Lesson Video: Laminar and Turbulent Flow of Viscous Fluids | Nagwa Lesson Video: Laminar and Turbulent Flow of Viscous Fluids | Nagwa

Lesson Video: Laminar and Turbulent Flow of Viscous Fluids Physics

In this video, we will learn how to model the motion of viscous liquids that can have uniform laminar flow or non-uniform turbulent flow.


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.

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