The chapter builds the pressure and external-force terms of the fluid equations, explaining how it is the pressure gradient (not absolute pressure) that produces a force, and how gravity or stirring add further force terms, arriving at Euler's equation of motion for an inviscid, frictionless fluid. It then traces the history behind that equation: Jean le Rond d'Alembert introduced the velocity field idea and showed it had to be divergence-free while working on a prize problem about drag, and Leonhard Euler, reviewing that submission, turned it into a Newton's-second-law equation for fluid motion. The chapter closes on d'Alembert's paradox, where applying Euler's equations to a sphere moving through fluid predicts zero drag, a result that contradicts real-world experience and points to physics the equations are missing.
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The original prize problem, set by the Prussian Academy of Sciences in the 1740s, asked how much resistance a body experiences moving through air or water, a question tied to contemporary interest in cannon and warfare.
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D'Alembert's prize submission was not awarded because the academy judged it too theoretical, even though it introduced the velocity field concept later used by Euler.
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The hosts compare this Euler-d'Alembert history to Euler's zeta function arising from the earlier Basel problem, later extended by Riemann into the Riemann zeta function.
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D'Alembert's paradox specifically shows that Euler's equations predict streamlines closing symmetrically around a sphere with no front-to-back pressure difference, hence no drag, unlike real fluids.
2,416 words · auto-generated from the episode video
1:02:08force? One obvious thing is pressure. Pressure is force per unit area. you know, higher pressure means that there's more sort of forces acting on um acting on your container. Like, you know, if if if you've got a um if if you've got a high-pressured gas inside of a container, that gas is pushing out on the walls of the container compared to a low pressure gas, it's not pushing out as much. >> Just hanging out. >> Yeah. Exactly. Like when you suck on a straw, what you're really doing is creating low pressure in your mouth and then the high pressure of the atmosphere pushes down on the liquid and forces it up into your mouth. That's a difference in pressure that you're doing. So crucially, what matters here? What is
1:02:48the force >> on the fluid? >> It matters what the difference in pressure is, not what the absolute pressure is. >> Right? There's absolute high like a,000 um 100,000 lbs per square inch right here >> um in this room. But I don't feel it because there's no pressure gradient. There's no difference in pressure, right? So I don't feel a force. Now all of a sudden if um you know outside there was super low pressure for some reason and we had like a a leaf blower in here um there would be high pressure in the room. Low pressure outside and we'd feel a wind that's going out. In weather systems you see this a lot right? High pressure the wind goes away goes away from the high pressure towards the low
1:03:28pressure. So one of the forces that we will have to worry about is the gradient in the pressure. How the pressure changes from one space to another. And it's negative because it's going from high to low. So the force is always from high to low, >> right? Right. Because it's going from a high it's always flowing from a an environment of high pressure to an environment of low pressure. So you can always have the negative in front of it because it's always going to be that way. >> Yeah. It's always going to go from high to low pressure. And so this is a force, right? The whole point right now is we're trying to describe what are the forces on the fluid. In this case, the forces on that velocity field are from negative to from from high pressure to low pressure. You're going to have velocities flowing, right? It's like if you're at a beach and you know those
1:04:10little videos where the surfers create >> Oh yeah. the the surfing thing on the from a beach into the ocean. Yeah. And it it's it's like >> the the water pressure is very high and built up at the top and then it wants to flow into the ocean which is a relative area of lower pressure in that context. >> In that context actually you're the next one that I want to talk about is the external forces on my fluid and in that case that would be gravity. >> Okay. >> Right. because the the um like the lagoon or whatever is slightly above the ocean, right? And so when you create that little tiny thing, gravity is going to force the fluid down that little canyon that you've built. >> So my example was actually it was not
