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0:00
Welcome to the 29th lecture video for Physics 4A. We wrap up fluid mechanics and all topics coverage of this semester with this lecture set. The first segment wraps up what we studied in the last lecture set—fluid statics—mainly analyzing arrangements of forces that are set up with the fluids at rest, here by looking at an application of hydraulics that results in a mechanical advantage. And we start and finish our brief coverage of fluid
0:35
dynamics by discussing the continuity equation, which expresses a kind of conservation of flow, and by looking at the textbook’s derivation of Bernoulli’s principle, a statement of conservation of mechanical energy in moving fluids. This lecture wraps up all the topics we will cover this semester. We do have one review lecture set in which we will do a recap and review of topics we have covered this semester, but not so much of the problem-solving techniques.
1:15
So I can kind of break it up into three different parts. One is something called Pascal’s principle. And do you guys remember “simple machine” that I have mentioned from time to time in this class? Does that phrase mean anything to people? Yes. Rathanan, what does a simple machine mean? Or do you remember any example of simple machine? Pulley. Pulley can be an example of a simple machine—what does a pulley do that you would call it a simple machine?
1:57
If I’m using a pulley as a way to build a simple machine, it would look like this: I would have a pulley here and a string that’s attached like this, and just so that I can pull it downward I might have one additional pulley here. I mean, it’s not necessary, but it makes things easier. Then this would go around, and I would be applying downward tension here, and I would attach a mass to this first pulley
2:29
here. This is an arrangement that someone might call a simple machine. What does this device do?
2:44
Let me ask you this another way. Let’s say you want to apply just enough tension so that this thing doesn’t fall. How much tension should I apply? Do I need to apply mg, or do I need to apply something else?
3:12
Asia? It’s less than mg. Here’s the quick way to get at it without doing a bunch of math. You have tension here; hopefully you have this intuition that the tension here is the same tension here, and it’s the same string—ignore friction or whatever—so this is also the same tension. On this pulley are two upward tensions of the same amount T,
3:48
which is going to equal the weight of this thing. So that means my tension must be half of mg. This is an example of a simple machine. Did we never go over this in class? It might have been in one of the sample exams, maybe, and some of your homework possibly. Well, I should've brought this up earlier. What do you think the purpose of a simple machine is? What does it accomplish that makes it an arrangement that you would want to use for something useful? Steven? "You can
4:31
lift more with less." Yeah, you can lift more weight by applying less force than the entire weight of that thing. Do you remember any other arrangement that accomplishes something like this?
4:46
We have seen maybe one more example that’s perhaps familiar. I guess we haven’t explicitly brought it up as an application of this. It came up in rotation, or it could have come up in rotation, or you would have seen something like this in rotation.
5:18
So this is why this is here. Let’s say I have something super heavy—actually heavy—and let’s say I want to lift this. The way this is, I would have to apply a force of about 2 kilograms to directly lift it. But if I set up a very particular arrangement, I can lift it—not entirely, but get it to move—by applying a force that’s less than 2 kilograms. The way I would do it is: I have this pivot point, I stick something underneath, and I have to do it so that it doesn’t slide out.
6:00
This pivot—something is holding this in place so that it’s not moving. I have to take this end and do it like this. Then the amount of force I have to apply at this end is way less than 2 kilograms. Do you remember seeing something like this before? Do you remember using something like this before? Right? "You use it. Those are the jacks that you use on a car." So, jacks use something different. I actually have a jack here. This is a jack,
6:28
right? A jack uses something different: it uses this screw thread that works kind of like a ramp does. But this is actually another example of a simple machine, because the only thing that’s operating it is your hand turning this, but this can lift a fair amount of weight. By turning this, I can lift—I guess I haven’t actually tried with this jack—but
6:57
depending on the car jack, you might have a lever thing that you move up and down, right? Yeah.
7:06
So at this point I shouldn’t waste too much time bringing up all the examples of simple machines. All these simple machines have really two common features. One of them is what’s called a force multiplier [mechanical advantage], and that’s what you have seen here. So all the simple machines—you can describe them by something you would call a force multiplier [mechanical advantage]. This is the entire point of a simple machine.
7:34
Why do you want to arrange pulleys this way? So that you can lift a weight that’s
7:40
heavier than what you can lift with bare hands alone. Why do you want to use a lever (a crowbar is another example of a lever)? It’s so that you can move this object by applying a force that’s a lot less. I think Archimedes is actually the one who said, “Give me a place to stand and a lever, and I can move the Earth.” Does that phrase sound familiar? No? Never mind. We don’t care.
