### Video Transcript

In this video, our topic is the
laminar and turbulent flow of viscous fluids. To get a sense for what this is all
about, we can imagine this somewhat silly thought experiment. What if, instead of swimming pools
being filled with water, they were filled up with honey? Besides being sticky and sweet,
honey is also much thicker than water. And this can begin to give us an
idea of what it means for a fluid to be viscous. Now, just a quick recap, we can
recall that a fluid is any material that flows. So a fluid can be a gas or a
liquid. Based on that, we can say that a
viscous fluid, and we’ll learn more about this term in a moment, must also be a gas
or a liquid.

To understand what a viscous fluid
is, that is a fluid with high viscosity, let’s consider again the two liquids we
were talking about on the opening screen, water and honey. We know from experience that if we
pour water out of some container, then it flows very easily. On the other hand, the flow rate of
honey is much slower. It’s harder to get honey out of a
container than it is to get water, and the rate at which this liquid moves is much
slower. When a fluid is relatively thick
and hard to get flowing, we say that that fluid is viscous. Honey is an example of this as our
cooking oils, like canola oil or olive oil. And in all these cases of viscous
fluids, what we find is a slow flow rate resulting from lots of friction within that
fluid.

And that really is the key to
understanding this term, viscosity. This describes just how viscous a
fluid is, and that means it describes how much internal friction exists between
layers of the fluid as it flows. Now, this idea of friction between
layers of fluid is actually something we can see when, for example, we’re emptying
honey out of a container. If the walls of the container for
the honey are clear, then as the container empties out, we can see a thin layer of
honey on the inside walls of the container that sticks there and doesn’t flow
out. And we can also observe that the
honey that does come out of the container’s nozzle typically comes from at or near
the center of the container, where it’s not in contact with the walls.

All this to say, fluids move in
layers. And for a viscous one like honey,
we can actually see that the speed of movement of those layers is different. To get a clearer sense for this,
let’s imagine that we have a tube here and that we’re passing honey through the
tube. So let’s say that the honey is
moving from left to right and that all these different lines represent different
layers of the honey in the tube. As this viscous fluid moves along,
there’s friction between all these various layers, and the effect of this friction
is to slow down the flow rate of this fluid.

Now, let’s think for a moment about
the walls of this tube, these blue lines we’ve drawn in. We know that these walls are
stationary. They’re not in motion. Because of the frictional forces
between these walls and the layers of honey that they’re in contact with. These layers, what we could call
the outermost layers of the honey, tend to match the speed of the walls they’re in
contact with. And then let’s think about the two
layers inside of these outermost ones. There’s also friction between these
two layers, and this tends to slow down the inner layers. And then this trend continues as we
move towards the center of our flow.

All this means that if we were to
map out the speeds of each one of these different layers, they would look something
like this. The fluid at the very center of our
tube is moving fastest. But then, as we move closer and
closer to the walls of the tube, that speed decreases. And eventually, as we get right up
against the walls of this tube, that speed goes to zero. A velocity profile like this is a
hallmark of a viscous fluid. When we see fluid velocity varying
like this across a tube as it flows through, what we’re seeing is the effects of
friction between layers of the fluid. So if this is what a viscous fluid
looks like, what about a nonviscous fluid? That is, what if we had the same
kind of stationary tube and a fluid flowing through it from left to right, but that
this time, there was little or no friction between the layers of the fluid?

Well, if there was literally zero
friction, then that would be an idealized case of a nonviscous fluid. But if we had that, then this is
what the velocity profile of the layers would look like. They would all have the same speed,
regardless of how near or far they are from the wall of the tube. In this comparison between
nonviscous and viscous fluids, one thing we can keep in mind is that no real fluid
is perfectly viscous or perfectly nonviscous. Rather, they exist somewhere on the
continuum between the two extremes of totally viscous on the one hand and totally
nonviscous on the other. And in fact, there’s a test we can
perform on a fluid to figure out where it lands on that continuum. We can figure out quantitatively
just how viscous it is.

The term for this is dynamic
viscosity, sometimes called viscosity for short, and we represent this using the
Greek letter 𝜇. If we have some fluid and we want
to determine just how viscous that fluid is — in other words, we want to measure 𝜇
— then here’s a way that we can do that. We can take two plates of some
solid material and sandwich between them the fluid that we want to test. The way the test works, we keep the
bottom plate stationery. But then, on the top plate, we
exert some known amount of force, we can call it 𝐹, acting in this direction. This force pushes the plate so that
it slides along the fluid.

Now, thanks to this force we’re
applying, our plate is going to pick up some speed. We can call that speed 𝑣. And then, just like before, if we
think of the fluid in between our plates in terms of layers, we can say that our top
plate, which is in motion, will influence the motion of the layer of fluid it’s in
contact with. This happens because of friction
between the plate and that layer, and that fluid layer in contact with the upper
plate will be influenced to match this plate’s speed. It won’t quite get there in
magnitude, but its velocity vector may look like this, moving in the same direction
as the top plate, but at a slightly lower speed.