1:04:50perfectly at apt because I was not yet accounting for gravity. >> Yes. And the next thing that we're going to do is account for external forces on the fluid. >> Okay. >> Like gravity, right? So on the right hand side we've got forces. >> Yes. the difference in pressure is one such force that is going to you know like if I were to increase the pressure here and decrease the pressure here and I started out with some velocity field those velocities those vectors are going to get bigger >> because there's going to be more fluid going through and the fluid is going to be moving faster similarly if I introduce gravity now the velocity is going to get bigger so that's a kind of force it's causing an acceleration right
1:05:30and so here we've already got somewhat of a Navier Stokes equation >> okay Right. Okay. We've got um a familiar example is gravity, but honestly that G could be a lot of things. >> That G could be um the stirring force, >> right? Like if I put a spoon in a piece of in a in a glass and I stir the thing, that's an external force that's acting on my fluid. >> So the the force can be a lot of things. >> Understood. It's not necessarily just gravity in the context we just talked about it. Mhm. Yeah. We just talked about gravity in that context, but I mean forces on fluids can be a lot of things. It can be like the the turbine in an airplane >> that's putting a force on, >> right? Because the thing is it's
1:06:11stirring a fluid and making it shoot out in one direction. >> Yeah. Yeah. Yeah. >> Um so >> that is about as far as one of our favorite mathematicians on this podcast, Leonard Oiler got. And I think this is a good point to talk about some of the history behind the Navier Stokes equation because that last equation that we had is one of the one of the oiler equations of fluids. So, so just and just to quickly kind of recap, >> we've started we're trying to describe um >> the acceleration of an individual entity. >> Mhm. >> Within a vector field
1:06:52from only having the context of the vector field >> and we are trying to describe it in the language of Newton's second law. Mhm. >> Force equals mass time mass time acceleration. >> Yeah. >> And we've built up now for an incompressible fluid. >> Yeah. >> All the way up to where Oiler >> Mhm. >> arrived in in this journey of trying to do so. >> Yes. He didn't get there on his own though. >> Okay. Oh, okay. Oh, >> okay. We had no drama till the end, but >> and and there is some drama here. He he there's actually some really weird drama that is very pertinent to today's drama of OpenAI versus all these mathematicians. So I thought it was it was kind of interesting to talk about.
1:07:33So Leonard Oiler, you know, the great Leonard Oiler, we've talked about him a lot in the Remon Zeta episode. We talked about how Leonard Oiler invented the Zeta function that then Reman later extended to create um the Remon Zeta function and the Remon hypothesis. It's a very similar story here. Oiler created the Oiler equations and then Navier and Stokes extended those equations to create the Navier Stokes equations that we know today. And obviously you want to name things after the second person who invented everything because you can't name everything after Oiler. But even Oiler inventing that first equation that wasn't on his own. Okay. There is a lot of shenanigans that happened in 1940s
1:08:14>> 1940s Berlin between >> 1740s. >> Sorry. Yeah. Seven of course. Yeah. 1740s Berlin between Leonard Oiler. He was um working at the Royal Prussia Royal Prussian Academy of Sciences. This is the German Academy of Sciences. Okay. And if you go to Berlin, you can find a little plaque for where Leonard Oiler used to live. So the German Academy of Sciences announces a prize problem about one of the great practical questions in the n in the 1740s. How much resistance does a body experience as it moves through air or water? Okay, it's a very pressing pressing question because I think around the time is when cannons are being used
1:08:56in warfare. So, it's like, you know, we want to know how far a cannon can go. It's it's their version of the DoD um creating, you know, a prize competition to try to understand how to do better warfare. Um, a guy named Jean Leon Dalm. >> Very good. For those who are longtime listeners, this is >> I usually do this, but I like Dalbe. So, um, he submits an essay. It's about fluids. He did not solve the resistance problem completely, but he helped introduce something that is arguably maybe a lot more important, which is this mathematical language of a velocity
1:09:36field to describe a fluid. He's the guy who invented that concept of u of the velocity field. You know this all this stuff that we've been talking about this new language of maybe let's not worry about all of the particles. Let's worry about the velocity field of this thing. That was Dal Bear >> and Okay. So what's interesting is I want to make a quick zoomin note to note that the the prize problem that he set out to answer. >> Yeah. was about how much resistance does a body experience when it moves through air or water >> and I just want to be that's the specific question he was trying to answer >> in that journey he did not answer that