8:12
So, in all those cases the main purpose is the force multiplier—you apply a smaller force to get a larger output force. But there’s always a price to pay. I mean, if you could get this force multiplier without anything else you pay in return, you would have hundreds of these pulleys and that’ll make you feel like a superman; you can lift 100 times the weight that you can normally lift. But there’s something you have to pay, which is why there’s a limit to where this
8:53
is useful. There’s a quantity that, in all these simple machine arrangements, you can always say: you might be multiplying force, but you are not multiplying this; this quantity remains the same. What quantity here, if you were to calculate, would remain the same? What one thing have you learned in this class that you cannot get for free? "Energy." Yeah, energy. So as you are using this simple machine arrangement, energy is going to be conserved.
9:37
However much work you put in here, the amount of work you get here will be the same—meaning you were able to apply double the amount of force at this end than what you’re applying. If you analyze this carefully, you will see that for every meter this can go up, I have to pull down 2 meters of string, so that the force times distance is always going to be the same. It’s the same deal here with the lever. You look at how much this end is moving.
10:14
Because of the geometric arrangement here, for something like 10 cm this far end moves, this end only moves like a centimeter. So if I’m multiplying my force by a factor of 10, then my distance goes down by a factor of 10, so that the amount of work you get out at this end is the same as the amount of work you put in here. And it’s the same thing with the jack, same thing with ramps, same thing with any kind of simple machine arrangement that you would have seen.
10:52
The reason I’m bringing up all of this is that fluids actually give you one more arrangement where you can get a simple machine. The idea of a simple machine is a general idea; there are many different arrangements you can utilize to get something that has a force multiplier and the same work in as work out, and the arrangement with fluids gives you one more arrangement where you can have this simple machine. You multiply force, but the amount of work you get out is the same.
11:38
Question? "No, I was going to mention hydroelectricity would be an example of a machine." Hydroelectricity is power generation. So you are thinking about electric dams and that kind of stuff, right? "Or the one with the wheels that capture the water?" Yeah. So that’s more of electricity and magnetism. I mean, energy is coming from the gravitational potential energy of water that’s going into turning the generator, but there we actually
12:09
have to talk about electromagnetic induction, which is a topic of Physics 4B—not this class.
12:17
Let me write down the name for it because some people might have heard of it. This arrangement of simple machine we call hydraulics, and actually it’s working part of a lot of different real-life machines. I may be misspelling it—hydraulics. If you are looking at any heavy machinery, it probably has some kind of hydraulics in there that helps people multiply force. You can use this instead of gears, maybe. Old-fashioned
12:59
power steering also used hydraulic power steering. I think these days power steering in cars is more of an electric power steering, but an old car like mine has hydraulic power steering. So let me describe just the basic idea behind hydraulics, and this is the kind of setup. Let’s use a car as an example. Let’s say you have a platform that has,
13:33
I don’t know, a 2,000-kilogram car sitting on it—a tiny sedan that’s 2,000 kilograms. And I want to say there’s some arrangement we can make where force that’s applied by a single person—let’s say you have a guy who weighs 100 kilograms—a guy who weighs 100 kilograms can lift this car. You might wonder how, and that’s the arrangement that you can make with hydraulics. Let me describe the sort of fluid arrangement that you would have so that this
14:23
100-kilogram person, using his weight alone, can actually lift this 2,000-kilogram car. This is the arrangement: you have this person standing on some kind of platform—let’s say that’s attached to something here. This is my representation of a standing platform. And this is already on a platform that’s going to be attached here, and these two platforms are connected by a pipe that contains a fluid. So they’re connected by an enclosed system that contains fluid.
15:15
Let me draw it this way. This is the pipe—the left-hand pipe of a smaller diameter—that’s going to be connected on the right side to a pipe of larger diameter.
15:35
It has to be full for this to work. It’s called hydraulics because it’s probably going to be filled with water. It needs to be filled with what people would call an incompressible fluid—a kind of fluid where, if you apply a significant amount of pressure, the volume doesn’t change much. That actually describes most liquids. So this just cannot be air; if it’s air, then it would just compress and not do what we want it to do.
16:07
I’m going to claim that with this arrangement, you can have a 100-kilogram person standing on a platform that’s going to apply enough force to cause this car to rise.
16:25
Let me try to describe this situation, and along the way I’m going to describe a principle known as Pascal’s principle; but without actually naming the principle, it’s something that will make intuitive sense to a lot of people. So let’s imagine: before this person is standing here, what would you say the pressure of the liquid in this spot is? What would you say the pressure is before there’s
16:57
a person standing here? What is the pressure of water here? Is the water here at any kind of pressure? What kind of pressure? So, water here—is that under any kind of pressure? "Under air pressure?" It’s under air pressure. So it’s at atmospheric pressure. I could say at this point, initially, the pressure would be atmospheric pressure. And in fact, could you say that for most of this—now, as you go lower it’ll be at higher pressure,
17:43
but as you go back up to the surface here—can you say that this is also at atmospheric pressure? Yes. Right. Okay. So that’s our starting point. (Also imagine there’s no car here.)
