Navier-Stokes Equations – Numberphile

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Navier-Stokes Equations – Numberphile

Navier-Stokes Equations – Numberphile

dude this is a first for number five on the ribs as well the pain of the rib steak that’s I’m gonna say this is a complete commitment to two Navia steaks and fluid mechanics why did you do that well what are the navier-stokes equations the first one grad dot u equals naught and the second one Rho dou u by DT equals minus grad P plus mu grad square u Rho like those mathematical equations perhaps they look quite scary but they’re actually universal laws of physics so this is why these guys model every single fluid we have on earth and I mean every fluid put a fluid in your head these guys tell you how it moves how it behaves what’s going on with that fluid and so a fluid I tend to think of it as something which changes shape to match the container that it is in of course a liquid but also a gas even some solids so like ice do you think about a glacier flowing down a mountain you watch those time-lapse videos kinda looks like a river so ice in that sense behaves like a fluid it is changing its shape to fit the valley the glacial Valley which is flowing it so the first one this is literally just saying that mass is conserved so this is just saying I have some blob of fluid it moves around you know with some velocity and maybe changes shape I’m not adding anything I’m not removing anything I would want the same mass of fluid to still be there mass is conserved it’s a pretty standard law of physics makes a lot of sense this guy is just Newton’s second law so this is just telling us that mass times acceleration is force the top one tends to be the mass equation conservation of mass or the incompressibility equation and the second one is the momentum equation or the small one in the big one quite like that let’s start with the little one so you here this is our velocity so this is just speed with a direction its vector you might say U is equal to UV W so we’ve got a component in the X Direction a component in the Y Direction a component in the Z direction how fast the water in the river is flowing how fast the air around a Formula One car is going thinking about aerodynamics how fast the custard is flowing off your spoon any sort of movement of the fluid the the speed and that motion is going to be encapsulated by the velocity that’s the key thing about its motion we’ve then got this other symbol here nabla we do love our Greek letters in maths so the symbol nabla is is telling us what to do to our velocity u so nabla is is a gradient it’s a derivative it’s telling us to differentiate our vector u in a particular way and the way it tells us to do that is we have our three components uvw and what we’re going to do is we’re just going to do do u by DX so differentiate the first bit with respect to its coordinate which is X it’s the first coordinate the x coordinate differentiate respect to ax then we’re going to add derivative of the second one with respect to our second coordinate dy and hopefully you’ve figured out the pattern we’re then gonna add on DW by DZ so this is the divergence of our velocity it’s just three derivatives so it’s saying how does the X component of my velocity U how does that change as I move in the X direction then how does V my Y component change in the Y direction and how does the third component W in the Z direction how does that change in the Z direction so this is equal to zero equation number one this just tells us mass is conserved now the second one the big boy so Newton’s second law in disguise so we’re expecting force equals mass times acceleration here we’ve got you our velocity and when we take a time derivative of velocity that’s exactly what acceleration is you’re going at a speed you increase your speed you accelerated your speed has changed with respect to time or you decrease your speed you decelerate it so that’s what the first term is going to describe and then we also need the mass you think of mass in this situation to be a density so it’s how you know fresh water has a smaller density than saltwater so saltwater is heavier in that sense but it’s you sort of mass and density are the same thing when it comes to fluids that’s how you you sort of work with those things so ro this Greek letter here that is going to be our density so this is our mass in that situation so this is mass times acceleration that’s our new to second law of the left-hand side and then all of this sort of stuff going on over here these are just all the forces so what we’ve got we have the first two terms these guys are what we call the internal forces so this is the force between all of those fluid particles hitting into each other crashing sliding grinding past one another there’s internal force there and then the third one it’s just capital F this is a bit of a cheat because we just say this is our external force F so this could be gravity is the standard thing you would normally you just replace F with G call it gravity that’s your external force in most situations and if you want to go really fancy you can put in electromagnetism and we can sort of combine navier-stokes with Maxwell’s equations and get magneto hydrodynamics and that is how stars form and galaxies form and that is just next-level how your Stokes is hard magneto hydrodynamics like try to model the growth of the Sun as hard as it sounds internal forces versus external forces so it’s just a sum of various forces what are these individual forces so let’s look at the first one this nabla P or grad of P so this is very similar to our mass equation up here so the gradient of P our pressure gradient it’s a vector representing the change in pressure when there is a difference in pressure a change in pressure high pressure over here low pressure over here the air moves from high pressure to low pressure if there’s a gradient there’s a difference in pressure between two points that causes the fluid to move between or along that pressure gradient so that is creating a force so then the final internal force this guy so this is viscosity kind of what it’s made up so you’ve got all that you can being in layers and it’s slightly those layers slide past one another they create a friction and how strong that friction is is is the viscosity so air super thin air particles they move around they do their thing it’s all good but if you’ve got like honey sliding past other bits of honey it’s very sticky it’s very thick much larger viscosity so that’s your second internal force where’s the problem with all this this is why these equations are so great because it’s mass is conserved it’s Newton’s second law it’s all just makes complete sense there is nothing we have said here hopefully that anybody can possibly disagree with it’s just Newton’s second law move my fluid around mass is conserved