Navier-Stokes Equations – Numberphile

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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