### Video Transcript

In this video, we’re going to learn
about viscosity and turbulence. We’ll learn what these terms mean,
what effect they have on fluid motion. And we’ll see how to work with them
practically. To start out, imagine that you are
a contestant on a timed obstacle course. The goal is to finish the obstacle
course in the least amount of time. And as you start, before you are
three different vats of various types of fluids that you need to swim through. To figure out which of the three
vats you should jump into and swim through, it will be helpful to know something
about viscosity and turbulence.

Beginning with the term viscosity,
this is a measure of the level of internal friction within a fluid. To get a feel for viscosity,
imagine taking a glass and filling it up with water compared to pouring maple syrup
on a stack of pancakes. Compared to the maple syrup, the
water flows much more easily. It’s less viscous than the maple
syrup is. Imagine that we zoom way in to get
a close-up view of the maple syrup as it flows across the top pancake. As it flows, the maple syrup, and
liquids in general, divides up into layers that move across one another. The speed at which these different
layers move depends on their distance from the fixed surface. The maple syrup layer right in
contact with the pancake moves most slowly, whereas the one on top moves most
quickly left or right. So there’s a velocity gradient of
the different layers as they move relative to one another.

We said that viscosity measures the
internal friction within a fluid as it flows. The friction arises at the
boundaries between the various liquid surfaces of a flowing fluid. When that friction is very high,
the viscosity of that fluid is high. This means that the various layers
of our flowing fluid have a hard time moving past one another. On a macroscopic level, this feels
like the fluid is very thick. If we tried to stir it with a
spoon, we would feel lots of resistance. That’s that internal friction being
experienced. What we call simple viscosity or
absolute or dynamic viscosity is represented by the Greek letter 𝜂. 𝜂 is a value in units of newtons
per meter squared times second, or pascal seconds. The viscosity represented by 𝜂 is,
as we said, sometimes called dynamic viscosity or simple viscosity.

There’s a second type of viscosity
called kinematic viscosity, which is equal to 𝜂, the simple viscosity of a
material, divided by its density, 𝜌. And this type of viscosity, to go
further with the Greek alphabet, is represented by the Greek letter 𝜈. Kinematic viscosity is useful to us
because it gives another level of information beyond simply its viscosity or
internal friction level. It accounts not just for the
thickness of the fluid, but also for its density.

Going back to simple or absolute
viscosity, this property is useful in helping us understand physical scenarios, such
as a fluid flowing past a spherical object. One of the things we might want to
know in such a situation is the force due to viscous drag on that sphere. If the speed of the sphere relative
to the fluid is 𝑣 and the radius of the sphere is 𝑟, it’s been discovered that, by
multiplying these two terms with the viscosity of the fluid times six 𝜋, we find
that viscous drag force. This relationship is named Stokes’
law, after the Scottish physicist George Stokes.

If we once again zoom in and look
closely at the fluid flowing towards the sphere in this example, again we see layers
of a fluid which flow past one another. Imagine that we speed up the flow
of this fluid so that the layers are moving along quite quickly. At some point, as the speed
continues to increase, the layers will not completely stay separated from one
another. But they’ll start to mix a bit. As the flow velocity continues to
ramp up and up, more and more the layers will mix with one another. And the flow will not be overall as
smooth. When this happens, we’ve moved into
a category of fluid flow described by turbulence. In general, a moving fluid can
either flow smoothly. That’s described as laminar
flow. Or it moves with rough motion,
described as turbulent flow.

Imagine that you take a small stick
and light the end of it on fire. If you’ve ever done this, you know
what happens. At first, the smoke that rises off
of the end of the stick is flowing quite smoothly. But then, as the smoke continues to
rise, its flow becomes disordered. The rising smoke moves from laminar
to turbulent flow. One interesting aspect of these two
states of fluid flow is that the transition from one to the other is not itself
smooth. Looking at an example like this,
certainly we can point to parts of the fluid flow and say that that’s laminar and
other parts and say that that’s certainly turbulent. But there’s an in-between region
where the type of flow is less clear.

To help us understand how a fluid
flows quantitatively, we can describe fluid flow using something called a Reynolds
number. This number, which we represent by
𝑅𝑒, tells us the ratio of inertial forces to viscous forces in a fluid. To calculate the Reynolds number,
we multiply the fluid density multiplied by its speed times the characteristic
length dimension of the scenario, perhaps the diameter of the pipe that the fluid
flows through. And we divide all that by the fluid
viscosity. When we say that the Reynolds
number tells us the ratio of inertial to viscous forces in a fluid, we’re saying it
tells us how likely a flowing fluid is to keep doing what it’s doing. Compared to how much internal
friction the fluid experiences as it flows.