and this is why these equations work so they’ve been around since the 1820s 1840s where navier-stokes both worked on them and this is why they work this is why we keep studying them but the problem is we don’t actually know if they always have solutions they can be used for almost anything you can think of involving the fluid so they could be aerodynamics of a Formula One car they could be designing new aircraft to go faster than the speed of sound this could be blood flow around the body for drug delivery maximizing the way the drug is deposited to do with the blood flow this could be pollution modeling climate modeling ocean modeling or anything involving a fluid it has to satisfy these equations so these are always your starting point but then the problem is and this is why you get the million dollars we just don’t really understand them mathematically in the sense that when you have a set of equations as a mathematician you want this that set of equations satisfy three particular properties first of all you want a solution to exist you know got equation you want to go right I already able to solve it that would be nice second of all you want a unique solution you know if you did an experiment with throwing a glass of water across the room and they did it again and it did something different no it’s like a loop-de-loop in the air it makes no sense and then the tricky bit is you want smooth solutions well-behaved solutions like I’ve made a tiny little change in how I started my and I want the result to also just have a tiny little change quantifying tiny but not to blow up to infinity that wouldn’t make sense because I’ve changed something so so small why have I got an entirely different solution we’ve also been taught that butterflies flapping their wings and cause cyclones the butterfly effects like a chain reaction is one big leads to another leads to another but in the in the sense of having an equation you input something into your equations like a function machine you input some initial condition it outputs what’s gonna happen next predicts almost like the future so you start with your fluid it has some velocity how are you some pressure some viscosity input it now your Stokes tells you how that fluid is going to move you know you can do these experiments so you know what’s gonna happen and you want the equations to give you that result if you tweak your velocity a tiny bit if you did that experiment you’d get almost the same result so you want that small change in the starting point to lead to a small change in the solution and this is what we don’t know about navier-stokes we don’t even know if a solution exists all the time so given an initial condition here’s a velocity here’s a pressure input it we don’t even know if the solutions going to come out so how come these equations are being used by climate modelers and by Formula One teams and all these things that you tell me are using navier-stokes it sounds like they’re using a really unreliable tool I wouldn’t say is unreliable because it’s based on standard laws of physics mass is conserved Newton’s second law so I think everyone’s happy that that makes complete sense we’ve got that bit right there’s no reason for that not to be correct but then the the sort of the mathematical complexity of it the tricky bit is we don’t know if a solution is always going to exist and so we kind of find ways to cheat so we might make simplifications we might make assumptions to reduce some of these terms or to remove time from the problem or you can get ways around it by making assumptions and making simplifications so that’s one way to use them and then another way is to what’s called averaging so rather than having a velocity field defined everywhere you can say well what if I just take a big circle of fluid take the average velocity in that circle and I want to know how that changes and how that behaves that we can do so that’s called averaging Reynolds averaging of the navier-stokes equations and that is the kind of stuff that we do in climate modeling because you can’t the computer power alone to model every particle in the atmosphere like it’s gonna take longer than the life of the earth to run that on current computers so you just say well I’ll just model the atmosphere as patches of let’s say ten kilometer squared bits are there as long as I know the average speed in that 10 kilometer squared I’m happy so it’s all about the averaging but mathematically it should in theory be able to be solved for each individual bit and that’s where we struggle I think the best way to figure out what you need to do to get the million-dollar prize to think about what we’ve done already so we have the equations they make complete sense with happy with them mass is conserved new to secular law great and then we also know that solutions exist and they are well behaved in two dimensions unfortunately we don’t live in a two dimensional world we have to read I mentions so if we were to ignore Zed and just have x and y are two dimensions we can do it we can show there’s always a solution it’s always well behaved it works but then when you go to three dimensions for whatever reason it’s just not working we can’t do it we have shown that weak solutions exist so these are sort of rather than being full solutions they’re similar to the averaging solutions not exactly but like some form of solution we can get we can get solutions when the initial velocity is really small we restrict to say we’re only going to move at tiny speeds to start with then we can show solutions always exist we can show solutions always exist for finite time so so up to some you know time equals 100 or something we can get solutions exist as well but we just cannot get that they exist for three dimensions for all of time for all possible initial conditions so Tom will the person who wins the Millennium prize be the person who explains why that’s the case like of course you can’t do it in three dimensions and this is why or could they possibly say yes you can do it in three dimensions silly-billy there was something you didn’t notice well what what will that person do or what are they being asked to do yeah so the the wording of the millennium problem is fantastic it just says further our understanding of the navier-stokes equations it’s it’s the most vague of statements but there are some attempts that qualify or quantify what that means it could be a case of show that a solution always exists in three dimensions for all possible initial conditions or it could be a case of as you said say well of course it’s going to blow up in three dimensions and it’s going to go to infinity and there’s we can’t expect solutions to exist so you can do it both ways the most recent progress in 2016 by Terence Tao he actually showed that for the average navier-stokes equations that you do get blow up in finite time the way he’s