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Fluid Mass, Flow Rate and the Continuity Equation

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  • 0:01 Fluids in Motion
  • 1:16 The Equation of Continuity
  • 4:30 Flow Rate
  • 6:14 Lesson Summary
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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.

Fluids in Motion

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.

The Equation of Continuity

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 π*(0.05 m)^2 for A1, and π*(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.

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