The chapter derives the acceleration of a single fluid particle from a velocity field, building the left-hand side of the Navier-Stokes momentum equation. Using a traffic analogy, it distinguishes the Eulerian term (how the flow pattern changes at a fixed point over time, like traffic slowing everywhere when rain starts) from the convective term (acceleration from moving through a spatially varying field, like approaching road work), showing that acceleration equals the sum of these two effects, written mathematically as the time derivative of velocity plus u times the gradient of u. Multiplying this total acceleration by fluid density gives the mass-times-acceleration side of Newton's second law applied to a fluid.
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The airfoil and traffic-accident examples show a velocity field can be completely static in time (zero Eulerian term) while individual particles or cars still accelerate, proving the time derivative alone is insufficient.
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The convective term is illustrated by comparing two cars approaching road work at 100 m/s versus 50 m/s: the faster car must brake harder, showing acceleration depends on both the spatial gradient of velocity and the particle's own speed.
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The two perspectives are explicitly named: the Eulerian view (the aggregate, chopper's-eye view of the field) and the Lagrangian view (the individual car or particle's experience).
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The convective term is written as u times the del (nabla) operator acting on u, i.e. u·∇u.
3,457 words · auto-generated from the episode video
34:34important distinction. Okay. So this is helpful because that's one half of what people say when they say uh the Navier Stokes equations >> and it's going to be the basis by which I'm guessing we're now going to build an understanding of the second >> of the second one. Right. We've we've sort of gotten in intuitive understanding of what types of velocity fields are allowed >> first of all. Right. and kind of what the velocity field represents, which is a bunch of these arrows that tells you how the velocity is at that moment in space and time >> in a fluid. >> In a fluid, right? In an incompressible fluid. >> In an in an incompressible fluid. >> Yeah. Okay. So now let's try to apply this velocity field to Newton's laws.
35:16Okay. Or the other way actually to we want to apply Newton's laws to this velocity field. Now that's easier said than done. Um the challenge now is how do we talk about Newton's laws which has to do with individual particles in the fluid right it's like it's like this water molecule is getting forces from other water molecules so it's moving around Newton's laws has to do with the stuff it doesn't have to do with the velocity field >> so we have to now >> put Newton's laws in the language that we've just invented >> and just to take a quick step back just for for for my brain part of like it's like you know >> if you have a bathtub with a little boat toy boat in it for a kid, right?
35:56>> We talked about Newton's laws in the con context of like the boat in the water. >> Yeah. >> And then when we talked about velocity fields, it's like the medium of the water. >> Yes. The water itself. >> It's the water itself. And these are like different. So it's like the discrete object versus the medium in which an object or many objects may exist in. And the lens in which we look at them is >> from different doorways. >> Yes. In in the first one, it could easily I could easily just be like, "What is the position of the boat?" Oh, it's like right there, >> right? >> What is the position of the water? What what do you mean? Which water? >> Yes, >> there's there's um an Avagadro's number of water molecules that we're talking about, right? So, instead, we have to reinvent this new language of the velocity field. And now, what we're
36:37going to do is build this step by step. We're going to apply Newton's second law, which is force equals mass time acceleration. We're going to apply that to our velocity field. The first thing we're going to do is try and describe what is the acceleration of a fluid particle. >> Okay. >> Okay. Like in this velocity field. >> Okay. >> Okay. >> Okay. >> And um we're going to we're going to use traffic as an analogy. >> If you live in LA, you uh are very much are intuitively familiar. >> Yes. And you're probably listening to this while you're while you're in traffic. >> Okay. So traffic is kind of a nice analogy for fluids because you've got these working um agents in traffic, i.e.