approached that could be a way of showing that the full 3d equations could also blow up in finite time so that would be that we don’t always expect a solution so we don’t really know which way it’s going to go I think one of the key things is about understanding turbulence so turbulence is this chaotic random motion of air particles of water think of two waves crashing into each other in the ocean that is almost as random and chaotic a situation as you can get if you do that again you’re not going to expect the same thing it’s it’s so so difficult to model and to understand and fundamentally fluids they just are turbulent not every fluid but like air water rapids that’s all it’s all turbulent and that’s sort of I think where the key problem lies because when we plug our data into our computers the computer averages things because it can’t solve for the turbulence it can’t solve for all those really small little bits and all those little interactions so it says I’m going to take this big square and average the velocity or average the length scale and that’s the way that we do it and it works but it’s not mathematically understanding the equations practically we are happy with navier-stokes we can use them the equations for basically anything we want it works it’s great it looks amazing it allows us to design all of these amazing aircraft to fly into space all of this stuff it works but it’s just from a math six point of view we just don’t have that proof it’s a classic case of maths wanting to know for certain rather than just knowing that it works in every case we can think of we don’t have that proof that it will always work or an explanation of why it might not if I gave you the dream computer that doesn’t exist and all the possible inputs you need at the start this equation would get you there it will just take a lot of power you make a good point you can import any bit of data any initial condition you want but then the computer may say oh but the solution goes to infinity the solution blows up and of course in a real situation we cannot have a water particle piece of air moving at infinite speed that just makes no physical sense but the computer is saying that’s the solution so there’s some disagreement there between the physical practicality of what can happen and what the computer is outputting and that would suggest that there’s something missing in our mathematical understanding so you telling me that you can input finite numbers into the Navi of Stokes equation like realistic numbers that would apply to something in the real world and Navia Stokes spits out impossible things like oh yeah your River is gonna be flowing at infinity miles per hour and things like exactly yeah so that there’s there’s a really the equations are wrong but but it’s conservation of mass that it’s due to the second law how can it be wrong this is the this is the sort of paradox it’s it’s everything we’ve done here makes complete sense because these these two laws mass is conserved Newton’s second law this will hold for any like this works for a solid it’s not just fluid mechanics you do the same thing with solid mechanics you get slightly different internal forces but it’s the same it’s the same set up so so there’s it doesn’t seem like that bit should be wrong you know it’s it’s conservation of mass its force its mass times seller ation there’s no reason to think that that’s incorrect so something somewhere is going wrong in the mathematical understanding there’s an example about the flow of a fluid around a right-angled corner like a canal basically a canal but we’ve got a really sharp right angle on the now you stalled navier-stokes this says that at this point this bright angle corner I have infinite velocity if I build this canal do I have an infinite velocity canal I’m going to guess I don’t so it’s it’s that there’s there’s no like everywhere else it works perfectly but at this little point it’s saying my velocity is infinite but it quite clearly isn’t that’s it in a nutshell the equations work for all possible practical uses so if you were building this canal you’d be completely happy you would know the speed everywhere except one single tiny tiny point but there there’s a repetition you’re like why is it infinity why is it infinity there I want to know and and that’s it that’s we’re obviously missing that mathematical understanding there’s some difference there between the physics of it and the equations we just do not understand it that’s kind of like how division like just simple division works and I use it every day it’s just this is one glitch where if you divide by zero it kind of goes a bit weird yeah that doesn’t matter because I never actually need to divide by zero exactly there were always ways around it there are approximately you know you know it’s a very large number so you just leave it at that so this is something that doesn’t really matter but it matters to you it doesn’t matter to any application of navier-stokes but it does matter to maths and it’s the classic case of alright if we solve it we have no practical uses but to actually figure out what’s going on here the max we’re going to uncover it’s gonna be brand new it’s gonna be stuff we’ve never looked at before thinking of the problem in entirely new ways and no doubt we will discover new maths leads to all kinds of other incredible new advances in all this stuff to do with fluids it’s just you know all of these things that we use navier-stokes for pollution modeling climate modeling blood flow aerodynamics all of these things if we really understand the equations those things are just gonna get better tell me how much you love these equations oh well they are my favorite equations and just just to emphasize the level of favoritism of these equations the full navier-stokes equations as written down in our piece of paper hang on which way around are we if we’re talking this if yeah so those should be exactly the ones I’ve written down there we go the little guy in the big guy dude this is a first for number five on the ribs as well the pain of the ribs like that’s I’m gonna say this is complete commitment to two Navia steaks and fluid mechanics why did you do that well I my PhD was studying fluid mechanics and these equations just they just model it so I spent four years of my life trying to not necessarily understand them but studying them and using them and it just felt like it felt like the right thing to get pissed what did what did the artist say when you read he had a lot of fun with it so he’s actually he’s quite an intelligent guy he’s got a load of like physics formulas and he has a portrait of Einstein tattooed on his back so he’s really into his physics and his maths himself so he was saying I’ve never seen these what are they so I sped the sort of two hours of being tattooed pretty much do what we just did and talking about navier-stokes

