Navier-Stokes Equations - Numberphile

[@numberphile]

Full Playlist: bit.ly/NavierPlaylist Part 1 (Navier-Stokes): kzbin.info/www/bejne/e4Olh3mZqtmfaa8 Part 2 (Reynolds Number): kzbin.info/www/bejne/raWsmYmthdeorbs Part 3 (River Water): kzbin.info/www/bejne/a56qmWOoaN92bLs

[@ssmith8828]

A mathematician named Agostino Prástaro claimed that he solved it, is it true?

[@MrMawnster]

Could that new bit of work by Martin Hairier who just won the Breakthrough prize be applied to turbulence for modeling that in these equations? Would that be the best we can get as an averaging the random turbulence? Maybe it's not totally deterministic and we're stuck with that's the best for along time kind of like the QM/QFT stuff. Lol I bet one day we find a connection there too in how those individual points behave. Some QM behavior in say modeling turbulence in fluids

[@RockBrentwood]

D00d?! Really? If you're gonna tatoo equations, at least put them back together. The Transport Laws: ∂ρ⁄∂t + ∇·(ρ𝐯) = 0 ∂(ρ𝐯)⁄∂t + ∇·(ρ𝐯𝐯 + ℙ) = ρ𝐅 ∂(½ρv² + 𝒰)⁄∂t + ∇·((½ρv² + 𝒰)𝐯 + ℙ·𝐯 + 𝐐) = ρ𝐅·𝐯 ℙ = p𝕀 - ½μ(∇𝐯 + 𝐯∇⁺) (∂/∂t + 𝐯·∇)ρ = 0 where ∇⁺ is ∇ applied to the left, and where the second to last equation is restricted to the "Navier-Stokes" stress tensor, and the last equation to incompressible fluids ... so those two equations go onto another part of the body and are impressed only with lick-on tattoo decals. Nobody's going to be solving the equations in generality by ripping them apart and restricting them to special cases. They gotta be handled all together as a single entity, not split and ripped. And you also have the transport laws for the other kinematic symmetry group Noether charges: angular momentum and "moving mass moment". Those figure prominently in fluid dynamics, too! (Think: vortices.) And dropping them totally muddies the picture. Oh .... if we solve them first, nobody's winning the prize. The bounty will be refused. Just to give you a head's up.

[@emanmoba]

@11:02 Reynolds Average N.S equations are averaged in time not space.

[@tadhgofogartaigh]

💕

[@DocteurZeuhl]

I spent my PhD working on Navier-Stokes, but instead of tattooing the damn formulas on myself, I celebrated the end of my PhD by writing them on a piece of cardboard and setting fire to it while drinking vodka. EDIT: This is my most liked comment of all time. The world is a silly place

[@michaeldamolsen]

Yes, I can see how that might be awkward to do with a tattoo.

[@Saxshoe]

"why'd you do it?" "Well......................."

[@TheEasyRail]

@Docteur Zeuhl I can see how that night could have ended up with the equation tattoo on the body

[@intfxdx]

My PhD also dealt heavily with Navier-Stokes :) I made a bonfire with old notes

[@IoT_]

Are you from Russia?)

[@numberphile]

More with Tom Crawford on this topic is coming soon.

[@esotericVideos]

Does he have a matching tattoo?

[@tonygrace2735]

Dr Hanna Fry's phD was about that...

[@MrMineHeads.]

Yes please

[@turkosicsaba]

I thought Numberphile was about maths and Sixty Symbols, about physics. Why did you decide to post a physics video on the maths channel?

[@frognik79]

no thanks

[@tulasdanslecubitus21]

So after his beef with Eminem, Machine Gun Kelly found a career in fluid mechanics.

[@saumitjin5526]

XDD

[@saumitjin5526]

Still wouldn't be able to beat Eminem's flow

[@ElTurbinado]

Lolll

[@AirNeat]

They don't even look close to similar

[@amitv9128]

@@saumitjin5526 this comment is more amusing than the navier stokes equation

[@adamgray9212]

Physicists and engineers: "Why can't you just be normal?" Mathematicians: *Screaming*

[@lopkobor6916]

RRREEEEEEEEEEEE

[@mellinghedd267]

Mathematicians: Everything must work together, that's how math works. Physicists: If the math doesn't work out then we need to reevaluate what we're doing Engineers: pi=e=3, g=10, f=ma, every body is rigid, and exponents can be approximated as multiplication.

