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AP Physics 2: Exam Prep27 chapters | 158 lessons | 13 flashcard sets

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Lesson Transcript

Instructor:
*Sarah Friedl*

Sarah has two Master's, one in Zoology and one in GIS, a Bachelor's in Biology, and has taught college level Physical Science and Biology.

Fluids rarely stay put, so we need to understand how they move. In this lesson, you will learn about how the movement of fluids is influenced by the space it passes through, as well as see examples of how to calculate aspects of this movement.

There's no doubt, fluids like to move! Just imagine a roaring river or a summer breeze blowing, and you can understand what I mean. Fluid dynamics is a tricky subject because there are different kinds of ways that fluids can move, and we don't even fully understand exactly how they do some of the amazing things they do. Because of this, we're going to talk about fluids in a very specific sense, making a few assumptions for this lesson.

First, we are going to assume that the fluid is **incompressible**, meaning that its density can't be changed. Try to condense 8 ounces of water into a 4 ounce container, and you'll see what this means!

Second, we're going to assume that the flow of the fluid is **laminar**, meaning that it is a steady, constant flow that doesn't change with time. Think of water running through a straight, narrow creek instead of pooling and twisting in small nooks and crannies along the bank.

Finally, we're going to assume that the fluid is **non-viscous**, meaning that there is no resistance to flow. **Viscosity** is the resistance to flow, so we would say that honey is more viscous than water because water flows much more easily than honey. But for this lesson, our fluid experiences no resistance.

OK, now that we've gotten that out of the way, let's look at how fluids move.

You know that you can make the water in a garden hose come out faster if you partially block the opening. This is because the same amount of water has to travel through that smaller opening as the larger opening.

This is true of any type of fluid and any type of tube or pipe - toothpaste coming out of its tube, blood flowing through your arteries, and water through pipes. The moving fluid can't be stored in the tube or pipe - it must travel through. And, the same volume of fluid that goes in must come out.

But when we make the area in the tube or pipe smaller, like we did with the garden hose, the fluid speeds up because the same volume of water has to go through a smaller area than before. This relationship between the area inside the pipe (the pipe's internal diameter) and the velocity of the fluid is expressed in **the equation of continuity**, written as *v1 A1 = v2 A2 *. Here, *v* is the velocity of the fluid, and *A* is the area that fluid travels through.

Because this is an equation, it means that the product of either side has to equal the product of the other. So if the area on the either side decreases, it means that the velocity on the same side of the equation has to increase accordingly.

It sounds simple enough, but let's work through an example to see how the speed changes depending on the area the fluid is traveling through.

Say that you have some fluid flowing through a pipe. At one end, the pipe has an internal diameter of 10.0 cm. But down the line at a second point, the internal diameter of the pipe is only 5.0 cm. The initial speed of the fluid moving through the pipe is 5.0 m/s, but we want to know what the speed is at the second point where the pipe is narrower. And here's where we can use the equation of continuity to help us figure it out.

First, we need to rearrange our equation to get *v2*, the velocity of the fluid at the second location, alone on one side. Next, we need to do some quick conversions to make sure we're working with the correct units. Since our velocity is in meters per second, we need to change our pipe diameters to meters as well. This gives us 0.10 m for the first point and 0.05 m for the second.

Finally, since we're working with the area of a circle, the pipe, our area values will use the radii of the openings instead of the diameters. This gives us Ï€*(0.05 m)^2 for *A1*, and Ï€*(0.025 m)^2 for *A2*.

Now all that's left to do is plug in our known values and solve! When we do, we find that *v2* equals 20 m/s. That's quite an increase! Can you see how the equation of continuity shows how the speed of the fluid is faster in narrower areas than wider ones? The same amount of fluid has to pass through, so it goes through faster to make up for the smaller area.

We can also understand fluid dynamics by calculating the **flow rate** of a fluid, which is the rate at which a volume of fluid flows through a tube. This is different from the speed - the flow rate is the time frame in which an amount of fluid flows, whereas the speed is simply how fast the fluid flows.

In equation form, flow rate is represented as:

*Q* = Î”*V* / Î”*t*

where *Q* is the volume flow rate, *V* is the volume of fluid, and *t* is the time in seconds. The Greek symbol Î” means change in, so we read this as: the volume flow rate equals the change in volume over the change in time.

Let's try an example with this equation. Say you have a garden hose that fills a 5.0 liter bucket in 10 seconds, and we want to know the flow rate at which the water comes out of the end of the hose.

Using our flow rate equation, we simply plug in our values and solve the equation. When we do so, we get: *Q* = 5.0 L / 10 s, which means that *Q* = 0.5 L/s.

Both the equation of continuity and this flow rate equation show us how the flow rate itself is constant at any point in the tube. One variable doesn't change without affecting the other, so as the diameter of the tube decreases, the speed must increase to make sure that the flow rate stays the same. If the flow rate is 0.5 L/s at one point, it will still be 0.5 L/s at the second point. In order to keep that flow rate constant, the fluid must travel faster to move the same amount of volume in a given time interval.

Fluids are dynamic. They like to move, but this movement is not always well understood. We can, however, describe fluid motion in terms of an ideal fluid when we make some assumptions. Assuming that a fluid is **incompressible**, has **laminar flow**, and is **non-viscous**, we can describe how fluids move through tubes and pipes.

Specifically, fluids move so that the same volume of fluid that goes into the tube must come out. This means that the fluid speeds up as it passes through narrower areas. This is described with **the equation of continuity**. This equation, written as *v1 A1 = v2 A2 * helps us understand how when the area *A* decreases, the velocity *v* must increase to keep the equation equal.

We can also describe a fluid's **flow rate**, which is the rate at which a volume of fluid flows through a tube. Expressed in equation form, flow rate is: *Q* = Î”*V* / Î”*t*, where *Q* is the flow rate, Î”*V* is the change in volume, and Î”*t* is the change in time.

Together, the flow rate equation and the continuity equation tell us that this flow rate is constant at all points in a tube. This is because the amount of volume entering the tube must be the same as the amount of volume leaving it, so in order to compensate for narrower spaces, the fluid must speed up to push on through.

Once you've completed this lesson, you should be able to:

- Describe three characteristics of an ideal fluid
- Identify the equation of continuity and the flow rate equation
- Explain how these two equations describe the flow of fluids through tubes

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AP Physics 2: Exam Prep27 chapters | 158 lessons | 13 flashcard sets

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