100 thoughts on Navier-Stokes Equations – Numberphile

  1. if the delta get too large a number, would it not be better to make them 'smaller' and relative to their neighbours. smaller sectors.

  2. of course nse is wrong! how can you get a right angle channel in a real life? plank's length is a limit for a nature while nse tries to solve it there in a perfect corner!? absurd! so this is a reason why you get a "division by zero" situation with an infinite speed! the nature is LINEAR in a microscopic scale! nse is wrong at all! moreover, there is no special relativity law relationship. a water particle can't exceed the speed of light! where is the connection between nse and E=mc²? that is why nse sucks!!! The science needs a new way to describe liquids in motion! mass conservation and momentum conservation is not enough here! a new law should also deal with special relativity laws and planck's length also!

  3. When we get infinity, like the corner of a canal, the scale at which we are calculating is too large to account for van der waals forces so the solution at that point seems infinite because the particles are close enough that the combined forces act elastic.

  4. I like how this video completely disproves all long term climate modeling. And proves the Climate Crisis to be a hoax. 👍

  5. How does the "little one" work for starts? When an blob of gas becomes a star and starts shining, E=mc² kicks in and its mass changes.

  6. The example of the corner made me think that this behaviour appears in singularities.
    This formulas are the differential expression of classical mechanics.
    But since you want somehow to put a smooth constraint on speed, that makes me think you maybe can try to modify the second equation to make it more "relativistic like".
    This way potential energy density should be costrained to rho*c^2, and speed modulus should be constrained to c as in relativistic mechanics.