[@is-ig4zh]

mathematiciants : AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

[@ipovaric]

@@mellinghedd267 Hahaha In theory, theory and practice are the same...in practice they are not. As an engineer, I laughed at your joke and have heard many like it. They illustrate the different pressures engineers in industry experience vs. a mathematician or a physicist working on a thesis or a paper. At the end of the day, the engineer typically has to deliver a product or a result, and deadlines have a way of stripping out all the information other than what's relevant to your current goal.

[@mmartin5816]

They an odd bunch.

[@HavasiP]

I'm jealous of this guy. Not of his intelligence (maybe a little) but of his passion and enthusiasm. I would like to know what that feels like.

[@davidobenitez3866]

It’s amazing, but im not a Navier Stokes Equations guy as much as I am Maxwells Equations guy

[@biashacker5542]

It feels like ever lasting curiosity

[@sharrick1208]

I'd just like to not be suicidal on a regular basis lol having passion and enthusiasm is like a dream

[@sharrick1208]

@Train 2noplace lol you have no idea how many I take

[@user-mv1hv5ce3b]

@@sharrick1208 I’m sorry you’re dealing with that /:

[@jmcbresilfr]

10:34 "We kinda find ways to cheat." Math in physics described in one sentence.

[@TheAkantor]

"exp(x) = 1+x for small x" - Physics in a nutshell

[@nicholaslau3194]

@@TheAkantor sinx=x

[@dynamight34]

I absolutely hate when they do that :(

[@vatsdimri3675]

@@dynamight34You're gonna hate it even more if they don't do it.

[@lynk5902]

You mean it's not ok to assume a horse is a sphere to make the math easier?

[@collintmay]

That image of ketchup at 9:30 is actually a special case of the Navier-Stokes equations. Ketchup is a shear-thinning fluid, meaning that, in simple terms, it's a special type of non-Newtonian fluid that becomes less viscous as it is disturbed, then returns to some baseline thickness when you stop disturbing it. There is a modified form of the N-S equations which can handle fluids that get thicker or thinner as they are handled, but it is not covered in this video. Food for thought!

[@zubin8010]

My guess is that the ketchup/lava images were added in post-production, because they aren't mentioned at all, just shown as accompaniment

[@nathanashley2693]

I'm currently doing research into non-newtonian shear thinning fluids and ketchup is the classic example.

[@Henryguitar95]

Are you sure about that? He said ice flow would count, even gas. So I’m not quite sure I trust you on this one.

[@rvsen5351]

@@Henryguitar95 It would work. I just guess that the part where viscosity is handled would get a lot more complicated, because it isn't constant as it is implied in the video.

[@markel-masry2389]

Interesting, at my uni, we are being not being taught that it could also describe non Newtonian fluids too.

[@ondermetu]

What a flow like narration, full of enthusiasm and fluent as much as nature taking its course

[@TomRocksMaths]

[@andrew9360]

The fluid guy has flow!

[@marcodesanti9304]

Navier-Stokes - Invokes Tears Anagrams are too real sometimes...

[@goyonman9655]

😂😂😂😂😂

[@goyonman9655]

Why isn't this more liked

[@liammathpinky]

Naive strokes

[@star_ms]

Goyon Man

[@marcodesanti9304]

@@star_ms what's that an anagram of?? It's been a while since I watched this

[@acerovalderas]

This mathematician is excellent. Extremely clear and joyful. I would like to see more videos of him.

[@Maniclout]

Who else thought his rho's look like integrals?

[@aianvigare1158]

Everyone.

[@andymcl92]

Who else thought he drew the partial derivative symbols from the wrong end?

[@wierdalien1]

@@andymcl92 isnt that a bit petty?

