we’re moving the
spoon with its mass more quickly. More than that, it’s because we’re
working against something called the drag force. This is a friction force that’s
applied to any object that’s in motion through a fluid. So, an airplane flying through the
air experiences drag force, a fish swimming through the sea. And we can feel the drag force
ourselves when we’re driving along in a car and put our hand out the window. We bring up this force because it
turns out that the magnitude of this force depends on the speed with which an object
moves through a fluid.

Getting back to our stirring spoon,
we saw earlier that when we moved the spoon more quickly, then we encountered more
resistance from the fluid, that is, more drag force. And therefore, we had to push the
spoon harder to move it faster. This is generally true of objects
moving through fluids. The faster the objects move, the
more drag force they experience. And here’s something really
interesting. If we consider the general case of
some object moving through some fluid, then typically there’s some speed, we can
call that speed 𝑣, where if the object is moving below that particular speed
through the fluid, then the drag force on that moving object, we’ll call it 𝐹 sub
D, is proportional to the instantaneous speed of that object, we’ll call it 𝑣 sub
i.

So, what we’re saying so far is
that if we have an object moving at less than this speed 𝑣 through some fluid, then
the drag due to friction that that object experiences is proportional to its
speed. So, if its speed doubled, assuming
it remains below this cut-off speed 𝑣, then the drag force experienced by the
object would double as well. But if the speed of our object is
greater than 𝑣, then in that case we can say the drag force it experiences is
actually proportional to the object’s instantaneous speed squared.

This means that once we’re beyond
this cut-off speed, if we were to double the speed of our object, then we would
quadruple the drag force it experiences. The tendency of drag force to
behave like this has some really interesting consequences. For one thing, when aircraft are
traveling at very high speeds — several times the speed of sound, say — then the
thrust force that the engines need to provide to move the aircraft just a little bit
faster goes up significantly. Because in this regime, the drag
force on the aircraft goes with the square of the aircraft’s speed. In other words, it gets harder and
harder to push the aircraft that much faster.

Now, it’s worth pointing out that
this cut-off speed 𝑣 is different for every combination of fluid and object moving
through that fluid. So, it’s not a fixed number that we
might be able to look up, say, in a table. So, anyway, as the speed of an
object moving through a fluid increases, so does the drag force on that object. And sometimes that drag force
increases with the square of the object speed. This may be what we experience as
we find the spoon harder and harder to move through this mixture at higher and
higher speeds.

Now, getting back to our mixture,
let’s see that we added a whole lot of flour to it. Our mixture, therefore, get
substantially thicker, that is, more viscous. And now, if we were to look at the
various layers of this fluid, we would see that there’s virtually no mixing between
them. That is, as our spoon moves through
this flour—water mixture, the layer boundaries stay intact. We could say then that our fluid is
flowing, albeit very slowly, in a very smooth fashion. This is the opposite of turbulent
flow.

And so, what we’re finding then is
that as the thickness or the viscosity of a fluid increases — that is, as the
friction between layers of the fluid goes up — then the tendency of that fluid to
flow smoothly also increases. When this happens, we say that the
deformation of the fluid is very low, where the deformation of a fluid is simply a
change in its flow. This change could happen, for
example, by layers of the fluid mixing together. A more viscous fluid has greater
resistance to deformation.

Knowing all this, let’s get a bit
of practice now through an example exercise.

If a fluid increases in viscosity,
in which of the following ways does this change the fluid’s resistance to
deformation? (a) The fluid has less resistance
to deformation. (b) The fluid has greater
resistance to deformation. (c) The resistance of the fluid to
deformation is not affected.

To get started here, let’s imagine
some fluid that increases in its viscosity. So, how about this? Say that we have a container that’s
filled with honey. We know that honey is already a
fairly viscous or thick fluid. And let’s say that we decrease the
temperature of this honey. We cool it down so that it gets
even thicker. By cooling the honey then, we’re
increasing the viscosity of this fluid. We want to know how this changes
our fluid’s resistance to deformation.

And we can recall that deformation
refers to a change in the flow of a fluid. One way we could change the flow of
this fluid, that is, deform it, is by mixing it, say, with a spoon. So, here’s the question: how does
the honey’s resistance to the spoon’s movement through it change as the honey’s
viscosity increases?

We know from experience that for a
thicker, that is, more viscous fluid, the resistance to a change in flow, that is,
the resistance to deformation, will increase. Simply speaking, it’s harder to
stir a thicker or more viscous fluid. And we see that of our three answer
options, option (b) describes this case. This tells us that as the viscosity
of our fluid increases, so does the fluid’s resistance to deformation.

Let’s look now at a second example
exercise.

Thin layers of equal area and
thickness of two different-colored oils are placed onto the central region of the
top surface of two identical solid disks, as shown in the diagram. The disks are then rotated with
equal angular velocities and the oils spread over the surface of the disks. The oils have the same density but
different viscosities. Which of the oils has the greater
viscosity?

Okay, so we see in our diagram
these two disks. And we’re looking at them in what
we could call before and after instances. First, we have the two discs
stationery and these two oils are deposited onto the center regions of each one. We could call the oil on the top
desk the orange oil and that on the bottom disk the yellow one. We’re told that the regions on
these disks that are covered by the oils are of equal area and equal thickness. This means that the oil volumes are
the same. So, there’s just as much orange oil
on this disk as there is yellow oil on this one. And they cover the same amount of
area at the center of each identical disk.

So, this is our before
scenario. And then, the disks are set to
rotating. Our problem statement tells us that
they rotate with equal angular velocities, in other words, the same angular
speed. And as a result of this, the two
oils spread out over the surface of the disks. The oils, we’re told, have the same
density, but they have different viscosities, that is, thicknesses. Based on how these two different
oils respond to the rotation of these disks, we want to identify which one has the
greater viscosity.

To begin figuring this out, we can
recall that viscosity indicates the internal friction of a fluid. That is, the more friction between
the layers of that fluid, the higher its viscosity is. We can also say that if the layers
of a given fluid have a lot of friction between them, then it will be unlikely for
those layers to move very much relative to one another. That’s because there’s such a
strong resistance to movement through friction. So, related to our question, the
oil with the greater viscosity will be the one that moves less or, we could say,
deforms less in response to the rotation of these disks.

That will be the oil that, we could
say, is held together more closely than the less viscous of the two oils. So, looking at this snapshot of our
spinning disks, after the oils have had a chance to expand across those surfaces, we
see that it’s the orange oil that has expanded less than the yellow one. That means this oil more strongly
resists deformation and that resistance, we can assume, is due to higher levels of
internal friction in this oil. In other words, this is the more
viscous of the two. For our answer then, we can say
that the orange oil has greater viscosity. And we say that because this oil
was deformed the least of the two.

Let’s now summarize what we’ve
learned about applications of viscosity. In this lesson, we saw that the
term viscosity describes the magnitude of internal friction of a fluid. We also learned that objects moving
through fluids experience what’s called a drag force and that this force is due to
friction. At relatively lower speeds, this
drag force, we can abbreviate it 𝐹 sub D, is proportional to the speed of the
object 𝑣. Whereas at relatively higher
speeds, the drag force is proportional to 𝑣 squared. Lastly, we saw that fluids with
greater viscosity offer greater resistance to deformation, a change in the flow of a
fluid.