  7. So if all these fundamental equations are continuous, they should describe the universe regardless of what scale you choose to approximate it by, right? So within a plack length/time tiny planets or other random stuff could exist that would just average out to some energy on our scale?

  8. Hmm… Speaking about it failing at corners… It'd be soo weird if it's just failing because it's assuming that corners are perfect edges, instead of the roughness of physical material.

    Seems like it'd be thought of already.

  9. Maybe the problems come from treating fluids not as tiny particles, but as infenitesimal pockets of space with a density. That might fail in corners…

  10. My guess: NS equations are very good, but not complete. Some component (or force?) is still missing which results into chaos (turbulence). Chaos is inherent to math (see chaos theory). So linking NS to chaotic dynamic systems might get us somewhere…

  11. This is pretty entertaining actually. We solved a simplified a version of Navier-Stokes known as Laplace's Tidal Equations and applied them to modeling the behavior of El Nino's for over the last 100 years. This is in our book Mathematical GeoEnergy and blogged on PeakOilBarrel yesterday.

  12. The underlined notation for vectors should be illegal LOL

    Imagine writing
    grad(some scalar field) rigorously for example and having to underline it

    I would slap myself

  13. One obvious physical mechanism missing from the equations is the dissipation of energy (velocity) in the form of heat. This will act to dampen the solution. Can be simulated computationally with a turbulence model (e.g. k-e, etc.), which will improve the solution, but this brings additional assumptions with it. This is the fun of non-linear equations… they can have islands of instability in the solution.

  14. There are specific circumstances involving fluid flow where the Navier-Stokes equations have analytical mathematical solutions. And example would be laminar steady state flow of a Newtonian fluid in a cylindrical pipe.
    The problems arise when the the flow becomes more complex (such as transitional or turbulent flow) and/or the fluid properties are Non-Newtonian (such as visco-elastic fluid flow, or time/temperature dependent viscosity effects, shear thinning, yield stress etc).
    In fact most of the fluid flow phenomena seen in nature or real life, do not have exact analytical solutions of the Navier-Stokes equations. In these instances, Numerical techniques such as Finite-Element analysis, are normally used to solve the Navier-Stokes equations. Complex fluid flow such as turbulence around a airplane wing or irregular shaped objects such as a stone falling through a thick starch-water mixture which is non-Newtonian do not have exact solutions to the Navier-Stokes equations in order to predict the resultant fluid flow behaviour. Perfect job for Finite-Element Analysis.
    It is more to do with a limitation in analytical mathematical techniques or tools needed to solve the Navier-Stokes equations, rather than a lack of understanding of the processes involved or an inadequacy of the Navier-Stokes equations themselves. (although a lack of thorough understanding of the physical processes involved in the myriad of fluid flow phenomena out there presents its own set of challenges and problems).
    In turbulent flow conditions, determining the pressure drop across a standard 90 degree elbow has no exact solution to the Navier Stokes equations – even when using simple Netwonian fluids such as water. This does not mean there are no alternative methods and numerical techniques available to accurately estimate this pressure drop and use it in practical or engineering applications. Finite Element Analysis using super fast computing produce astonishing results that are validated by observation and experimentation (the basis of the scientific Method)

  15. To me, it kind of makes sense that at the corner there would be an infinity. In nature there's no such thing as an infinitely sharp corner and yet mathematically that's what they're using. On the very, very small scale there are no sharp edges. Maybe it's not a problem with the equation but with the boundary conditions we're using to model real world scenarios.

  16. Chemical engineer here, so I spent plenty of personal time with the Navier-Stokes equations in Transport Phenomena my junior year. I know mathematicians love pure answers, but using these equations is all about making simplifications and assumptions and setting the right conditions to reduce them to something usable. And doing that requires making extremely smart choices. Probably the most memorable thing from that class for me was going through a famous reduction of the equations, removing insignificant terms and making various assumptions, until the equations could actually be solved analytically. This was first figured out way before computers. The elegance with which these brilliant engineers reduced these equations to a solvable form was, in my opinion, legitimately beautiful. It's similar to how the Schrodinger Equation can be solved exactly for hydrogen. These people had incredible minds.