Now, that top layer of fluid will,
through friction, influence the movement of the layer below it. And that layer will pick up speed,
and its velocity vector can be drawn to look something like this. As we move down through the layers
in our fluid, the same effect takes place. The faster moving upper layers
influenced the lower layers through friction so that the speeds of each one of the
layers in our fluid will vary something like this. By the time we get to our layer,
which is in contact with our stationary lower plate, we expect that layer to be
motionless, just as we expect the speed of our top most fluid layer, the one in
contact with our upper plate, to have a speed that effectively matches the speed of
that plate.

So we have this smooth variation in
the speeds of the different layers of our fluid. And the fastest moving layer, we
said, has a speed 𝑣. Now, in order to calculate the
dynamic viscosity 𝜇 of this particular fluid we’re testing, in addition to the
force 𝐹 we apply to our plate and the change in speed experienced across the layers
of our fluid. There are two other parameters
we’ll need to know. One is the area 𝐴 of the plate
that’s in contact with our fluid. The bigger this area is, the more
friction will exist between the plate and the fluid layer. And then the other parameter is
simply the vertical extent of our fluid between these plates. That, too, has an effect on the
dynamic viscosity we measure. And this viscosity is equal to 𝐹
divided by 𝐴 times 𝑦 divided by 𝑣.

So the dynamic viscosity of our
fluid 𝜇 is equal to the force we’re applying to the upper plate divided by the area
of that plate multiplied by the height of our fluid, that is, its vertical extent
between the two plates, divided by the amount that fluid speed changes. We could consider the total speed
change from zero up to 𝑣. Or assuming there’s a uniform
change in speed from one layer to another, we could also just consider the speed
change between adjacent layers. Whichever way we choose, it’s
common to use Δ𝑣 instead of 𝑣 in our equation to emphasize that we’re talking of
speed change. In a similar way, to indicate that
the fluid height we’re speaking of is really a change in height, we might see this
written as Δ𝑦. This is the form of the equation
we’ll use for calculating a fluid’s dynamic viscosity.

As we mentioned earlier, this is a
way of quantitatively indicating the viscosity or the resistance to flow of a
fluid. Now that we’ve talked about fluids
and viscosity and how to calculate that. We can get on to the two types of
fluid flow that our lesson is focused on, laminar and turbulent. And here again, we find some terms
we’ll want to define. A good way to understand laminar
and turbulent flow is to take an up close view at the layers in a fluid as they flow
along.

So let’s say that, once again,
we’re looking through the side of a tube that’s carrying some amount of fluid. We can model the fluid as having
different layers, some near to the walls of the tube and some farther away. And assuming that this is a real
fluid — that is, it’s not idealized — that means it will have some nonzero dynamic
viscosity. In other words, there will be at
least some friction between the layers of the fluid and the layers of the fluid in
the walls of the tube. That friction might be very
minimal. It could be small enough that, by
and large, these layers of fluid move along smoothly.

When this happens, that means fluid
in these layers flows along without mixing. That is, fluid within one layer,
say this one right here, doesn’t cross the boundaries between this layer and the
ones above and below it. A way we could test this
practically is to use an eyedropper and drop a bit of color dye into the fluid at
some point. If that dye moves along, contained
within the layer where it started, then we have evidence that the fluid layers are
not mixing. This is the definition of laminar
fluid flow. So when our flow is laminar, if we
were to drop a little marker into the flow at some location, that marker would stay
within the fluid layer where it began. This is in contrast to turbulent
fluid flow.

In this case, friction between
adjacent layers of the fluid is strong enough that they do mix. And so if we were to track the path
of a marker dropped into a given layer, as it traveled along, it would move in and
out of other layers in the fluid. The smooth even flow of the fluid
has been disrupted. This is what turbulence means. Now, it’s worth pointing out that
both laminar and turbulent flows can happen for any fluid with any dynamic viscosity
𝜇. Recalling our two fluids from
earlier, water and honey, we can say that water would have a comparatively low
dynamic viscosity, while honey’s is fairly high. And yet both of these fluids, under
the right circumstances, can flow in a laminar or a turbulent way.

All that said, there is a
mathematical connection between dynamic viscosity 𝜇 and the turbulence of a
fluid. The turbulence of a flow of fluid
can be measured quantitatively, using something called a Reynolds number. The Reynolds number of a fluid flow
is typically abbreviated like this: capital R, lowercase e. While we won’t get into the
equation for Reynolds number in depth, there is a relationship between this number,
this measure of turbulence in a flow, and the dynamic viscosity 𝜇 of the fluid in
that flow. It turns out that the Reynolds
number of a flow — that is, the measure of its turbulence, where higher Reynolds
numbers indicate greater turbulence — is proportional to one over the fluid’s
dynamic viscosity.