Practically, the Reynolds number is
a rule of thumb we can use to classify whether fluid flow is laminar — smooth — or
turbulent. While there are no guaranteed or
sharp cut-offs between which type of flow is which, in general, when a Reynolds
number is less than 2000, that flow is often characterized as laminar or smooth. There is a transitional zone when
the Reynolds number is between 2000 and roughly 4000. And when the Reynolds number is
above 4000, that flow is typically safely characterized as turbulent. This means that calculating the
Reynolds number gives us a clue as to which type of flow our fluid is in. Let’s get some practice using these
ideas through an example.

At what flow rate might turbulence
begin to develop in a water main with a 0.200-meter diameter? Use a value of 8.94 times 10 to the
negative fourth pascal seconds for the viscosity of water. And assume that a Reynolds number
of 2000 corresponds to the onset of turbulence.

In this exercise, we want to solve
for a flow rate we can call lowercase 𝑓. In this example, we have a water
main of diameter we’ve called capital 𝐷 of 0.200 meters, allowing water with
viscosity 8.94 times 10 to the negative fourth pascal seconds to flow through
it. The flow rate 𝑓 we want to solve
for corresponds to the onset of turbulence in the pipe. This means that the speed at which
the water flows increases until the Reynolds number of that flow is equal to
2000.

We know that, in general, flow rate
is equal to the area through which a fluid is flowing multiplied by the speed of
that fluid. If we call the cross-sectional area
of our pipe capital 𝐴, we can solve for that area based on the diameter of the pipe
𝐷. But we still don’t know the rate of
fluid flow 𝑣. To solve for that speed 𝑣, we can
recall that the Reynolds number describing fluid flow is equal to the density of
that fluid times its speed multiplied by a characteristic length 𝐿, all divided by
its viscosity 𝜂.

When we write out the Reynolds
number equation for our scenario, the characteristic length 𝐿 is the diameter 𝐷 of
the pipe. We notice that fluid speed 𝑣 is in
this equation for Reynolds number. So when we rearrange to solve for
𝑣, we find it’s equal to the Reynolds number multiplied by fluid viscosity divided
by fluid density times pipe diameter 𝐷. If we substitute this expression in
for 𝑣 in our equation for flow rate 𝑓 and then replace the cross-sectional area 𝐴
with the area of the cross section in terms of diameter 𝐷. We now have an expression for the
flow rate 𝑓 in terms of known values, except for the density 𝜌 of the water. The density of pure water is very
nearly 1000 kilograms per cubic meter.

So with that, we’re ready to plug
in and solve for 𝑓. Just before we do though, we notice
that a factor of the pipe’s diameter 𝐷 cancels out. Now when we’re entering the values
of these variables, when we plug in for the diameter 𝐷, the Reynolds number of
2000, the viscosity of water, and its density 𝜌. When we enter these values on our
calculator, we find, to three significant figures, 2.81 times 10 to the negative
fourth cubic meters per second. That’s the flow rate at which
turbulence begins to develop in this water main.

Let’s summarize what we’ve learned
so far about viscosity and turbulence. We’ve seen that viscosity is a
measure of the level of internal friction within a fluid. Think of those various levels of
the flowing fluid and how they interact with one another. We’ve seen that simple or absolute
viscosity is represented by the Greek letter 𝜂 and that a second type of viscosity,
called kinematic viscosity, equals simple viscosity 𝜂 divided by fluid density
𝜌.

We’ve also seen that Stokes’ law
describes the viscous drag force that acts on a spherical object in a flowing
fluid. As an equation, that viscous drag
force is equal to six times 𝜋 times the radius of the sphere multiplied by the
viscosity of the fluid it’s moving through times their relative velocity. We’ve also seen how fluid flow is
characterized as either laminar — smooth — or turbulent — rough and disordered. And finally, we’ve seen that the
Reynolds number is a tool for quantifying the type of flow a fluid is in, whether
laminar or turbulent. And we saw that the Reynolds number
provides the ratio of inertial forces, density times speed times length, to viscous
forces, viscosity, in a fluid flow.