The convective term in the Navier-Stokes equations makes them nonlinear, because it multiplies the velocity field by its own derivative rather than treating each contribution separately. This is contrasted with Maxwell's equations, where divergence is linear so separate electric fields (like different radio or wifi signals) can simply be added together without interacting. In fluids, combining two velocity fields produces cross terms that make them interact rather than pass through each other, which is why merging flows produce mixing, rapids, and whirlpools instead of just superimposing.
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The traffic example is extended to show two separate contributions to acceleration: the spatial change in velocity across a region and how fast the fluid was already moving as it enters that region, and their combination is what produces nonlinear cross terms.
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A real-world example given is the confluence of the Bhagirathi and Alaknanda rivers at Devprayag in the Himalayas, where visible whirlpools form at the meeting point, illustrating the nonlinear crashing of merging flows.
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52:18even start talking about the other side of that equation which is the force. I want to linger on this acceleration term a little bit more especially that convective term because that convective term you might have heard that um the Navier Stokes equations are nonlinear. >> Yes. >> Right. Compared to linear differential equations the Navier Stokes equation is a nonlinear differential equation. >> That second term the convective term there >> has everything to do with that. Okay. So let's linger on that for a little bit longer cuz I want to show you exactly what all the trouble is. And just to reiterate here, the convective term that we have on the right was when we were talking about the car in the yellow or in the red moving from the 100 meters per second to the 80 meters per second describing the acceleration in this
52:59slice of the larger >> system. Yeah. And exactly. And crucially, it had to deal with its own velocity and the spatial >> difference in the velocity. Right. That's why it's nonlinear because there's two there's a compounding effect that is happening. Okay, let's let's take a step back and let's look at for example Maxwell's equations in a vacuum. Okay, these are the equations of electricity and magnetism. They're described for example Gaus's law for electric fields is that um the divergence of the electric field is equal to how much charge there is. You know how before I was telling you that like fluids are divergence less the divergence is zero. But I showed you those two examples where the vector
53:40field was going out or the vector field was coming in. And that's because you can have negative charge and positive charge, right? That's Gaus's law for electricity. Okay? It just says that del E the divergence of the electric field equals to some charge. Now, crucially, the divergence of stuff is a linear operator. Meaning, if I've got one electric field and another electric field and I want to find the divergence of the two of the electric fields put together, I can just find the divergence of one and find the divergence of the other and I can add them up separately. So I've got if I want to combine two electric fields, I just do the math for one of them. I do the math for the other one and I combine them. And that's totally fine. This is why, for example,
54:21if I've got two charges, a negative charge and a positive charge, and I want to find the electric field everywhere, all I do is find the electric field due to one, the electric field due to the other, and I add it up. >> Okay? This is also why telecommunications works with electric fields, right? Like in this room, we've got Wi-Fi, we've got Bluetooth, we've got radio, we've got um our phones, we also have the light from the studio, right? All of these are electric fields, but they're all just moving through one another. They're they're like ghosts that are just like moving through one another. And if I can tune my radio to one thing or the other, I can listen to one station or the other. Or I can like my Bluetooth isn't
55:02randomly like interfering with the Wi-Fi. Like the Wi-Fi is coming from the router. There's a Bluetooth that's probably in my phone, but it's not like the Bluetooth signal and the Wi-Fi signal is like crashing. >> Mhm. >> Right. >> Mhm. >> Yeah. They they don't interfere with each other. >> Yeah. Yeah. Even like um in a radio, right, in in the car, like if I'm tuned to 91.5 KUSC classical radio, um the car is being inundated with all of the radio stations. There's also 89.9 coming at me and Kiss FM >> and also the AM, you know, talk radio. >> Not sponsored. We need our check. Yeah, these these are just some of my favorites that I listen to in LA. But like
55:43>> my point is the car is being inundated with all of these radio stations, but my radio can tune in on into a specific band. And it's not like the KISS FM radio signal is interfering with the KUSC signal is interfering with Rush Limba or I he's not. >> I think he's still on radio. >> Yeah, he's still on radio. >> We I think he's still on radio. He might be on YouTube now but or a satellite. But I think so part of what you're saying is like I don't have to account for the other radio stations. >> Exactly. >> It doesn't matter what And if I want to account for all I do is add it, >> right? If I want to know what is the electric field here, all I do is add up all of the contributions. It's totally fine. Mhm. >> Um in fluids we do not have that luxury. >> Okay. And that is because of that
56:25convective term. >> Mhm. >> That convective term is nonlinear. And this is what it does. If I add up two velocity fields that are separate suppose, you know, and before I was I was adding up E1 and E2, the electric fields. >> Yes, >> it's totally fine. >> Fine. >> Suppose I have two velocity fields, right? I got two hoses and I'm like crossing streams or something. U1 plus U2. Well, that's I shouldn't have, you know, whatever. like two two things of water uh and I and I and I add up, right? Like U1 and U2. >> Suppose I I go through the math with those. >> Okay. >> If it was linear, I would only get the terms in the green. >> Mhm. >> Right. Because I could just do the math for one U1 and I could do the math for U2 and I'd be fine.