37:21the cars, right? The cars can be the individual fluid particles and the highway is like the receptacle and the entire flow of traffic is the fluid. >> Okay. The challenge now is that we want to have a description for how the individual cars accelerate or de accelerate if they break or if they press on the pedal based on the stuff that we can see from a news helicopter because the news helicopter is up there and it's seeing the velocity field of these cars, right? It's seeing like, oh, over there cars are moving fast. Over here the cars are moving slower. And like that's how they like if you're on the radio they tell you, "Oh, the 101 is backed up from Sepulva blah blah blah
38:02blah blah, right?" Um, so they're actually watching the the cars, but they're watching an aggregate mass of cars. It's not like they're following a single car and seeing how that how how stuff is happening. So what we want to do is describe the acceleration of a single car based on what we see from from a chopper. Right? Now, first let's think about how cars can accelerate. Cars can accelerate and de accelerate as a response to the traffic itself changing at a specific spot. For example, suppose you're in the chopper and you notice that everyone is moving at like 80 m hour, right? It's a nice day and everyone's moving at 80 mph and all of a sudden it starts raining.
38:43>> Uh I was going to say 80 m hour in LA. Never happens. Never happens. >> This is an unrealistic example. >> Yeah. Yeah. Uh that's totally fair. Anyways, that's what I'm stuck with in the visuals. So, we got 80 mph and then all of a sudden it starts raining. >> Okay. And within 10 seconds, everyone slows down to 60 mph. Obviously, the cars have de accelerated, right? They've used their brakes to slow down. This 20 mph reduction happened in 10 seconds, let's say. >> And so, the car is accelerated, de accelerated at -2 mph per second. They're slowing as a response to what the traffic itself is doing. The entire
39:25traffic slowed down and so we slowed down. It's kind of like seeing like, you know, in the in the traffic map of LA, everything was green >> and then rain started happening. Everything turned yellow. >> Well, that means all of the individual cars also de accelerated. So, that's one way that you can de accelerate. And this is called the oilarian perspective because it's asking how the field itself is changing as a function of time at a given spot. >> And this is represented by the derivative of the velocity field with respect to time. >> This is makes sense, right? It's a velocity field. If I take the derivative of velocity, I get acceleration. And there you go. This is an acceleration >> of my particles, right? In this case,
40:08the thing is slowing down. But you could easily imagine after the rain ends, the things go back to 80 mph and it's going to speed up again. This is represented by a derivative of my velocity field. >> And so the idea is is here's a way to describe the velocity field as this system unique and different from the discrete individual car in traffic. >> Yeah. Yeah. And we can use the description of the velocity field and how the entire field is changing in time to describe how an individual car is experiencing acceleration. That's the key right? >> So, good, we've got at least some of acceleration down, but that can't be the whole story. Okay, that cannot be the whole story. For example, let's take a look at this air foil. This is a
40:50airplane wing where the air is coming in from the left and it's going over the wing and under the wing and then to the right. Okay. Now, crucially on the on the lower right, you're seeing the velocity field representation of this airplane wing. Okay. So the arrows above the wing, the velocity is very high because the air is like getting pushed up and then there's high pressure. So it's like >> these are the red arrows. >> Yeah, those are the red arrows above the wing. So there's higher pressure above the wing, lower pressure below the wing. And so there's slower fluid below the wing. >> Mhm. And and just the length of these arrows is meaningful here in that there there's more >> Yeah. It's faster. >> Faster. Yeah.