18:05
Suppose I’ve placed a person on here. Then there will now be additional pressure here due to the weight of the person. So at this same spot, pressure after the person gets on is going to be my initial pressure—the atmospheric pressure—plus the pressure due to this person. One way to express that would be: if I know the cross-sectional area here—let me call that A₁—then the pressure due to the person would be whatever force this person is applying,
18:53
call that F₁, divided by area A₁. That’s how much pressure this person is exerting by standing here.
19:08
This is what I want you to think through. If the pressure in this part of the liquid is increasing, what do you think might be going on with the pressure in this other part? Can this remain at the same pressure it was at before? Can this remain at simply the atmospheric pressure, or will it have to change to something else? Change. Why? Because of the external pressure. Think of this iteratively: as the man stands on here,
19:54
that exerts additional pressure here, and let’s say for the time being that’s the only change that happens—nothing else changes. Then next, as you pass around,
20:06
imagine these fluid elements: pressure on one side of them is greater than the pressure on the other side, so this element is going to get pushed, and the one next to it will get pushed, and so on, until that change in pressure propagates through the entire fluid. At this end you will end up with the same change in pressure. The pressure at this end of the fluid will be the atmospheric pressure plus the same additional pressure that was applied here—so F₁/A₁.
20:52
This is what’s known as Pascal’s principle: in a static fluid, any change in pressure in one part of the liquid gets transmitted throughout the liquid. It’s because liquid can flow. If there’s a pressure difference from one end to the other at the same height, then that pressure difference will cause the fluid to flow until the pressure equilibrates—it becomes equal throughout the entire thing at the same height.
21:24
Now, this is what I want you to look at: you have this additional pressure. What is the additional force on this platform due to the additional pressure? Is it simply F₁, or is it going to be something different? You know you have pressure; what do you need to do to get to the force on this platform? Multiply by area, right. So is it this area that we are multiplying? No, it’s the area of this surface that we are multiplying by.
22:07
To get this force F₂, you multiply by the larger area here. So F₂ = [this additional pressure] (F₁/A₁) × A₂. Here the force multiplier is the ratio of these two areas, A₂/A₁, times F₁. So here, with the car being 20 times heavier, you would want this area to be 20 times the area of this. Then the weight of the 100-kilogram man is enough to lift the entire weight of this 2,000-kilogram car.
22:58
So this is another simple machine: you have a force multiplier that’s given by this ratio of the areas because of the way pressure works. The challenge would be—this is why I mentioned the limitation of a simple machine—an arrangement like this has no motor, no external source of energy. Whatever energy this car gains must come from the energy loss of this 100-kilogram man. Can you see that the amount of work done on the car will be equal to the amount of work
23:40
that the man is doing? Or the easiest way to talk about it is in terms of potential energy. However much potential energy this gains is equal to how much potential energy the man is losing. Arjun, can you explain how that makes sense—how does that happen? "Change in height." Okay. So you are arguing the change in height of the car is less than the change in height of the man. How? What aspect of fluid would you bring up to say: because of this, the change in height of the car
24:19
must be less than the change in height of the man—probably by a factor of 20? Steven? The displacement of the water. Yeah, displacement of water. So whatever volume is gained here must be the volume that’s pushed out here. Their cross-sectional areas are different, but their volume must be—volume = cross-sectional area × height. So this A₁ × h₁ must be equal to A₂ × h₂.
24:56
So the reciprocal of the ratio of these areas is the ratio of the heights. If the area is 20 times greater here, then the height changes 20 times less. So this is another arrangement of simple machine that conserves energy. There’s no external source of power. The only thing it does is multiply force, but it does it at the cost of reducing the displacement, which means the amount of work done is the same. So this is one useful thing, and if you want to give it a name,
25:35
it’s really the biggest application of what some books call Pascal’s principle.
25:49
There are really two ideas I want to talk about that have to do with fluid in motion: something called the continuity equation and Bernoulli’s principle. I’ll try to give you some intuition for them, because especially the continuity equation sounds fancy but actually describes a really simple idea that you kind of knew already—you just maybe didn’t know the implication of. Let me start with a little demo.
26:17
You might have seen this—it’s a popular demo dealing with Bernoulli’s principle. I have this flat, bendy piece of paper, and I can make this paper bend up without touching it. You might say, all right, blow into it—that’s one way. But I’m going to make it bend up by blowing over it—blowing air over it. When I do this, it comes up. This is a result of Bernoulli’s principle:
27:01
when you have a fluid moving at some velocity, there’s a related difference in pressure that occurs. As I pass fluid here, that lowers the pressure in this region; pressure here is higher, so that difference in pressure pushes the paper up. This is a demo that you might have seen at the Exploratorium as a child. If you didn’t—now you saw it. I want to get more mathematical in a way you might not have seen in other science classes.