[@andymcl92]

@@wierdalien1 Petty? I was just surprised. It's like how Michael from Vsauce draws 8 as two circles, as in draw the top circle then draws the bottom circle.

[@gregoryfenn1462]

@@andymcl92 What do you mean? You start in the middle and spiral outwards, that's what he did..

[@Rotem_S]

"describes any fluid on earth" *uses the incompressible formulation*

[@wansichen3743]

exactly my thought when i see a math channel doing physics topic

[@wansichen3743]

and a bit fail at it tbh in my opinion

[@SO-dl2pv]

Actually, besides of that, the given equation is valid only for Newtonian fluids.

[@blue-pi2kt]

One must always start somewhere. It is only step by step, you can complete the impossible.

[@wolfyklassen]

​@@SO-dl2pv Which is funny since there was a bottle of ketchup in the cartoon when he said "think of any fluid"

[@Garbaz]

Funnily enough, Nabla isn't a Greek letter. It's a made up symbol named after the Greek word for a harp. Sorry, had to be that guy :/

[@hugoburton5222]

Yeah. Well it is derived from upper delta.

[@DocteurZeuhl]

Some people still call it atled, actually.

[@davideranieri5553]

*the Hebrew word for a particular kind of triangular harp, the "nebel". So, it's got nothing to do with Greek at all (unless you call it "anadelta" but nobody does that).

[@unflexian]

@@davideranieri5553 That doesn't mean it comes from Hebrew, it could be the other way around, like how the word נרקיס (Narcissus) comes from greek mythology.

[@Zersetzor]

Don't feel bad. Someone had to do it, might as well be you.

[@catherinebrower3560]

"Further our understanding of Nevier-Stokes equations" translation "Plz halp" - science

[@andy-kg5fb]

Navier* not nevier

[@johnbiluke8406]

@@andy-kg5fb Pointless comment.

[@andy-kg5fb]

@@johnbiluke8406 -->

[@adi8oii]

I mean really all open-to-research questions can be described as a call for help, no?

[@TheRealGuywithoutaMustache]

0:02 When you want to cheat on your test but used permanent Sharpie instead.

[@mathevideos9909]

I lol'd 😂

[@AcieVODS]

*Tattoo gun

[@entangledmindcells9359]

if your going to try and cheat.. give the tattoo guy a clue and tell him you need it small and inconspicuous

[@lPlanetarizado]

damn this guy is everywhere lol

[@MrTVx99]

Wtf I see you on every bodybuilding and gym channel and you are here too

[@hansb1337]

You know you're dealing with serious science when you have to ask which way round to read the equation.

[@athitham96]

🤣🤣🤣after all the explanation man asked which way to read

[@billyjames3046]

Omg this is my tutor at Oxford 😆 he’s amazing btw

[@brainmind4070]

He seems like a cool guy, but the format of these "-phile" videos kind of makes me hate him a little.

[@samgentle]

@edward Lol yeah he might have a PhD in mathematics but I have a PhD in not getting a tattoo so who's the genius now?

[@brainmind4070]

@@billyjames3046 Um, I think you're going after the wrong person, Billy. Sam was being sarcastic.

[@Cory_Springer]

My cat has a PhD in not getting a tattoo.

[@WorldisArt]

...so is your tutor single? 😆

[@gabrielsetyohandoko7178]

I am preparing for my undergrad thesis defense 1 hour from now... This video really helps me to understand the governing equation i use on my research. Thanks Numberphile!!!

[@TomRocksMaths]

You're very welcome - hope it went well!

[@PetraKann]

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) Cheers

[@aerpk]

Petra Kann I agree with you. For many practical problems there are numerical methods to seek for a solution, but in the end at some point the models come to their limits. It may be turbulence model's limitations or grid density or something else in the modeling that simply is not anymore accurate. It doesn't mean the simulations would be rubbish. There are just areas that model is not able to represent accurately.

[@shaxxshelmet1938]

Yes, but FEA (CFD in this case) analysis is averaging just as he described. It’s taking small little cubes or squares and averaging the properties and parameters of them, running them through the Navier-Stokes (or similar relevant equation) and getting results that are so close to the real answer that it doesn’t matter if it’s a little off. If you could take the little finite elements and make them infinitely small, you would be truly solving the equation for every particle, but that isn’t really possible as we know it because it would take an infinite amount of time to solve.