41:30>> Um and Okay, I'm already seeing where this is going. Okay. So, so this is the on the left is your particles that are moving, >> right? Like our flow of traffic. >> Yeah. Our flow of traffic. And on the right hand side is the velocity field. >> Mhm. >> But if I were to take a derivative of that velocity field, if I were to ask how is this velocity field changing with respect to time, the answer would be zero. >> Right? Because this is a static velocity field. If you go back to that picture that I had with the tornado, even though the the the particles were circling around >> velocity field, >> the velocity field was static because in the in this point the velocity is this way and then as you it goes around it's
42:11like coming around, right? And so so what does that mean that the particles are not moving around? >> Mhm. >> Another way of putting it with the chopper idea is like you know you have you have let's say an accident over here. Okay. on the 101 a way behind way way way ahead of the accident the um the the speed is very high but as you approach the accident it's very low right but the accident isn't going anywhere >> because the police are very far away. It just happened and so the police haven't gotten there. So the accident is not going anywhere which means that ahead of the ahead of the accident the speed is very high. Before the accident the speed
42:52is very low and that's not changing. >> The traffic pattern is not changing. It'll still be red here, green here. If I come back 5 minutes later because of how slow the response is, it'll still be red here and green here. So the traffic pattern has not changed in time. And yet, I think you would be um remiss to say, yeah, the the cars are accelerating once you get past the the accident and you're done rubbernecking. Yeah. You're like, "Okay, I got to get on with my life. Like, let's go." Right. >> Okay. A and so I just want to make sure that I'm I'm following this here because what part of what we're trying to say is >> we're trying to go from the velocity field >> and then say can we describe an
43:34individual object within that velocity field >> and how it accelerates >> and how it accelerates based on the the mapping we see in the velocity field. Exactly. And part of what you're pointing out here is there are contexts like the air foil or the traffic accident example you just described where the velocity field equals zero. >> And so that means you can't >> the the derivative the velocity field is not changing. >> Sorry. Right. So the right the velocity field it's static. >> Yeah. It's not changing which means you can you in some cases you may not be able to accurately describe certain particles within that context because >> yeah using just the time derivative of the velocity field you have to figure
44:15out something else about the velocity field is doing this and let's just think about this I think we've got in the next one you know the highway slowdown let's say there's a let's say there's um >> next one >> in the next one exactly so let's say there's road work ahead right >> um a car is going to slow down. >> Mhm. >> Now, all of the cars near the road work are going to be going slow and the cars before the road work are going to be going fast. The road work doesn't change where it is. >> But what is making the car slow down? It's the fact that it is approaching the road work. There is a spatial difference >> in the velocity field. >> You see, there's the time difference that we talked about earlier where the the rain came in. So, everyone slowed
44:56down. People slowed down here. If there's only a patch of rain here and everyone else is fine, but this patch of rain, people slowed down. Right? So that's a single spot, stuff is changing. But here I am accelerating or de accelerating because I'm moving through the traffic pattern. Right? If I go from a red to a green, I'm going to accelerate in my map. >> If I go from a green to a red, I'm going to deacelerate. Even though where the green and red are are staying exactly the same, the fact that I'm moving through it is making me accelerate or de accelerate. >> And so I think part of what you're trying to get at here is that the time derivative of the velocity field is not
45:36sufficient to describe the acceleration of an individual object. >> Yeah, it's part of the story. >> It's part of But it seems like the distance >> is potentially the other ingredients in the pie at least based on the example that you just brought. >> Exactly. Yeah. And that's a big part of it. It's the spatial difference. It's it's how the velocity field changes as I move from one space to another. Now, >> in that case, what I showed you is like, okay, you might think, well, um, what if, you know, I took a time derivative before. What if I take a spatial derivative of the velocity field? So, that tells me how much the velocity is changing from one spot to another, right? If I move here and then I move like um 100 ft over there, >> if I've slowed down, that tells me how
46:18much I should slow down. M >> that's not everything. >> Okay. >> Okay. There is a secondary effect. >> Okay. >> And to to understand that, let's let's look at that scenario again where you've got road work ahead. >> Okay. >> But now I've got two different cars that are approaching. The first car approaches that road work at 100 m/s. >> Okay. >> Okay. And the second car approaches that road work at exactly half the speed, 50 m/s. I think you'll agree that the guy that is moving faster is going to have to break harder. >> Yeah. >> Right. Because he's going to he's going to come upon that road work way faster than the guy who's slowing, >> right? He's going to have to be like,
46:58"Oh, there's road work. >> You know, fines are doubled. I better slow down a lot." Yep. >> So, there's a secondary effect. It has to do not just with how fast the traffic pattern is changing but also how fast the individual car is moving. >> Mhm. >> Yes. >> Yes. Because so and it again trying to use the terminology again. We talked about the time derivative matters. We talked about uh the the spatial matters and also the individual objects acceler uh velocity velocity speed itself >> is the third uh item in our uh uh cookie
47:39recipe here. Like all of these three things are relevant >> in order to be able to go from >> looking at a velocity field and describing an individual object within that velocity field's acceleration. >> Acceleration. Exactly. And so for the second scenario, we have to take into account both the velocity and we have to take into account >> the change in velocity, >> right? >> Okay. The change in velocity as a function of distance is like how how far away is the road work, >> right? >> And my own velocity as I approach that also has something to do with how far how fast I need to break. >> So it's going to be a multiplication of the velocity field.