27:35
The first thing I want to address is a consequence of something called the continuity equation. There are at least two different ways of expressing this that I know of: one involves differential calculus—I’m not going to use that; the other looks more like an algebraic relationship, and I’m going to use that. It
28:09
expresses a very simple principle that you would figure out as you think through this demo. This shows a flow of some kind of fluid—that’s what the simulation is simulating—and it has some adjustable parameters. I can adjust the flow rate, have less water flowing. Pay attention to the unit: it’s liters per second, measuring how much volume of water is flowing per unit time. I can change the flow rate. Let
28:44
me set it back to 5,000; that’s the default. I like it enough—don’t need anything else.
28:52
I can put sensors to measure what’s called fluid speed—fluid velocity,
29:05
regular velocity. It measures, given this particle, how fast it’s actually moving: it’s moving at 1.6 m/s. The simulation is set up so that the speed at which this is moving relates to the flow rate. If I increase the flow rate, one way is to have the particles moving faster, and the speed is greater. Good. All of this makes sense. There’s a pressure here, but we’ll talk about that in a little bit; we don’t need to talk about it right now.
29:40
Here’s the prediction I want you to make. I’m back down to a flow rate of 5,000 liters per second, and I’m going to make two changes: I’ll make this part of the pipe narrower, and I’ll make this portion of the pipe wider. How do you think the fluid speed, as measured here or here,
30:17
will change as I make these two changes? As I make one section narrower, how should the fluid velocity change? “Narrower is faster.” Wait—I heard “increase.” Did you say faster, Abdi? “Yeah, the narrower, the speed is faster.” Okay. Anyone say a different answer? “Same speed.” Same speed? Okay. Let’s try it and see. I’m going to make this narrower. Let’s see—this much. Yeah, you can see with the particles that it’s faster. Good job—you have the correct answer. A lot of
31:00
people might intuitively answer that it slows down. At least for me, that would've been my answer; I’d think of an analogy to traffic: if you have a narrower portion of the road, traffic usually slows down; the cars don’t suddenly start going faster in the narrower portion. So, Abdi, can you explain why you predicted that the fluid would be faster here? "Because the cross-sectional area is narrower." Okay—area is narrower; that’s part of this being
31:42
narrower. “The pressure coming from that area is higher…” I think this is what you are thinking of, which is a good intuition to have. If you have a water hose and you want the water to come out faster, you narrow the end; then the water comes out faster, right? That’s what you’re thinking of. By blocking it off, pressure builds up here, so it makes it go faster. Yeah, that’s consistent. But I want to look at it from a different perspective.
32:21
So all of what Abdi was saying is correct. Since I have the simulation, let’s see it. Let me bring this back to what it was, and I can put this pressure meter here to measure the pressure here. All right, it has some pressure—120-something kilopascal—and you will see that as I make this narrower—wait,
32:47
it doesn’t increase; something else must have happened. Oh, pressure here decreased. Is that right? I guess that’s kind of right, and actually we’ll talk about this with Bernoulli’s equation.
33:09
One thing that is correct is: there’s a difference in pressure. Right now the pressure between here and here is more or less the same. But when I made this part narrower and made particles move faster, that somehow decreased the pressure way here. We’ll look at that later.
33:37
For now, I want an explanation that doesn’t have anything to do with pressure, because what’s happening with the pressure is a little bit weird. How would you explain why the particle is coming in relatively slow, it speeds up here, and—funny thing—doesn’t it look like they’re slowing back down here? Yeah. In fact, let me do this second part: let me make this portion of the pipe wider and see if they slow down further. Yeah—I make the pipe wider,
34:13
and you can see that the water actually slows down more. Once again, that’s counterintuitive. You think in traffic when your road widens up, you speed up—you don’t slow down. So what’s going on here? Why does water speed up in the narrower portion and slow down in the wider portion? Once again, I don’t want to deal with pressure because what was happening with the pressure was weird; I don’t want to have to explain that portion. In this wider portion,
34:51
pressure increased. So pressure starts out from 120-something, decreases, and then as you go out here pressure increases again. Something’s going on with the pressure, and we’ll talk about it with Bernoulli’s equation—Bernoulli’s principle—but that’s going to take some more work. We will come back to that in a bit. How would you explain it without referring to pressure? “Because of the volume.” Okay.
35:18
"From the smaller, when you come out to big, you have to fill it that big." You have to fill it that big. Okay, what do you mean by volume? Volume of what? “Volume of the pipe. Volume becoming bigger.” Okay, that’s closer. I do want us to focus on volume, but it’s not the volume of the pipe that I’m concerned with (although that’s part of the picture). The volume that I’m really concerned with is the volume of the fluid. And that’s
35:59
the principle behind what people call the continuity equation. You see it in fluids. You see—when you deal with electricity and magnetism at upper-division level—the thing expressed in a continuity equation is always the idea of
36:19
conservation. It could be conservation—in this case it will be conservation of volume, because the kind of fluid we have here is an incompressible fluid. So here it’s conservation of volume. If you’re dealing with a gas—something that can be compressed—then we might be talking of conservation of mass. Or in electricity, which you might deal with in upper-division electrodynamics, that’s conservation of charge.
