19. Fluid Mechanics

Intro to Pressure

# Anderson Video - Bernoulli's Equation

Professor Anderson

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>> Let's talk about the big equation for this chapter which is Bernouilli's equation. Okay. Bernouilli's equation, that might be one of those equations that you've heard thrown around before. Whenever they talked about fluid flow, lift of an airplane, drag, somebody's going to mention Bernouilli's equation. So this comes into play in a lot of physics problems. Let's see how we can model it. And to do that, let's go back to our pipe and let's flow that fluid uphill. So here's our pipe. We're going to flow the fluid up this hill. And let's write down some parameters for this pipe. So we said that there's a mass of water, m. That same mass has to exit the pipe if we have continuity. This is moving at v1. This is moving at v2. But now there's a height here, y1. And there's a height here, y2. And these occupy some length, delta x1, some length, delta x2. And now there has to be some driving force to push this thing up. We'll call that F1. It's going to act on the end of the pipe, cross sectional area A1. And there might be some force pushing back against you, F2, acting on the end of the pipe, A2. Those are all the parameters that we need to now derive Bernouilli's equation. Bernouilli's equation is really not so complicated because it is completely analogous to conservation of energy. It's basically conservation of energy for fluids. So when you have fluids moving, you have kinetic energy. Anything moving has kinetic energy. When you have them going up a hill, you have gravitational potential energy. When you're applying a force, you're doing work. All of that stuff is energy. And so really Bernouilli's equation is nothing more than conservation of energy. So let's set up the problem and then we're going to skip a little bit of the math and we're going to jump to the solution. And then we'll see how we can apply that solution. Okay. The work involved in moving the fluid is going to be what? Well, we have F1 delta x1 pushing this fluid up. There's some work of this one trying to go the other way, but the fluid is still moving up and so we have to make a minus sign F2 delta x2. All right. But we know that force relates to pressure. This is P1 times A1. This is P2 times A2. And A1 times delta x1, that is a volume. A2 times delta x2, that is a volume. What's the volume of this fluid? What's the volume of this fluid? Now right off the bat you should look at this and go, "Wait a minute. That sounds like something I learned about in chemistry." Work equals P delta V. P times V, that is work. All right. So you might remember this from your chemistry classes. But the volume here is the same, right. The mass going in has to be equal to the mass going out. The density doesn't change. And so the volumes are in fact equal. And we will just define that as V. And so work is P1 minus P2 times V. That's one of the terms that we're going to need. Let's see if we can calculate the other terms that we're going to need and we'll put it all together. All right. Let's separate the terms. We've got P1 times V. I'm going to move this v1 term over to the other side. So we have one-half m v1 squared. And now I'm going to move the y1 term over to the left side. So I have mgy1. And now all the stuff on the right I want to leave as number two. So let's move P2 over to the right side. And we have a one-half m v2 squared. And we have an mgy2. All right. That looks pretty good except I don't really like this volume V. And I tried to write capital V and small v because this one's a speed, this is a volume. So let's divide everything by V. If I divide everything by V, what's going to happen? That goes away. I put a capital V down there. I put a capital V down there. That goes away. This, we put a capital V down there. And that we put a capital V down there. And that looks pretty good because mass over volume is density. And so this becomes rho times v1 squared. Mass over volume, that's density, rho gy1. The right side P2 plus one-half rho v2 squared plus rho gy2. And guess what? That is Bernouilli's equation. Conservation of energy applied to fluids, this is what you end up with. It says at one end of the pipe if you have all these terms, pressure 1, speed v1, height y1, you know that that is a constant. And if I go to the other end of the pipe, those same things have to add up to equal the same number. Okay. Bernouilli's equation is very, very powerful.

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