48:19>> Yeah. >> Times the spatial derivative. That's the under upside down triangle. It's called nambla. I call it dell. But it's u dodll time u. >> Okay. >> This is called a convective term. And this what this convective term is doing now, it's giving us in the example we just talked about here. It's giving us the 100 m/s, 80 m second, right? That okay. multiplied by the delu which is the spatial how fast how how how much time do I have before the road work I mean how much distance do I have >> before the road work gets there right how much time do I have is dependent on how fast I was getting there >> yeah yeah yeah okay and that's okay
49:01>> make sense >> yes got >> okay and both of these terms combined give us our acceleration >> we've got the oilarian perspective which is how is the traffic pattern itself changing for example >> is it going from green to red if rain came down or something or if there was an accident, right? Then at a specific spot that highway would go from green to red on our Google maps >> and this >> and that's the first term >> that that's our velocity the derivative of the velocity field >> with yeah with respect to time right the veloc the field itself is changing the time >> the color is changing on the on the Google maps right >> right on the right hand side the Google maps color is staying the same but I am moving through
49:41>> that portion of the yellow or the red >> and so I'm going to be breaking or accelerating based on that >> and a more and effectively a granular way. There's like a bird's eye and then there's a like a a a zoom. >> Yes. And both of those are matter. And that's why it's called actually the oilerian perspective because that's kind of the bird's eye view. And then the lrangeian perspective is like the the nitty-gritty what's happening there. And that's the conceptual trick. Okay. And >> in in in the Navier Stokes equations, we're writing Newton's laws for the person in the car for the car itself based on the data that's collected by the chopper. The chopper is only >> finding you. The chopper only knows the the green and the red and and everything
50:22like that. It's seeing these aggregate >> traffic patterns. But from those aggregate traffic patterns, I could tell you if a car is over there, how fast is it going to have to accelerate or decelerate? >> Mhm. Mhm. >> Does that make sense? >> Yeah. We're trying to um again describe the acceleration or deceleration of an individual thing in an incompressible fluid >> from a vantage point that is only the vector field. >> Yes. >> Which which um the point being the vector field in and of itself does not have enough to describe the acceleration of the individual object until you >> until you do all of this. Until you do
51:03all of this. >> Yeah. Uh, is that the right way to say that? >> I think that is the right way to say it. Yeah. Exactly. And so now we have an acceleration. Right. >> Right. And all we need for that other part of Newton's laws, right, is F= MA. We already have the A, the acceleration. What is the mass? Well, the mass is just for a fluid, it's like the density. >> Yeah. >> Right. It's just um I could do a F= ma per volume >> and then just say, okay, density multiplied by acceleration. That's my >> that part of Newton's law. >> That's my ma. >> Yeah, that's my ma. And that's exactly what we get. We've got the left hand side of Navier Stokes equations. Okay. The first Navier Stokes equation. >> Right. Because density multiplied by acceleration. >> Right. Right. Uh because our
51:43acceleration is everything in this in this larger bracket >> which is what we just described. >> Yeah. >> Um which is trying to describe the acceleration of an individual object from the helicopter. >> Exactly. >> Uh in Okay. Got it. So now again we're trying to go we're trying to use Newton's laws as a language to describe stuff that's happening in an incompressible fluid. >> Yes. >> Um and we've now have half of Newton's force time mass equals acceleration for an incompressible fluid with the description that we've just gone through. >> Exactly. And now I want to before we