36:56
In all these examples, the important idea is that there is some conserved physical quantity. Once it exists in some place, it cannot simply disappear; it has to move through space.
37:19
Here one big hint is that, through all these changes in the pipe diameter and changes in the fluid speed, one thing actually remains constant: the flow rate. Let me show you with the measurement. I have one thing that measures flow rate—ah, flux meter. This measures the flow rate. So flow rate starting from here—that’s the specified 5,000 liters per second. As I go to different parts of the pipe, I’m not going to pay attention to “flux”; the only thing I’m
38:07
going to pay attention to is flow rate. Flow rate remains the same: 5,000 liters per second comes in through this section here; 5,000 liters per second has to go through this section, because there’s no space in between where it can accumulate; and as they go through, as they come to the wider portion here, still 5,000 liters per second has to flow. This is a place where it’s nice to have an expression for flow rate—as in,
38:42
how does this flow rate (some volume per second) relate to the fluid speed? That will help me explain why fluid is flowing faster here in a way that keeps the flow rate constant. Yeah. I don’t want to spend too much time on this argument, so I will just give you the answer.
39:11
This type of stuff is something we would spend more time with in thermodynamics in Physics 4B, but for this class we’re just talking about something like this.
39:23
This is the standard way to think about it. I’m thinking of flow rate—how much volume of water will cross over some space. One way to express that is: given this boundary I’m thinking of,
39:42
how much volume will cross over this? Let’s extend it out a little bit—maybe it’s about this much volume that will cross over this boundary in the next time interval. Good. When I’m looking at this, there are two quantities to worry about. What is the amount of volume?
40:12
I would take the area and multiply by this distance Δx. That would be my volume, right? Okay. So let me express that volume that way: area × Δx. And then it’s a question of: in how much time is this whole volume going to cross over? I’m looking at the duration of time—call
40:45
it Δt. That’s how much time it would take for the last of the particles here to finally cross over. So I can write down my flow rate this way. Flow rate would be volume per time. Right? That’s flow rate. So flow rate = (area × Δx) / Δt. Do you see some of these terms being combined in a slightly different way that would give different physical meaning to some of these quantities? Asia. I have this Δx/Δt, right? The meaning I can assign there is that, for a particle here,
41:39
it’s crossing this distance Δx in time Δt. So these particles must be moving at some speed given by Δx/Δt. That’s my fluid velocity. So I can call this v—my fluid velocity v. So this is the expression we get for flow rate: flow rate = area × fluid velocity. You will see that if you write down the units, all of that works out. In this particular context, this is what we would say based on the continuity principle: my volume is conserved,
42:28
meaning my flow rate across all these different cross-sections has to remain constant. If it doesn’t, then I have this weird situation where more liquid flows in than leaves, or more liquid leaves than flows in. And that doesn’t happen with liquids because they cannot accumulate. So because my flow rate is the same throughout the entire pipe, I can write down something that looks like a conservation-law equation.
42:59
I would write it down this way. I’m going to imagine writing down my flow rate at a particular point along the pipe, and write down your flow rate at some different point along the pipe. Because the flow rate is the same here as here, I would have to say A₁ × v₁ = A₂ × v₂. And let’s say A₂ is greater than A₁. So let’s say I’m looking at the fluid velocity v₁. If you solve this for v₁, this is what you get: v₁ = (A₂/A₁) × v₂. So this is a number bigger than one. The velocity in the
43:57
narrower section is greater by the ratio of the areas. This is one of the conditions that has to be enforced when you have a liquid like water flowing through a pipe. This is called the continuity equation, and it’s a starting point to talk about some of the equations and expressions that fluid in motion has to obey. This is one of the simplest ones because it’s based on the idea of conservation of matter—really, it’s based on conservation of matter.
44:39
Now we have to think about: okay, if velocity increases, how is it increasing? Because increasing velocity relates to increasing kinetic energy. The kinetic energy of these particles here is lower than the kinetic energy of the particles here. So something must be set up externally—something must be set up in a way that energy is still conserved. That’s the next set of things for us to worry about. But at least this
45:10
is a starting point: because the volume has to be conserved, no matter what else happens, at least this has to be true. Yeah. All right. So that’s the continuity equation.
