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Physics: Bernoulli's Equation

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  • Type: Video Tutorial
  • Length: 12:57
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 139 MB
  • Posted: 07/01/2009

This lesson is part of the following series:

Physics (147 lessons, $198.00)
Physics: Fluids (13 lessons, $13.86)
Physics: Fluid Dynamics (4 lessons, $5.94)

This lesson was selected from a broader, comprehensive course, Physics I. This course and others are available from Thinkwell, Inc. The full course can be found at http://www.thinkwell.com/student/product/physics. The full course covers kinematics, dynamics, energy, momentum, the physics of extended objects, gravity, fluids, relativity, oscillatory motion, waves, and more. The course features two renowned professors: Steven Pollock, an associate professor of Physics at he University of Colorado at Boulder and Ephraim Fischbach, a professor of physics at Purdue University.

Steven Pollock earned a Bachelor of Science in physics from the Massachusetts Institute of Technology and a Ph.D. from Stanford University. Prof. Pollock wears two research hats: he studies theoretical nuclear physics, and does physics education research. Currently, his research activities focus on questions of replication and sustainability of reformed teaching techniques in (very) large introductory courses. He received an Alfred P. Sloan Research Fellowship in 1994 and a Boulder Faculty Assembly (CU campus-wide) Teaching Excellence Award in 1998. He is the author of two Teaching Company video courses: “Particle Physics for Non-Physicists: a Tour of the Microcosmos” and “The Great Ideas of Classical Physics”. Prof. Pollock regularly gives public presentations in which he brings physics alive at conferences, seminars, colloquia, and for community audiences.

Ephraim Fischbach earned a B.A. in physics from Columbia University and a Ph.D. from the University of Pennsylvania. In Thinkwell Physics I, he delivers the "Physics in Action" video lectures and demonstrates numerous laboratory techniques and real-world applications. As part of his mission to encourage an interest in physics wherever he goes, Prof. Fischbach coordinates Physics on the Road, an Outreach/Funfest program. He is the author or coauthor of more than 180 publications including a recent book, “The Search for Non-Newtonian Gravity”, and was made a Fellow of the American Physical Society in 2001. He also serves as a referee for a number of journals including “Physical Review” and “Physical Review Letters”.

