# Lesson Video: Viscosity and Turbulence

In this video we learn the definitions of absolute viscosity and kinematic viscosity, as well as laminar and turbulent fluid flow and how Reynolds number characterizes fluid flow.

11:23

### 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.