45:30
So, this is the second attempt—or another attempt—at lecture material that does the proper derivation of Bernoulli's principle. I’ve done this in the past, and I’ve got to tell you:
45:47
you’ll see it if you watch this lecture video. Every time I do this I kind of mess it up. I try every semester; I make some subtle mistakes, and I got through this somehow. I’m okay with the edited version that’s here; it doesn’t have any serious mistakes in it at the end. But I’ve got to tell you: the length of it is highly unsatisfactory to me. It shouldn’t really take 40 minutes. So let me give a second try at this lecture using the textbook resource. Instead of trying to derive
46:31
this from scratch—which I’ve done here and I can try to redo the exact same thing—let me
46:40
read it through the textbook, as a model of how I think you should be benefiting from
46:49
this condensed written material that’s available as your required course resource. This is from Chapter 14 which, if you recall, earlier we skipped: after Chapter 13 we went straight to Chapter 15 because, of all the 17 chapters in this long textbook, the one chapter that I thought we could rush through and be okay is Chapter 14. If you go into a field like mechanical engineering or civil engineering where fluid mechanics is important, you will
47:37
learn so much more than what we can cover here. So it’s okay to skim through it—rush through it. Within Chapter 14, Section 14.6 “Bernoulli's Equation” is where Bernoulli's equation is introduced and derived. We’ll read through this together and I will pause and comment where I see something. I’ve got to tell you, I haven’t actually read through this in detail before—maybe I should have—but one benefit is you’ll get genuine modeling of
48:14
what it looks like when someone’s reading through dense technical material like a physics textbook. I do have some benefit of knowing some of this stuff already, but I’ll try to role-play someone who’s seeing this for the first time. Section 14.6, Bernoulli's equation. It says, “As we showed in this figure…” I have a feeling it’s a figure in the previous section. Let me right-click and open a new tab so that I don’t lose my place here. Yeah, it’s in the previous section. Okay,
48:51
I think I’m okay with that preview of it there—something narrowing and velocity changing. Good. I think they probably covered what we call the continuity equation. “The speed of the flowing fluid increases. That’s dictated by properties of the fluid and geometric constraints on it. So when a fluid flows into a narrow channel, its speed increases. That means its kinetic energy also increases. The increased kinetic energy comes from the net work
49:25
done on the fluid to push it into the channel." Right? The work–kinetic energy theorem that we covered way back when we were introducing energy. "Also, if the fluid changes vertical position, work is done on the fluid by the gravitational force. So there must be some kind of potential energy associated with it.” As I’m reading it, I’m connecting it to the things that I’ve learned previously. You’ve heard me say physics is cumulative—this is how it’s cumulative.
49:58
“A pressure difference occurs when the channel narrows. This pressure difference—" I think it’s basically a statement of experimental fact, and hopefully we’ll see some derivation that justifies theoretically why this should happen. “This pressure difference results in a net force on the fluid, because pressure times area equals force.” Okay ... "and this net force does work." That makes sense conceptually. “Recall the work–energy theorem.”
50:29
Yeah—what I was saying earlier: the difference in kinetic energy comes from the net work. “The net work increases the fluid’s kinetic energy. As a result, the pressure drops in a rapidly moving fluid, whether or not the fluid is confined to a tube.” Okay, that’s interesting. I don’t know if that makes full sense, but I’ll move on. When you are reading dense technical material, you have to strike a balance. You can’t speed-read
51:05
through it. I can't read, “There are many common examples of pressure dropping in rapidly moving fluids…” and with the last sentence—“This effect was observed as far back as the mid-1800s when it was found trains passing…”—I can’t speed-read. Don’t do that. You should take your time reading through it. But at the same time, if a particular sentence doesn’t make sense, ignore it and keep reading. It might be necessary to come back and read it a second time. I’m advising you to
51:40
ignore it and keep reading because, as you keep reading, it may make sense to you eventually why they said this. You don’t want to get stuck at a single point just because of what might have been a minor misunderstanding or a smaller degree of understanding than what we are hoping for.
