Navier Stokes Equation | A Million-Dollar Question in Fluid Mechanics

[@coolhersheys]

As a fluid dynamicist, I congratulate you, sir, for the quality of your videos. You manage to convey the meaning of the topics you present, in a clear and concise manner, and the beauty of mathematics. More content like this is needed. Keep the good work.

[@VeteranVandal]

You look tense in your profile pic. Get those tensors out of your life man!

[@theludvigmaxis1]

I also specialize in fluid dynamics and concur this video is wonderful.

[@reimannx33]

@@theludvigmaxis1 yeah, I specialize in spotting the specialists who fake it.

[@pipi0290]

I'm also a fluib dynansmicsist and it's ok

[@primenumberbuster404]

As a Food dinomasochist I can relate.

[@nickmyers4681]

I’ve discovered a truly marvelous solution to the Navier-Stokes equation, which this comment is too narrow to contain

[@rjthescholar177]

Publish it, then.

[@sthamansinha243]

@@rjthescholar177 You missed the reference.

[@nickmyers4681]

@@rjthescholar177 en.m.wikipedia.org/wiki/Fermat's_Last_Theorem

[@bramilan]

😄

[@rituparn1]

Myers Last Solution

[@ayushthapliyal2462]

The best video on navier-stokes equation

[@Aleph0]

Thank you!! Really appreciate it :)

[@DJVARAO]

Indeed

[@jivanvasant]

Ayush Thapliyal > Math as Theology = Sacred Geometry? [6:54 -- 7:01] Aleph Zero: "In a very secular sense, one can say: To know Navier-Stokes is to know the mind of God."

[@DJVARAO]

@@jivanvasant “Mathematics is the language in which God has written the universe” ― Galileo Galilei

[@aVoidPiOver2Rad]

Nah bro. This video ist dogshit in comparison to the Numberphile Video

[@LostAlienOnEarth]

Why didn't Navier and Stokes solve their own equations, damn it?!

[@archockencanto1645]

😂

[@ihato8535]

They did. But the margin was too small.

[@kartikkalia01]

@@ihato8535 lmao

[@maxwellsequation4887]

@@ihato8535 you are a Legend

[@neosokretes]

Run out of paper 😀

[@kirach3734]

for the very first time in my life...i feel alive after watching this video....wow thank you

[@z-beeblebrox]

There's something profoundly recursive about using a fluid simulation to illustrate what aspects of fluid motion prevent us from solving Navier-Stokes...an equation that would allow us to accurately simulate all fluids (obviously fluid simulations are using shortcuts and it's just for demonstration purposes, but it's still brain twisty)

[@mranonymous2729]

I agree with you completely and what beautiful way did you use to present that haha

[@Moodboard39]

Cuz is from God ...

[@mmsrkmax5820]

I have been studying fluid dynamics for last 7 years and I must say this presentation was spot on!!! Great job!!!!

[@AnakinSkywalker-zq6lm]

I’ve been studying fluid dynamics for the past 7 minutes. And it’s true!!!

[@AndrewPa]

I study it for 25 years and can say it is mathematical point of view on physical problem.

[@HadiM-rb7yo]

i'm really happy i discovered this channel

[@Aleph0]

I'm glad too! Welcome.

[@PacoOtis]

The "Stokes" is an ancestor of mine who developed an equation for the rate of fall of a sphere through a viscous medium. I never have figured out what use the formula was. It's nice to see educational videos and thanks for sharing.

[@reimannx33]

Yeah, my ancestor proved your ancestor tried to pull a failed hoax. So, you inherited a failure. So, how it feel to walk into a bar knowing that the ladies think you inherited a failure ? :)

[@EmilM-pb2hn]

@@reimannx33 My ancestors of vikings made your ancestors beg for their lives. So, how does it feel to walk into a bar knowing that the ladies think you're a weak beta male?

[@reimannx33]

@@EmilM-pb2hn Civilized intelligent Man may be physically weaker than the brutes and beasts, but it is our wits and IQ that led us to the moon, and create science & technology rather than the reptilian-brained brutes. So, while you may flout your low IQ and drum your empty cranium to make noise, the rest of the world laughs at your folly characterised by ignoramus being your "Dream"y bliss.

[@cara-seyun]

My ancestor made a song that almost no one cares about anymore but was featured in an old movie, so that’s cool 😎

[@IODell]

@@cara-seyun We might be related.

[@Bton0711]

Love the video! Maybe interesting to know: In CFD (Computational Fluid Dynamics) we solve it by averaging the turbulente fluctuation of the velocity. Therefore different turbulence models are being used and improved every year. That's why flow simulation, around airplanes for example, is possible. So we can't solve it at all, but we became really good at simulating it!

[@OwenMcKinley]

Dude, your channel is gonna blow up. Content and presentation is awesome! Really hope to see more from you. You make math feel visceral, not "far away," if that makes any sense

[@Aleph0]

Thank you!! That's very kind :) I try to show the core intuition behind the topics when making these videos.

[@BZ_Tam]

Bro he didn’t explain anything about the mathematics of the equation u simply found the explanation U were looking for to the degree of difficulty that U were looking for but this is simply a history lesson on the problem

[@OwenMcKinley]

@@BZ_Tam Where in my comment did I indicate that I felt as though I understood the massive scope of this problem with an elite level of precision? It's enjoyable to watch a well-made, digestible presentation on these enormous math problems; nothing more, nothing less.