1:10:17question he did not answer the question but he invented he gave us the language he gave us the language for it right and his submission didn't win because the academy of sciences was like oh it's um it's it's too it's too theoretical he also showed that the that velocity field has to be diverted meaning that douals like the fact that it's incompressible and things like that. So he already established one of the two Navier so equations dam um he submits this thing the the academy decides that it's not going to give out the prize. Um no prize was awarded but here's the thing one of the people that was on the committee was Leonard Oiler. So Oiler saw an early submission of Dalbert's work and he's reading this
1:10:59thing going yo this velocity field >> is a kind of nice idea. He takes that idea and not long afterward he presents his own general theory of incompressible fluid motion and that's the equation that we see on the right hand side. So on the left hand side Dalambert has already invented >> the velocity field and he's shown that the velocity field has to have zero divergence meaning it's incompressible no sources or syncs. Leonard Oiler takes that concept and says well I can actually make a Newton's law version of that. M >> so very similar to remon zeta where the bassel problem existed beforehand right the one plus 1 over4 plus 1 over 9 plus 1 over 16 so on and so forth
1:11:40>> leonard oiler saw the bassel problem solved it and then presented the zeta function in this case he saw dalbert's velocity field and he's like wait this is a good idea and he just ran with it and created the derivatives which are huge like the the >> the the derivative term that acceleration is massive to create that material derivative and then also say that okay the forces on it are going to be the gradient and the pressure whatever external force >> and just say what you just spoke to that was the convective term that we talked about earlier. >> Yeah. Yeah. >> Um Yes. Okay. And this is so interesting because it's um it it's always funny how these things
1:12:24basically we have Dalam who started the >> who created the vision to think about it this way >> and then Oiler was like let me take Newton's >> framework >> Dalbear's vision >> and combine these two things >> yes and he gets an equation of motion right he gets the first equation of motion for a fluid this is for an invisid fluid as they say because these are >> as we'll find out frictionless fluids. >> Okay, now obviously that can't be everything. And Dalbert sees this um it's it's kind of like a mathematical twist that's so perfect it's got to be scripted. Dalambert looks at the original competition he's like wait a minute that it was about drag. If I
1:13:06apply Newton's I mean sorry if I apply Oilers's equation to the concept of drag and I ask what is the drag on a body? I can exactly solve Oilers's equations around let's say a ball moving through a fluid >> and there's no drag >> like in in in the next thing we'll see this is called Dalbert's paradox okay he uses Oilers's equation that's up there and he applies it to a ball a sphere that is moving through fluid >> according to Oiler's equations if you were to solve with all those boundary conditions the streamlines just go up and then they come back >> there's no difference in pressure >> from from the front to the back. And so there's no drag. But obviously, as we
1:13:46know from our FIFA World Cup episode where we talked about the drag on a soccer ball, there is a lot of drag on a soccer ball, right? >> Just as just to also note, this was Adidas's peer-reviewed study on their new balls for this year's World Cup, and they did incredible deep dive studies into the literal drag of how they designed. It's great episode. Definitely go watch it. >> Exactly. And what we can see already is in real fluids, the streamlines aren't just meeting up right after. >> Yeah. >> Right. There's a lot there's this like region where there's turbulence and all this weird stuff that's happening, >> chaos, >> right? >> Oilers's equation does not account for that. >> So basically what we're saying is Oilers's equations only accounts for
1:14:27what we see on the left, which is effectively >> uh a system that's not actually exists in real in real context. Exactly. The military can't use it. >> Exactly. Yeah. And so 20 years after the whole competition and everything, Dalbert Bert says, "This is a paradox. This is not everything. There's got to be more." And this is the part that Navier and Stokes >> help us repair. >> Okay? They're the ones that come in and