    I would actually watch a dozen or more videos of different simplifications for the N-S equations depending on the context. It requires great ingenuity and can go off in all different directions. That's probably a little equation-y for Numberphile, which is fine, but if some other channel wants to do that, I'm all for it.

  17. Well, you can't build an exact square edge like that in the real world because the atoms it's made up of will have some uncertainty in their location so it's kind of fuzzy?

  18. Is it the case that it's tricky in 3D because you have 3 variables but only 2 equations? Over-simplification, but that makes sense to me on the general solving simultaneous equations front.

  19. btw: i think the glitch is, that conservation of mass is not satisfied, because we are not dealing with closed systems. Never!

  20. Clearly these equations must not be complete if under some conditions they give different results to those observed in the physical world. I think it probably has to do with them not accounting for both relativity and quantum mechanics. Relativity is what would stop a particle from moving at infinite speeds and since (at least in the canal example) there's only a single point in which this occurs that would translate to the quantum scale. Unfortunately we still don't understand how these sets of laws interact but it must be in a way that avoids all of those weak solutions.

  21. 2 questions:
    1. isn´t the example of the infinitely fast point inside the 90° turn already showing that the equation does NOT work ? i thought the whole video was about mathematicians being UNSURE about whether or not it works? or was it only about "having" solutions, no matter if they are wrong?

    2. how does the equation answer a specific question? as far as i can assume, it only gives me a single number as a result. based on many numbers about the effects on a fluid. now what does the number describe? the velocity of some random particle within that fluid? the temperature of the fluid? the overall surface area of the fluid ? ? ? i do not understand how the equation can be put to use.. ?

  22. nice piece – I had not heard of the 'upside down triangle' called a NABLA. I am used to calling it a DEL (operator) must be a British thing. Anyway lovely program

  23. Hello Brady, thank you for the amazing content as always.
    I have a couple questions.
    Firstly, my understanding is navier Stokes is always correct, irrespective of weather a computer is solving or you're solving it by hand(assuming there is a solution).
    Computer gives us unrealistic answers because it is solving the discretised navier Stokes equation which is only a numerical solution to navier Stokes equation which can cause divergence when solved by computer.
    Secondly, navier Stokes is non linear partial differential equation, unlike equilibrium equation solved in solid mechanics. That's the reason there is no divergence in solid mechanics whereas in CFD divergence is an issue.
    Let me know if my understanding is right.

  24. The N-S takes a general descriptions in the form of density and viscosity of the materials, which are themselves derivatives of other phenomena on different scales. Why do you expect these equations to work to every scale? It is like trying to define what every tree looks like with a general description of a forest.

  25. Wait a min, why (u*nabla)u term was in the tattoo but not in the explained equation ?
    As far as I know, this is the term which causes headache for people trying to model these equations, because it's nonlinear. Due to nonlinearity, usual infinite superposition methods (say, Fourier transform) cannot be applied. Also, nonlinearity brings chaos (here, turbulence) into the picture.

  26. Here a few questions:
    1) You say that the velocity blows up, however I'd say that this would be highly unphysical violating the conservation of energy, which here isn't even explicitly stated. When you derive the Navier-Stokes-equations from the Boltzmann-equation you get an infinite hierarchy of differential equations. However you stop this after the second order equation — the energy equation using the equation of state. I suppose, this should fix unphysical velocities?
    2) The Navier-Stokes-equations are just the classical limit of the relativistic hydrodynamical equations. If you get high velocities, densities and energy densities shouldn't you naturally switch to the relativistic equations?

  27. ca. 18:00 do you get "unphysical" results only for unnatural ("unphysical") conditions? There's no way to really build a "really sharp right angle" on a corner. The closest you can build one is with a (line of) single atom(s) at the corner – and that will be rounded down immediately when having a flow. And voila: no "really sharp" corner anymore…

  28. I get that these equations are purely mathematical and don't care about the speed of light. But Newton's 2nd Law in the form F = m dv/dt doesn't hold at arbitrarily high velocities. So if you're interpreting the second equation as the 2nd law, how can you declare infinite solutions inadmissable because they're unphysical, when the equation doesn't describe nature at very high speeds? I just don't understand what exactly is wrong with divergent solutions.