52:05
“There are many common examples of pressure dropping in rapidly moving fluid. For instance, shower curtains ..." Oh, I could have sped read through this; this is just a bunch of examples. "... have a disagreeable habit of bowing into the shower stall whenever the shower is on. The reason is that the high-velocity stream of water and air creates a region of lower pressure inside the shower, whereas the pressure on the other side remains at the standard atmospheric
52:30
pressure. This pressure difference results in a net force pushing the curtain inward.” "Similarly, when a car passes a truck on the highway, the two vehicles seem to pull toward each other. The reason is the same: the high velocity of the air between the car and the truck creates a region of lower pressure between the vehicles, and they are pushed together by greater pressure on the outside." I guess that might be true. I’m trying to imagine
52:57
this in the reference frame of these two cars. I guess it does look like there’s a higher velocity—maybe. I don’t know. Fluid dynamics stuff: I will tell you that a lot of it is not intuitively understandable to me. But they’re saying this effect was observed; if it’s experimentally observed, then yes, I trust that. Experiment, I do trust. Theoretical reasoning—maybe, maybe not. "—when it was found that trains passing in
53:26
opposite directions tipped precariously toward one another." Maybe. I don’t know. But I’m again doing the thing that I described: whether that effect, this situation they described, is true or not, whether I understand it or not, I don’t think it matters too much. So we’ll keep reading. "Energy conservation and Bernoulli’s equation. The application of the principle of conservation of energy to frictionless laminar flow leads to a very useful relation between
54:01
pressure and flow speed in a fluid. This relation is called Bernoulli’s equation, named after"—oh, I didn’t know his first name—"Daniel Bernoulli." Oh, there are a bunch of Bernoullis, so I’m not going to get too stuck on Daniel, although I guess this is one of the Bernoullis "... who published his studies on fluid motion in his book Hydrodynamica (1738)." "Consider an incompressible fluid flowing through a pipe that has a varying diameter and height as
54:36
shown in the figure below. We’ll look at that in a bit. Subscripts 1 and 2 in the figure denote two locations along the pipe and illustrate the relationship between the areas of cross-sections A, the speed of flow v, the height from ground y, and the pressure P at each point." So we have one, two, three, four variables that get subscripts because there are two different locations we’ll be looking at. "We assume here that the density at the two points is the same. Therefore density
55:10
is denoted by"—this is not P; look how different this letter looks from the P above; you should not be confusing these two—this is the letter rho (ρ), a Greek letter that sounds like “r.” I usually write it with a single stroke ρ, "without any subscripts. So density is constant throughout the fluid because we assume it’s incompressible." "And since the fluid is incompressible, the shaded volumes must be equal." Volumes of fluid, I think
55:47
that’s what they’re talking about. Yeah, I think I see what their reasoning is. What they must be thinking is: if it’s like air, the geometric volumes being the same—okay. But they might be drawing the distinction that, if the fluid is compressible, then given the same quantity of fluid its volume could change. Maybe. Again, I think these nuances don’t matter too much. Let me zoom out a little bit to show the whole thing. Yeah. So it’s marking one position as
56:23
subscript 1, another position as subscript 2, and all the dynamical quantities—they’ve given them a subscript to distinguish between quantities over here and the quantities over here. Okay. "We also assume that there are no viscous forces in the fluid." Yeah, this is an important part. "So the energy of any part of the fluid will be conserved." It’s important, both for the derivation and for observation, that Bernoulli’s equation isn’t always true.
56:54
With water you can see significant differences, and that’s because water does have viscous forces. "To derive a Bernoulli equation we first calculate the work that was done on the fluid." So, they have this quantity they are labeling with dW—d means infinitesimal, or very small amount. So, very small amount of work done. They are saying that’s the force on this fluid element F times—right, force times displacement—dx₁.
57:35
So that’s how much work you would do on this if you push it for that distance. And they’re subtracting—this is the situation they have in mind in the rest of the derivation. Think of your portion of the fluid that’s here. You can think of this fluid as doing this movement over some infinitesimal amount of time. This fluid starts out containing this element here; after some infinitesimal amount of time, the whole fluid has moved, and the way it’s moved
58:26
can be characterized this way: this portion has come off, and this portion has been added.
58:38
Now you might object: what about this—the fluid that entered from here into here? That’s not the same fluid that came out this end and into here, right? It’s not the same fluid. But when we are considering the change of energy of this portion of the fluid, whether the fluid that came out at this end is the same fluid that came in at this end doesn’t matter. So we can do that switcheroo without affecting any mathematical expressions.
59:17
So in this first equation that they write down, what they are looking at is: for this element that moves in, there’s going to be a positive work being done by this force as it moves this distance to enter this portion of the volume—that’s positive work. And as this portion squirts out on this end, this force is going to be doing negative work because it’s pushing against the direction of motion. With that negative work done, it’s going to take out some energy, and
59:51
that’s why there’s a minus sign there to indicate all that. Okay, that expression makes sense to me.
1:00:00
When you look at mathematical equations, you’ve seen me do both. In the past two sessions you’ve seen me basically skip over equations because it made my head hurt—sometimes I do that. What I will tell you is that when it’s a derivation and you are seeking to understand the derivation, you should do this exact thing I’m doing: take enough time with each equation and make sure that you
1:00:24
understand it before you move on to the next. It looks like I’m doing different things at different times, and that’s because, yes, I am—there’s an element of judgment. If you’re not sure, the easy thing to do by default is skim through it first, because the things that were mysterious, maybe the second time you see them they’ll be less mysterious. If you’re not sure what to do, skim through it first; you can always come back later. That’s the advantage of written material.