[@StuntpilootStef]

I'm pretty Stoked about this video. But joking aside, I've been waiting for you guys to cover Navier-Stokes. Thanks for doing a video on it.

[@vodkacannon]

U could have said i'm pretty stoked about this video, and it would have been funny. U didnt have to say im joking

[@curtiswfranks]

I was surprised that it took so long!

[@StuntpilootStef]

@@vodkacannon I could have, but I wanted to show genuine appreciation for the video. It's a fascinating equation and it's great Numberphile has finally done a video on it.

0:53 "Something that changes shape to match its container." So… a cat?

[@Trias805]

Well, it's a known fact that cats are liquid.

[@HolyAvgr]

I'm positive there was a meme somewhere about this, yes. Something about expressing a cat as fluid.

[@ThePrimevalVoid]

@@HolyAvgr There's a paper that discusses the rheology of cats.

[@alakis]

Well, yes, I think you could model the cat's shape over time using Navier-Stokes. The force that the cat's muscles produce, have to be taken into account as part of the external forces. I don't go to parties so I don't need to be funny.

[@stulora3172]

@@ThePrimevalVoid correct. it's rule ᔭƐ.

[@jppagetoo]

I love this stuff. As an undergrad I specialized in numerical solutions to fluid dynamics and heat flow problems solved by numerical analysis. But... I have forgotten too much after all these years (33 years since I got my math degree). Channels like this on KZbin has been great for keeping up on the world of math and physics I was once deeply involved in.

[@not_potaytoes_hobbit]

I love seeing people with enthusiasm teaching stuff, so inspiring!

[@TomRocksMaths]

[@crystalrogers9012]

Seriously, his enthusiasm and actual knowledge pulls me in more.

[@moreaufamily437]

I'm an ME and fluid mechanics was one of my favorite subjects and I loved working on Navier-Stokes. It's hard work though. I remember working for hours on a single simple problem. Have fun explaining the tattoo ;). There is some truth in the differences between engineering and math. As an engineer I get the answer I want and keep working. A mathematician though knows that there may be more than 1 solution or inconsistencies that need to be explained.

[@sentientmgtow1039]

In fact, history verifies how an engineer and a mathematician work, knowing how Stokes (mathematician) and Navier (engineer) came to the development of their equations. I found it fascinating.

[@SonnyBubba]

The difference between engineering and math, the engineers would be satisfied with an answer that’s accurate to three decimal places…

[@labibbidabibbadum]

@@SonnyBubba For government jobs we just use the little one.

[@lalapt127]

Tao has proposed a program for ns-equation millennium problem. May be numberphile can to a 2nd part on the proposed methods.

[@Maharani1991]

+

[@cwaddle]

I think i read it but it seemed so complex to me and probably way too techical for numberphile

[@frtard]

@@cwaddle "I think i read it" If you're not sure of even _that_ then yeah, your're probably right lol

[@Smonserratm]

Only a painful equation could be tattooed in a painful area. Fitting.

[@AlphaNumeric123]

Tom Crawford did his undergrad at Oxford, his PhD at Cambridge, and now he’s at Oxford. Watching this video, I’m not surprised at all!

[@puckry9686]

What about Masters

[@hoixthegreat8359]

@@puckry9686 He did an MMath at Oxford for his undergrad, so he did his masters at Oxford. Most top universities let you stay on do a masters course right after your bachelors for mathematics (you normally have to decide by the end of your second year though), including Oxford and Cambridge.

[@gertorrrrrr6]

Fabulous explanation. If more teachers were like this much more kids would study math.

[@ninawii5318]

I just passed my fluid mechanics course so now I can fully enjoy this video without stressing myself beacuse I might fail due to Navier-Stokes and me not knowing how to use it properly also, because of all the pipe losses

[@danielmarkkula3004]

I was hoping for graphs but this is cool too.