  29. Conservation of mass.. e=mc2, conservation of energy through Noether theorem into time translation symmetry.. which is broken by time crystals.. is the conservation of mass still as predictive as it should be in fluid dynamics??

  30. Three thoughts on this:

    "Clearly a solution where the velocity blows up to infinity is nonsensical" makes me immediately think of relativity. Could that possibly what's missing to ensure a well-behaved solution always exists?

    "With a perfectly right-angled channel, the velocity is infinite at the corner" – but of course we can't have a perfectly sharp corner in reality. Maybe if we could, then the velocity would blow up in reality too. Perhaps well-behaved solutions only exist for well-behaved (i.e. physically plausible) input, of which the perfect corner is not one.

    And finally, if in 3D we know that a well-behaved solution exists when the velocities at t=0 are "small", why can't we start with no velocity at t=0, and then use the external force F to manipulate velocities to a given desired state by t=k, and then consider the resulting solution as a solution for the desired velocities, but using (t-k) instead of t?

  31. So, has anyone come up with a relativistic version of the Navier-Stokes equations? Not that relativity is immune to having things blow up to infinity, though.

  32. it would be kinda cool to take some of the mathematical calculations out by pairing it with pattern recognition so you don't need to solve every equations every time just each type as another way of averaging.

  33. Navier-Stokes only supply 3 scalar equations (velocity) out of 7 unknowns needed to define a flow field (pressure, density, internal energy. Temp, and velocity components). By only using continuity and N-S he is assuming thermal effects are unimportant

  34. I wonder if the corner being infinite velocity has something to do with the fluids being incompressible and that the "container" is not (able to be) modeled to flex or for the fluid in the canal to simply jump up vertically because the part in contact with the air can do that instead of constrained moving with a perfectly flat surface

  35. The first equation is the conservation of mass for an incompressible flow, it is not one of the NS equations. There are only 3 NS equations.

  36. You claim conservation of mass and Newtons 2nd law are universal. I agree, as long as speeds are not close at all to the speed of light. Now, your computer model spits out "infinite velocity". Obviously, in such a situation, relativity is going to come into play in the physical approach.
    Even though realistic velocities are much too low to take relativistic effects into account, turbulence is such an unstable pattern that including relativity may still affect the output.

    So my question is: are there Navier-Stokes equations in which the "big"formula represents Newtons second law, but then modified to take into account relativity? Or is there a definite proof that relativity is not the cause of the discrepancy between Navier-Stokes simulations and real world observations?

  37. What if the answer could be fine by unifying Relativity and quantum mechanics ..? Would be amazing
    Great video by the way !

  38. What way of writing rho is that with the bottom end pointing left?

    It’s not like with delta, where one writes ∂ instead of δ for partial derivatives.

    There are two standard ways of writing rho: ρ and ϱ. Why invent a new one?

  39. Suspect them to find out that it's something simple and not produce new math. Like; given the formula's use an infinitesimal small points instead of atoms, one would expect them the approach zero mass in the formula's. Using any force on a zero mass particle would give it infinite speed since newtons second law does not encompasses relativity. Also forces are not instant in realty; they move at light speed, thus pulling and pushing at larger speed wil also be different from te formula.

  40. The problem is with "F=ma". Mass is not only misunderstood, but it is not formally defined in newtons physics. If we treat mass "m" as a value representing the resistance to a change in velocity rather than a static and unchanging value, the equations will always solve no matter the conditions and no matter the duration of any simulation.

  41. In that canal wouldn't the velocity be zero at the corner, requiring infinite acceleration, rather than infinite velocity?

  42. Shows air flowing around a fast moving object, while using incompressiblity assumption…. Really?
    The first equation is actually div (rho*u) = d(rho)/dt. THe second equation has the rho inside the derivatives where it appears, instead of a constant multiple. This is much much harder to solve, and for most liquids (NOT GASES) density is effectively constant, so it simplifies to the form shown in the video.

    @Numberphile When you look at equations from physics, please make sure you check the assumptions of your equations before selecting examples. It's very misleading when you do things like this

  43. For fox sake.
    I was going to do a navier stokes tatto next month as well. I've worked with it in my final papper and it's so gracefull.

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