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Understanding the steady flow of a fluid requires more than just the equation of continuity. The equation of continuity merely tells you that if you've got a flow of fluid, that when the area that the fluid is flowing through gets smaller, the velocity gets bigger. That's just a statement that the amount of fluid coming is the amount of fluid going out, when you have an incompressible fluid. It's useful, but there's more that you can think about and talk about. When we were talking about hydrostatic, we were very interested in pressure. What's the pressure at different spots in the fluid? And it's the same story when you've got flowing fluid. If you've got pipes, and they're changing in dimension and going upstairs and downstairs, you would like to know what's the pressure in those pipes at various different places. It's kind of like asking questions about dynamics - about forces - rather than asking questions about kinematics, like velocity alone.
So how are we going to approach this? In ordinary, non-fluid, physics, when we wanted to understand about dynamics, we used Newton's second law. And you can do that with fluid flow. It's a very difficult problem, because you have to look at little chunks of fluid and it's just a nasty business. It's like trying to use Newton's second law with a complicated system with lots of pieces. The way really to approach such problems is to use conservation of energy - energy principles. It's true that when you use energy principles, you're not learning everything there is to know about the system, but you're learning a lot. It teaches you a lot of physics that's very useful, and that's what we're going to do here.
The equation that we're after is called Bernoulli's equation. And Bernoulli was the first one to really clearly lay out how is the pressure connected to the speed at different spots in the flow. So I'm not going to work through the proof in detail, of Bernoulli's equation, but let me just tell you the ideas and then we'll talk about the equation. So here's a flow tube in some flowing system. This could just be a chunk of a flowing system or it could be the whole thing. This is a streamline here on the outside. And at time 1, I've got some system of water, which I'm going to just define arbitrarily. It starts here at point 1 and ends here. And then a little time, t, later, I'm going to follow that fluid as it's moving, and it's a steady flow. So this end has moved over to here and this end has moved over to here. So now I have a system, which has just shifted a little bit. And the way you can think about it is the liquid that was here has moved over to here. And, of course, this is a steady flow. There's still other liquid coming in to take the place here and there's liquid being shoved out of the way here.
So this is the picture, and here's the velocity of the flow at this spot, running through a cross-sectional area, A[2]. Here's the speed, v^1, direction this way, running through a cross-sectional area, A[1]. If I want to understand what's going on here, I can think about mass. I can think about the fact that the mass in this chunk equals to the mass in that chunk. That's how I got the equation of continuity. Area x Speed = Constant arose just from arguing to myself that the mass that comes in equals the mass that goes out. But supposing instead of thinking about just mass entering and exiting, I think about work being done. In fact, let me use the work/energy principle. Work net is equal to change in kinetic energy. This theorem applies to anything and it certainly can apply to this little system that I'm looking at.
So, I'm not going to go through all the steps, but the basic idea is this. The work net is the external work being done. What external works are being done on this system? Not very many. There's a fluid on this side, which is pushing. So there's force this a way, and it's moving through a distance. So there's an external work force times distance. And what's the force? Why, it's just pressure here. Pressure times area - P[1]A[1 ]x X[1]. So that's a work. And then there's a negative work being done over here. Why is it negative? Well, it's the work being done by the external fluid, which is pushing us this way but we're moving that way. And so the work done when the force and the displacement are opposite directions is negative. And it's going to be P[2]A[2][][][][]X[2].
There's one more external work being done and that's the work of gravity. Somehow or another, from the beginning situation to the end situation, really, all that's effectively happened is that this mass of water (M) here has moved up to here. So there's been work done by gravity, which is negative Mg times the change in height. It's negative because gravity's down and the displacement of the chunk is up.
So those are the three things that you have to write down on the left-hand side. And K is just the change in kinetic energy, and that's going to be (MV[2])^2 minus (MV[1])^2, where again M refers to the mass only of this little chunk. There's only one mathematical trick that you need to use to really finish this story. And that is, over here, the work done involved P[1]A[1][][][][]X[1]. And what's A[1][][][][]X[1]? That's just the volume of this little cylinder over here. And volume of a cylinder - could call it V[1] - remember volume is just . I usually think of that formula slightly different. I usually think as density as . But mass is just this constant mass. It's the same M here and here. And density is the same here and here. I'm assuming an incompressible fluid. And so instead of writing A[1][][][][]X[1], let me write it as . And now it's really just algebra.
You've got to divide everything through by M and multiple everything through by and add terms to both sides to make it look pretty. And what you'll discover, when the dust settles, is really not so bad. It looks like this. P[1] - that's the pressure at the left point, where you entered - plus (V[1])^2 + gy[1] - I don't bother putting a subscript on the density, because I'm assuming it's constant - is equal to the same expression with twos - T[2] + (V[2])^2 + gy[2]. In other words, T[1 ]+ (V)^2 + gy is the same at position 1 and position 2. And of course, position 1 and 2 are totally arbitrary. No matter where you are in the flow of the fluid, pressure plus (V)^2 + gy = constant. This is Bernoulli's equation. Bernoulli derived this and now, we really just need to talk about it, because the derivation is nasty. The formula itself is a little bit intimidating looking, and there's lots of physics in here.
Let's look at some special cases just to try to get some sense of this. Supposing that we have static, so there is no V - nothing's moving. So this says pressure plus gy is the same everywhere. Or if you think about that, it means P, between two different points, is equal to g times the difference in height between those two points. That's that old hydrostatic equation that we were using. The pressure difference is times g times the difference in height. You have to think about the signs, and it really does agree with our old equation.
So that's nice. This is in agreement with hydrostatics. How about if y is zero? All right, in this picture, I was imagining a pipe going upstairs, but you can have a flat tube and you could have a flat river - lots of situations where you might not worry about changes in height. So what this is saying is that pressure plus v^2 is a constant. I don't know what that constant is, in any given problem. I need to figure out pressure and velocity somewhere, and then I'll know it everywhere else, or at least I'll know the connection between them everywhere else.
Remember that you always have this equation as well. So you really have two equations, so if you know about the shape of the flow, that tells you what the area is, then you can deduce how the velocity is changing and then this formula can then be used to tell you how the pressure is changing. So the two of them together are often very powerful combination.
Supposing that, as I said, there's no change in height. So P + v^2 = constant. What is that telling me, physically? If the speed of the fluid is bigger somewhere, the pressure is lower at that spot, because the sum has to stay constant. So you're in the shower, and there's air in the room, inside the shower and outside the shower. And if you've got an old-fashioned shower with a curtain, the pressure on the outside and inside is equal and the curtain hands straight down. And now you turn on the shower. The water drags some air with it. So now there's flow of air past the curtain on the inside and there's no flow of air on the outside. That's just atmospheric pressure. So what this formula says is that on the inside, where the v of the air is big, the air pressure inside the shower is a little bit smaller. So there's high atmospheric pressure on the outside, a little bit less on the inside. And the shower curtain attacks your leg, because of the pressure difference, which is pushing it, and it's very light.
Here's another example of that same idea. Here's a piece of paper, and it's drooping, because gravity is acting on it. Let me blow some air over the top. So there's going to be a v on the top and therefore, the air pressure on the top, by Bernoulli's equation, will be reduced a little bit. So high pressure below, lower pressure above, and the thing should rise up. Works like a champ. Very surprising phenomenon - I can't imagine how I could have derived that, but it comes straight out of Bernoulli's equation. It's really conservation of energy principles and nice to have this to just apply and solve problems like that quickly and easily. Technically, this is really supposed to work when the density is constant - incompressible. I was using air here. This formula actually works decently for air, but you have to be careful. It really was derived for constant density.
Let me just show you one last problem. Supposing you have a tank, and it's filled with water. And you want to know how fast is the water spewing out when there's a hole in the tank at distance H below the top. It's a direct application of Bernoulli's equation. You know, I've shown some turbulent flow here, so this equation only works for laminar - steady - flow. So it's right here. Right here, you're outside of the tank. The pressure outside of the tank is atmospheric. So I have P[atm] + v^2 - that's the density of the water - times the velocity where it's pouring out, plus gy, and let me just call this y = 0. That's Bernoulli's equation at this spot. And it should be equal anywhere, so it should also be equal to the pressure, etc., up at the top. So what have I got up at the very top? I've got, at the very top, P[atm] plus - there's no velocity up here. The big tank - the water is screaming out here, but it's hardly moving at all up there. So I'm adding zero. But I do have a g times y, which in this case is h. So here's my equation. P atmospheric cancels, and I can immediately solve for velocity, in fact it's immediately easy. The density cancels out. . It's the same velocity you would get if you just dropped a drop of water from the top to the bottom. That's an interesting thing to think about. That's conservation of energy and so is this.
Bernoulli's equation takes a little practice. You've got to think about what's the pressure in flowing fluid - what's the velocity. And you've seen a couple of problems, you discover that it really saves you a tremendous amount of grief when you're trying to understand the physics of flowing fluids. It doesn't tell you everything. It's really just relating pressure and velocity and height and that's all. But it's certainly handy. If you're designing systems, trying to understand flow of fluids through pipes, it's really nice to have.
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