So this means that as dynamic
viscosity decreases — that is, the fluid becomes less viscous — the Reynolds number
for that flowing fluid goes up. That is, the likelihood of
turbulent flow increases. And then, on the other hand, as the
fluid’s dynamic viscosity increases, then the Reynolds number decreases. So this is interesting. It means that a more viscous fluid,
like honey compared to water, will have a lower Reynolds number and therefore is
less likely to be flowing turbulently. That is, it’s less likely to flow
in such a way that the layers of the fluid mix with one another.

We might expect the opposite to be
true, that for a fluid with a low dynamic viscosity and therefore less friction in
between the layers of that fluid, we might assume that it would be harder for those
layers to mix. But actually, it’s easier. Friction helps to maintain layer
boundaries. And this is why thinner fluids —
that is fluids with lower viscosity like water compared to honey — are more likely
to have higher Reynolds numbers, meaning that they’re more likely to flow
turbulently. Knowing all this, let’s look now at
a quick example exercise.

A volume of a nonviscous fluid
is contained between two parallel horizontal plates, as shown in the
diagram. The plate above the volume of
the fluid moves horizontally at a speed 𝑣 one. Horizontal layers of the fluid
move at speeds 𝑣 two to 𝑣 six. Which of the following
correctly describes the relationship between the speeds of the layers?

Before we get to our answer
options, let’s consider our diagram, which shows us a stationary plate separated
from a moving plate by five layers of a fluid. We’re also shown that the
moving plate has a speed 𝑣 one, while layer one of the fluid has a speed 𝑣
two, layer two has a speed 𝑣 three, and so on, all the way down to a layer five
with a speed 𝑣 six. We want to consider the
relationship between the speeds of these different layers. And a critical fact to keep in
mind is that the fluid we’re working with is nonviscous. A nonviscous fluid is an
idealized case where there is no friction between the layers of the fluid.

This means that layer one
exerts no frictional force on layer two which exerts no frictional force on
layer three and so on through all the fluid layers. Knowing this, we want to
identify the correct relationship for the speeds of these layers, in other
words, 𝑣 two, 𝑣 three, 𝑣 four, 𝑣 five, and 𝑣 six. So let’s now look at our answer
options.

Option (a) says that 𝑣 two is
greater than 𝑣 three is greater than 𝑣 four is greater than 𝑣 five is greater
than 𝑣 six. Option (b) says that 𝑣 six is
greater than 𝑣 five is greater than 𝑣 four is greater than 𝑣 three is greater
than 𝑣 two. Option (c) says 𝑣 four is
greater than 𝑣 five, 𝑣 five is equal to 𝑣 two, 𝑣 six is equal to 𝑣 one, and
𝑣 three is greater than 𝑣 two, while (d) says 𝑣 four is less than 𝑣 five, 𝑣
five is equal to 𝑣 two, 𝑣 six is equal to 𝑣 one, and 𝑣 three is less than 𝑣
two. And lastly, option (e) says
that all the speeds of the layers are equal.

Now, the key fact in all of
this, as we saw earlier, is that we’re working with a nonviscous fluid. This means that the fluid
layers don’t influence one another through friction. And that means it’s impossible
for any one of these layers to move in a way that’s different from any of the
others. To see why that’s so, let’s
pick a layer, let’s pick layer two, and let’s imagine that this layer is moving
along left to right faster than layers one and three. If that was the case, if these
layer speeds were unequal, then layer two would exert a frictional force on
layers one and three. It would have to because the
molecules in this layer of the fluid are moving faster. But because our fluid is
nonviscous, that can’t be.

None of the layers exerts a
frictional force on any of the others, which means that rather than thinking of
this fluid as five separate layers, we can really think of it as one single
layer. It all moves together and all
at the same exact speed. Therefore, whatever the speed
of layer one, for example, that’s 𝑣 two, this must equal the speed of layer two
and the speed of layer three and four and five. And we see that out of our
answer options, it’s option (e) which claims that all the layers move with the
same speed. This must be the case for a
nonviscous fluid.

Let’s now summarize what we’ve
learned about laminar and turbulent flows of viscous fluids. In this lesson, we first saw that
fluids are materials that flow; they’re liquids and gases. We learned further that viscosity
is a measure of the internal friction of a fluid. Related to this, we learned this
term, dynamic viscosity. This is a quantitative measure of a
fluid’s viscosity, and we calculate it by finding out how the speed of a fluid
changes in response to friction between the layers of that fluid.

And lastly, we looked at the
differences between laminar and turbulent fluid flows. We saw that laminar flows are cases
where the layers of the fluid do not mix with one another, while in turbulent flow
situations, the layers do combine. The degree of turbulence of a flow
of fluid is given by something called the Reynolds number. And this number abbreviated Re is
inversely proportional to the fluid’s dynamic viscosity 𝜇. This is a summary of laminar and
turbulent flow of viscous fluids.