1:00:58
Okay, this equation makes sense, so I’m moving on to the second line. The infinitesimal amount of work—okay, this looks similar to that. So I will bet they are expanding out F₁—yeah, F₁ is pressure times area; they have done that there. And here F₂ is pressure times area; they have done that there. They are taking this expression—cross-sectional area times the displacement—and expressing that as
1:01:26
a volume element dV, and as they are expressing the volume element they drop the subscript; instead of saying dV₁ and dV₂ they’ve said dV, because they’ve constructed this situation so that these two have the same amount of fluid. That’s how I was describing: some fluid comes in here; the same amount of fluid goes out here. So, okay. The second equality makes sense. And now that they’ve gotten dV in the same form,
1:02:00
they’ve factored it out. So it’s pressure at one point here minus pressure at the other point here. Okay, let’s keep going. That line makes sense: dW is equal to difference in pressure times the volume element. It’s quite an elegant derivation, and I think that’s the part that I keep messing up when I’m trying to do this on my own. “The work done was due to the conservative force of gravity…” Wait, what? Let me keep reading.
1:02:37
“…work done was due to the conservative force of gravity and the change in the kinetic energy of the fluid.” All right, I’m going to keep reading. "The change in the kinetic energy of the fluid is equal to ..." this. Okay—difference in kinetic energy for this infinitesimal amount the fluid is moving is ½… So they’re taking the amount of mass here—so they’re doing the same thing they did when they were setting up the equation.
1:03:17
Taking this amount of fluid that you start out with, they are imagining adding this amount; that’s the part of the fluid that squirts out. So plus ½ m₂ v₂², minus—and they are imagining taking out this portion of the fluid, the part that comes into this common shared portion—so minus ½
1:03:45
m₁ v₁². And because the density is constant and the volume is the same, these two masses are actually the same amount. So they factor it out as ρ dV. I think this expression for change of kinetic energy makes sense to me. "The change in potential energy is ..." Again a small amount of potential energy change, and they’re doing the same thing here. There’s a portion that squirts out; there’s a potential energy here that’s associated with that height—that’s m g y₂—minus the
1:04:26
potential energy associated with this amount subtracted, since that’s in the common shared part, m g y₁. Express mass in terms of ρ dV and express g—same constant. And I guess if they were being consistent this should have been m₁ and m₂, but it’s fine; it’s the same quantity anyway—sorry, m₂ and m₁ is what it should be, but again, it doesn’t matter here. Okay, so with these two equations I’m fine, and ... "The energy equation then
1:05:02
becomes—" They say this—okay, I agree: the infinitesimal amount of work done is numerically equal to the infinitesimal change in kinetic energy and the infinitesimal
1:05:19
change in potential energy. What I have an issue with—and that’s the reason I was pausing—is the kind of causality assigned here. “Work done was due to the conservative force”—that’s good; I don’t have an issue with that. “Change of kinetic energy” doesn’t cause the work to be done; it’s the other way around—work done causes the change of kinetic energy. But the nice thing about mathematical equations is that equations don’t care about causality. It’s equal on both sides;
1:05:51
you can reverse them—it’s fine. So this equation is valid; no problem with that. Now that we have this, we can use the three pieces derived above: plug in the expression for infinitesimal amount of work; expression for infinitesimal change in kinetic energy; infinitesimal change in potential energy. You see a lot of things that cancel out: dV—the common volume element—cancels out, and… oh,
1:06:19
I guess that’s it. So the difference in pressure is equal to—and this is really the reason for introducing density in this context—because if you just kept this as mass, that’s the simplest form it would remain, but by expressing mass as density times volume element, you
1:06:41
can cancel out the volume element and have something in terms of a constant quantity. So when you rearrange this equation, that gives you Bernoulli’s equation,
1:06:55
and that’s the derivation you can think of. This form of equation is nice because I hope this reminds you of a conservation law. You have some quantities measured at one point that are equal to the same combination of quantities—pressure, this kinetic-energy-related quantity, this potential-energy-related quantity—then measured at another location, and it’s showing some form of conservation. That’s because, yeah, it comes from conservation of energy.
1:07:30
So, yeah, that’s the derivation, and I think this is definitely shorter than my other longer, meandering lectures. I’ll replace that one with this. And, yeah, the relation states… and you can read through the rest. “Special note must be made here: pressures at the same height in different parts of the fluid may be different…”—yeah, if they have different speeds of flow. So like here you can imagine y₁ being equal to y₂, but if the size of the tube is different, then speed
1:08:04
will change and that will cause the pressure to change. Yeah. And they have some examples, which I encourage you to look through. I won’t right now because I want to do the other thing. Yeah—entrainment. That’s fun. You might have seen a demo of… I believe there are two practical things. There’s a Dyson bladeless fan—I think they use some sort of fluid-dynamics phenomenon (I don’t know if they call it entrainment; might be—not sure). And,
1:08:52
yeah, they give you other examples too. So just let that be, I think.
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