[@harrysvensson2610]

Don't you mean simulations?

[@MsSlash89]

How do you graph such equations?

[@trololollolololololl]

@@MsSlash89 u do

[@trololollolololololl]

@@MsSlash89 u can graph it in bath xD or see weather forcast

[@photinodecay]

There was a "graph" of the right angle turn solution. The problem is that what you're looking for is a 6-dimensional object so it won't look like any simple x versus y graph you've seen before

[@SM321_]

Are you going to make videos on other millenium problems? So far we have only Riemann, Poincare and Navier Stokes now

[@hafizajiaziz8773]

They already have P vs NP by Simon Singh.

[@danieljensen2626]

Those are the only ones you can really understand without spending like 10 years studying the specific area the problem comes from.

[@zennincasl9425]

They also have poincare but that's been proved

[@jackozeehakkjuz]

@@zennincasl9425 And don't forget Poincare.

[@zoraizmohsin935]

Ayy. What about poincare?

[@igNights77]

"Why did you do that?" "To cheat on an exam."

[@tomhanlon1090]

honestly pretty cool to see a young guy with tats & piercings on the channel. anyone can do math.

[@brainmind4070]

I don't know if I'd say that. Having 'alternative' style choices doesn't preclude one from having mathematical talent, though.

[@sakki3378]

@Niels Kloppenburg please tell me you're joking

[@eliasgallegos3058]

This is one of the best videos I've seen in a while! Time felt like it passed in a second!

[@delanask]

I really wish he talked about the assumptions used in the equations: continuum and incompressibility. If there is a reason they don't work it comes from one or both of those. Especially continuum, it actually slightly explains the 90 degree bend issue: there are never any particles at the point which has infinite velocity

[@jhonbus]

This guy Stokes my Navier.

[@kyakarogenaamjankar898]

Bahahaha!

[@y__h]

Yeah. I've Navier been getting this Stokes before.

[@rafaeligmpraciano4123]

He got me stoked.

[@WritingMyOwnElegy]

*strokes

[@devilliar3786]

Ewww

[@LogicraftRedstone]

I loved studying this at uni, still look over all of my notes and coursework in pride :D

[@kdawg3484]

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.

[@SonnyBubba]

All those different directions is why the equations get so complicated. A pure closed form solution to Navier Stokes would give you the motion of a 5 cm eddy within a hurricane, as well as the motion of the hurricane.

[@paradoxparade1]

You're missing the point though. It's not about practicality, which is "solved". It's about the edge cases where the laws of physics should be making sense but they don't, such as the inifinite speed example at the end of the video. I agree with the interviewee that there's probably a new form of mathematics, which we just haven't developed yet, which will explain those edge cases. Which could also be applied to the Riemann hypothesis and likely the other unsolved Millenium problems. If there exists such a new form of mathematics, it can then also be applied to physics and engineering.

[@tobiasgehring2462]

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?

[@msrodrigues2000]

The second thought is interesting, but if we could actually accomplish the free manipulation of single atoms, we could make a perfect corner, so my guess is that we get those results because of the way we treat the problem, the viscosity in the problem is not a function of temperature for example, which itself should be a function of velocity, so we are using average viscosity in each area of the flow, this of course leads us to a blow up, what may be occuring is that maybe the single atom of this corner in a well know time reaches high enough temperature so that its viscosity goes low enough for it to make trough the corner without breaking the relativity.

[@msrodrigues2000]

But again, this only show us the discrepancy of what we consider and what really happens

[@simpletongeek]

Velocity cannot be infinite for something that has mass, right? So, there's that limit. Also, something with light speed will split atoms. So, doesn't that describe the erosion of smooth river stone?

[@tobiasgehring2462]

@@msrodrigues2000 even in the case where you're manipulating single atoms, an atom has a radius, so you still don't have a perfectly sharp corner. There's also the fact that presumably as the velocity at the corner tends to infinity, so will the force exerted by the fluid on the container, which in reality would eventually get to the point where you break something off the corner and round it off further.