57:06>> Mhm. You do the divergence for U1. >> Yeah. Divergence for U2 multiply by U2 and U1 and that'd be it. Okay. But because it's u multiplied by the divergence of itself, >> right? Or it's multiplied by the gradient I should say of itself. Not divergence because divergence has a dot. Anyways, the the point is I'm taking the the velocity field itself and I'm multiplying by the derivative of itself. >> Because I'm doing that, I have to do the distributive property. >> You have to do it twice. >> Yeah, I have to do it twice. And so I'm going to get these cross terms on the right in the red circle. >> Okay. And those cross terms are what make this whole thing nonlinear. And that gives us all of the richness of fluids. This is why fluids
57:48don't just go through one another. >> They crash. There's rapids, right? There's when when two rivers meet together. They they mix and they they they have turbulent flow. Mhm. >> You know um when I when I was in India last time um we visited D praag in the Himalayas which is the birthplace of the Ganga where like you you have like Bhagirati coming from one side and Alakananda coming from the other side. There are two rivers from the Himalayas that meet in this very holy place in the Himalayas that begins the Ganges river. Um and you could you could see the whirlpools forming like right at that spot. It was it was amazing. And and the the reason for the whirlpools and the reason for all of the chaos in the birthplace of the Ganga is because of
58:29those two terms on the right hand side. >> You know what I mean? >> Yeah. And so now you can already see that there is trouble with this equation, >> right? And I I just want to take a quick step back to make sure I'm internalizing what we're saying here. um when we talked about Maxwell's equations >> uh and we have you know a positive and a negative charge and we can we can just add them up. It's because part of what we're saying is is it's because we don't have that extra level of complexity when we talked about the convective term that's true for fluids. >> Yeah. Yeah. In the Maxwell's equation it's just d right. It's a single E. >> Right.
59:09>> Here there's two versions of the vector field. There's the vector field itself. Mhm. >> multiplied by the derivative of the vector field >> vector field. >> Right. And can we can we just briefly try to connect it back? >> I'm trying to connect it back to our yellow red traffic example just to make sure it lands in my brain because I'm I'm I'm having a hard time finding the leap from I I understood the convective term. >> Yeah. Um we have this idea that the traffic is red in one part, yellow in one part and we can look at it from the helicopter and we can look at the whole vector field. >> Yeah. And if a car is going from red to yellow or from red to green then it is going to accelerate even though red and
59:51green have stayed put. Mhm. And so would it like from that perspective, would it be like the equivalent of saying if that was it and that was the only level of complexity? It would be similar to Maxwell's equations and that if all I had to do was take a take a spatial derivative, >> right, >> of red and green and be like, oh, there's red here and green here and they're 100 m apart. So, you know, the difference in the velocity is divided by 100 m. That tells me the spatial derivative of my velocity and I'm fine. That's fine. But crucially, it also has to do with how fast or how slow I was going into the thing, >> right? If I'm transitioning from red to green and I'm moving at a snail's pace of like 5 mph and all of a sudden now I have to get up to highway speed, I got to floor it.
1:00:31>> But if in the red I was already going at 30 >> and then now I got to get to 60, I don't have to floor it as hard. >> So there's two contributions, right? And that's that's what's giving us this nonlinearity. Yes. >> Is because >> fact that there's two contributions that we have to account for >> as we look at trying to do the mathematics for this. And that's where we're getting this is where we're getting having to do it the we have the green but we have to have the addition of the red because of that second >> because because of the fact that yeah there's two of them. So I have to do these cross terms. >> Understood. When I combine velocity fields when I combine let's say a flow in this direction and a flow in this direction I can't they don't just ghost
1:01:11past one another >> because they're interacting. >> Yeah. Yep. Okay. >> Make sense? >> Yes. Tracking. >> Yeah. And so that's why the Navier Stokes are so hard. >> Mhm. >> That's one of the reasons. Okay. Um so now let's get back to our Navier Stokes equations. Finally, we can now start asking what is the force that is acting on the fluid. So far we figured out how to describe the acceleration >> and we've figured out how to describe the mass, >> right? Um yes, >> but in order to have an equation of motion, we have to describe the forces that cause this acceleration. And then if we write it out then perhaps we can solve it >> right in the same way that for the
1:01:51spring we could solve mass times the acceleration. In that case the acceleration was super easy. It's just the second derivative of the position and then the force was kx. I had an equation of motion that I could solve. In this case already the acceleration is is quite difficult. >> Right? But now let's ask what is the