[@sheabrown]

Perhaps developing a perfect corner is the key to lightspeed travel 😂

[@michaellavy3269]

Just finished a fluid mechanics class and it feels awesome to fully understand a numberphille video that I wouldn’t have understood beforehand

[@camelectric]

Precision: this is the equation for incompressible fluids, not all fluids

[@landsgevaer]

Yep, incompressible and newtonian on top of that. So no gasses, no ice, no ketchup. He's enthusiastic, but somewhat inaccurate, if you ask me. (Wouldn't have been so picky if not for calling nabla a greek letter too...)

[@Smonserratm]

@@landsgevaer Just sauce, raw sauce.

[@OtherTheDave]

And even “incompressible” fluids are only incompressible in a “sufficiently normal” environment.

[@antoniogarest7516]

Quick maths

[@Thalario]

Do I understand it right that the “small” equation states that sum of volumetric changes zeroes out, which is where incompressibility comes from?

[@FryuniGamer]

"There is nothing we have said here, hopefully, that anybody could possibly disagree with" You have way too much hope in humanity.

[@IBITZEE]

Thanks Tom... always a pleasure to hear from someone who can explain the subject to a 5 year... when this happens... means you really master it... 🧐🧐🧐

[@caseydouglas3671]

I love how excited he is talking about his field and this problem, it makes me excited to learn about it.

[@afterthesmash]

If I'm not mistaken, Navier-Stokes assumes continuum mechanics, and does not finally apply for molecular systems in any case.

[@sebz.2756]

Yes, you are right.

[@neologicalgamer3437]

Neither does fluid dynamics in general. Try applying their equations to sand. It works more or less the more grains you add in

[@randybobandy9208]

I was thinking that as you get to the molecular scale maybe there are uncertainty effects that come into play.

[@SonnyBubba]

Quantum effects would mean that these equations don’t describe the motion of fluids in a nanometer scale. But for a centimeter to kilometer scale, they work, in the sense that what they predict is in close agreement with what’s observed. But, again, there’s that bit about averaging.

[@TheSecretmirror]

The best thing about maths is really seeing people almost explode of excitement cuz well, the maths behind it is just too fckn cool

[@peanutboy41]

Perfect, I have a course in fluid dynamics starting next month. This was a perfect introduction!

[@FM-rb3kq]

If it's your first course on fluid dynamics you might not have to worry about NS. That's usually Fluids 2 material

[@Ny_babs]

I love Tom’s excitement, and passion for the topic. Bravisimo!

[@sumilidero]

"Oh yeah, your river is gonna be flowing at infinity miles per hour"

[@devilliar3786]

@@anonymousone5245 just stop

[@johnnymichaelnoobmaster69]

By far the coolest guy to appear on this channel. We need more of this.

[@bcaudell95]

This was my first introduction to Tom, and I want to see a thousand more videos from him. He seems like such an awesome guy. Also giving me some inspiration for future tattoos.

[@pinnacleexpress420]

Ive never been so ok with not fully understanding the concepts discussed. this was great.

[@stefanschacht3322]

Thumps up for the remark that star-formation is due to magneto-hydro-dynamics instead of gravity! Finally...

[@whalingwithishmael7751]

I quite fancy the style of this man

[@eve8372]

Tom Crawford has furthered my understanding of the Navier-Stokes equations, can he win the prize?! 😂

[@subhasish-m]

This is the same guy that made the video called Equations Stripped: Naviers-Stokes. Worth a watch

[@leopardus4712]

I felt like thanos when I knew how to derive the navier stokes equations

[@threesixtydegreeorbits2047]

but, our troops!

[@AdityaGhosh50]

You're not the only one cursed with knowledge

[@budtastic1224]

@@AdityaGhosh50 yes.. perfectly balanced as all things should be

[@AdityaGhosh50]

@@budtastic1224 It will be balanced when you FIX THIS DAMN DOOR!

[@jackozeehakkjuz]

You mean deriving as in writing down the stress tensor in spacetime and then just requiring zero divergence?

[@VEER4L]

I'm an engineering student and I would really like to thank you sir a lot for making me understand the equation and it's problem so so easily.

[@ciroalberghi]

I'm watching this amazing video while my MHD simulations are running. Thank you Numberphile, great work!

[@josephelmes2165]

Finally, some fluid dynamics!! 👍🏻

[@abdelmalekghassen563]

Two of the most elegant equations that I've ever seen. Well them and Maxwell's electromagnetic equations

[@wilderuhl3450]

Tom Crawford is now my favorite numberphile guest.

[@SunayH01]

The best Numberphile video of all time. GOAT!

[@KX36]

I think this is the clearest numberphile video ever! Good job Tom.

[@fourierbird]

TOM IS FABULOUS AND I LOVE HIM ALREADY

[@cesarangulo1402]

Absolutely an outstanding video! I was totally focused for the 20 minutes. Great!

[@numberphile]

That’s great to hear. Thank you.

[@cesarangulo1402]

it's true! I enjoy the 20 minutes and almost didn't notice when finish. and I believe a grasped a lot@@numberphile

[@sahin8780]

Not kidding, I am in love with y'all, all I need is math and you as math talkers

[@ewenchan1239]

This is a FANTASTIC, and easy-to-understand explanation of the Navier-Stokes equations!

[@TomRocksMaths]

Thanks Ewen, that's great to hear!

[@ewenchan1239]

@@TomRocksMaths You're welcome. I've been running/performing computational fluid dynamics simulations for about 12 years now and this is the most succinct explanation of the Navier-Stokes equations I have ever seen. It would be nice to see if you have a longer, more in-depth explanation about the Navier-Stokes equations (and what makes them so difficult to solve), from a mathematic point-of-view.

[@TomRocksMaths]

@@ewenchan1239 That would be exactly the subject of my 1-hour talk on the topic... I've given it at a few universities in the UK so if you can get me an invite I'd be happy to come along!

[@ewenchan1239]

@@TomRocksMaths Unfortunately, I'm not affiliated with any post-secondary or higher education institutions. Sorry.

[@QUANNGUYENVOHOANG]

Hi, I'm studying Master degree in Computational engineering and just found your video. Many thanks for your effort on explaining this!

[@hiimapop7755]

That's the most rad-looking mathematician I've ever seen...

[@ersatz_cats]

The poke-ball tattoo is a nice touch.

[@gabotron94]

Love seeing modified people on the STEM fields!

[@ViratKohli-jj3wj]

@@Son-Of-Gillean no u

[@gustavoexel5569]

1:22 This equation not only says that mass is conserved, but it also assumes that the fluid is incompressible, so if it isn't, the equation is already wrong. That menas that we went from modelling all fluids, to only newtonian fluids, to only incompressible newtonian fluids. Say goodbye to modelling any kind of gas (including the air)

[@hao2000ki]

Just learned this in my Intro to Fluid Mechanics class last semester so this is cool to see

[@ethanderagon7907]

I've been waiting for you to this for years brady! I'm so excited

[@matthewherald1905]

This is now my favorite video of all time

[@matthewconcepcion1756]

Hey! Can you make a full playlist explaining these millennium prize problems? :D That would be awesome :))

[@EtzEchad]

Now that I know a little bit about this, I'm really Stoked by it!

[@MrSabba81]

Nice one! I am an ecologist working on Eddy Covariance as a post - doc at the University, that's inspiring!

[@gertorrrrrr6]

His passion and motivation are so amazing. It makes me so jealous to see someone like that.

[@mfcoats00]

This was 2/5 of our first test in Transport Processes II, the class averaged a 46. Continue with these videos long enough and eventually even America will have a free education system.

[@521Undertaker]

A beautiful explanation. Thank you for expanding my knowledge!

[@adamhrabovsky5485]

I’ve been waiting for this video for years!! Thank you, Brady! Thank you, Tom!

[@inam101]

I love how much this guy loves Maths. I would love to have a similar tattoo, but I am going first to find my equation first.

[@jeromeofmarmite8914]

Find it?

[@inam101]

@@jeromeofmarmite8914 Sadly not yet. But I am studying maths slowly and steadily and I am sure I will find my beloved equation.

[@abstract2807]

​@@inam101 keep it up man... 😊

[@RupertLazzano]

Perhaps the N-S equations fail at the corner of the canal because in reality there is no such thing as a perfect 90-degree bend terminating at a point on the corner. Even going down to the level of atoms, there would be some distance at that corner, not a point. I'm not a mathematician so please forgive me if this has already been considered or is not relevant in the first place. Very interesting stuff. I hope I live to see someone figure it all out.

[@amabilispalmiery2798]

everytime i solve a navier-strokes problem, i take a hit of any fluid i want, wine is my top pick

[@henrique7612]

It doesn't model any fluid. Only newtonian fluids, for other fluids you would have to change the viscous term

What's a Newtonian fluid? I know I could google it, but let's tell it here for the lazy?

[@Libcio97]

I was looking for that comment, thanks.

[@lolwutizit]

Žiga P. Škraba It’s a fluid in which the shear stress and the gradient of velocity are linearly dependent. The coefficient of this dependence is what we know as ‘viscosity’.

[@Jesse__H]

And if it's NON-newtonian, that means it gets solid if you slap it. facts. 🙃

[@Mrt-dy2kq]

There are also shear thinning non-newtonian fluids, for example is ketchup

[@darkermatter125.35]

The last guy I met with a physics equation tattooed on them had F=ma. It is nice to see someone have a more complicated equation on them like me, also in fluid dynamics no less XD

[@soccerbels7947]

Lol

[@cactuslovesballoons8581]

I Navier did math but this Stokes me all the same.

[@adaikhsan]

finally, I found the best video that explains the Navier-Stokes equation, thank you Tom

[@TomRocksMaths]

You're very welcome :)

[@godesacher2388]

His overall explanations are great in the video, but in 17:00, I think his explanation is quite vague and not sufficient to understand for the problematic, arisen in conversation, so I'll add some comment on this. The physical claim that NS equation itself makes alone has physical fidelity to the actual nature indeed. The reason that the computer outputting nonsense results is because computer solves the equation with discretization, rather than doing it analytically. When the computer calculate for the partial differential equation(PDE) "numerically", it inevitably introduces discretization to the original formula by its own limit that works with 1s and 0s, and that is where the problem arises as with numerical artifacts, numerical instability, truncation error of numerical scheme, round off error which induced by its own computation, etc. This is "a fundamental limit" in CFD that makes a trade-off in calculation, which a lot of researchers trying to improve.

[@KC-dw6yz]

I appreciate the use of the theme from 'The Right Stuff' at about 15:23 :) Have been waiting forever for Numberphile to do Navier Stokes. Would be great to see more fluid dynamics, stuff like Couette flow or pipe flow or trailing vorticies, it's a really pretty science when shown visually.

[@carolinewilhelm7672]

Thank you for such a clear explanation! Love your ink too. You found amazing artists.

[@darklink1113]

Absolutely fascinating. Great video

[@f1tech249]

You are the rockstar of Thermodynamics OMG THE WAY YOU EXPLAIN CONCEPTS ARE AMAZING THANK YOU SO MUCH THANK YOU THANK YOU THANK YOU!!!

[@WorldWaterWars14]

My favorite thing about this video is the multiple desk lamps sitting on the shelf in the background

[@mastersasori01]

Terry Tao been working on these for quite a while.

[@PaulPukite]

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.

[@blessedowo1958]

Will have a look thanks :)

[@KirbyTheKirb]

13:55 Terence Tao is everywhere. What a genius.

[@vtfresh]

Finally a video on my favorite equations!

[@npip99]

3:50 So, the reason why it's saying that mass is conserved here, is because you can imagine that du/dx = 5 means that 5 units of water is coming in from the x direction, and dv/dy = -5 means that 5 units of water is leaving on the y directions, describing a fluid making a 90 degree turn for example. And now we see that this defines an incompressible fluid of course! The water in = water out, so this equation requires that there are no regions that are water-consumkng black holes where water goes in and does not come out, and the equation also requires that there are no regions that open up into holes of vacuum in the water as water leaves but does not get replaced by more water.