[@rumfordc]

@@OwenMcKinley you indicated it at the words "not far away"

[@OwenMcKinley]

@@rumfordc Ah, perhaps; I agree. But I don't believe that I understand the full scope of the problem. There's a reason why it still hasn't been cracked.

[@playerscience]

I have no words to describe......... How beautiful this explanation was......

[@davidmyers7508]

I just want to gush about this guy's videos to every person in my life. Their quality is singular. Thank you Arts-and-Crafty Storyteller Math-Man for expending your focus in this way.

[@12jgy]

What I find even more beautiful about Navier-Stokes is that when actually think about it, it arises in a relatively "simple" manner, being Newton's Second Law applied to fluid mechanics, but it still so incredibly difficult. Nonlinear parcial differential equations are so rough to handle, but at the same time they appear in so many places in the study of nature, I guess this is a testament to just how complex nature really is. Anyway, very good video, and a question for you, do you plan on covering any other specific differential equations, and if yes, which ones?

[@Aleph0]

Couldn't agree more! The simplest equation becomes unsolvable once applied to fluids. It's crazy that we can't solve even the most basic non-linear PDE's; it shows just how far we need to go in understanding nature mathematically. I might make a video on some other DEs: likely the Einstein Field Equations of General Relativity (tensor calculus on curved manifolds, very interesting), or maybe the three-body problem. Thanks for the comment!

[@carloserazoramirez4286]

This is one of the most beautiful videos I've seen on youtube. Dude, youre freaking amazing and you've got me subbed so bad

[@michaelmello42]

It's one thing to understand a difficult subject but quite a different matter to convey one's understanding with this level of clarity. It is a gift.

[@cchakrapani3574]

Brilliant. This is totally awesome. Way to go, Aleph!

[@Aleph0]

Thanks uncle!! That's very kind of you :)

[@oot007]

This is the best explanation of Navier Stokes I've seen. Well done.

[@imaduddinalawiy3426]

you explained everything so clearly, thank you

[@Aleph0]

thanks so much!

[@ariuwu1234]

when i finally grasp the concept by the end of the video, especially if it ends with such quotes, i get chills down my spine, i love your content

[@AryanSingh-og7ke]

A pattern to noticed in likes and dislikes, likes can be described as squares and dislike as sum of squares Like : n^2 Dislike : (n+2)^2+(n-1)^2

[@almalucia2650]

Thanks for the great video. I have watched a handfull of videos to understand the Navier Stokes equation, and yours is the first one that actually managed to teach me something about it.

[@verrug4]

This is so deep. The analogy to the human mind really caught me off guard. Thanks for the content.

[@the13thTone]

I see that trajectory, well deserved, you'll quickly rise to the top. I love your enthusiasm and passion for the subjects, not to mention the production and execution of the content are top-notch. Hats off.

[@osemudiame123]

Best of luck getting this Chanel up and running

[@Aleph0]

Thank you!! Appreciate it :)

[@BYJUSExamPrepGateEseEEECINCS]

Hey I must appreciate ur work behind d screen..good job👍👍

[@Mohd53000]

😅😅

[@bythetimeyoufinishedreadin9083]

I'm a second year in aerospace engineering, and this was very interesting to watch. Super stoked to learn more about this in further detail (no pun intended)

[@luk45ful]

This is definitely the kind of content I was looking for! So good!

[@Enthropical_Thunder]

This started like 100 Second Physics, went on like Today I Found Out and ended like VSauce. XD

[@michalfishkin9200]

I am so happy to see Ladyzhenskaya's work mentioned here! Excellent video, as always.

[@mariaameddah5983]

Damn this is like the best video I have seen about this topic, very well explained. Thank you!

[@jackl331]

soooooooo moved by your fascinating presentation...... mind blowing... thx

[@juanfelipe9983]

Just Woww!!! It was quite simple for You to explain the problem behind Navier-Stokes equations!! Congrats!!

[@tamilolidurai6043]

BESTEST Video ever in internet about N.S equations. Million thanks for this. Also, Solutions for N.S. equations doesn't exist like the word BESTEST :)

[@HitchHawk]

These videos are amazing!! I really love the presentation along with the explanations. Phenomenal work dude!

[@phuongly7811]

the best video about the Navier Stokes Equation. "Solving Navier Stokes Equation is like solving a personality" ...wow

[@joeljacob2234]

Just beautifully put together!

[@Aleph0]

Thanks Joel!

[@farhanfuad6045]

Your explanation, presentation, comparison with nature life god everything...everything is awesome

[@anirudhsingh3507]

One needs a certain level of passion to make such video!!!!! Thanks man , it was a great! explanation.

[@Manjeetemo]

Really nice video on Navier-stokes eqn. Will be waiting for furthur uploads..Keep it up dude.

[@faizanka5088]

Wow, that was beautiful explanation of navier stokes equation. This channel deserves more attention

[@BHAVYAMATHURBEE]

Dude. Awesome. Keep doing what you do. Ill be watching your future with great interest.

[@rousselrobin6060]

Loved the video! This made me realize that there isn't much vulgarisation content on youtube about functional analysis and pde theory compared to topics like algebra and topology. I'm guessing that might be because the subjects seem to deal on the surface with relatively easy topics like multivariable calculus, and it feels hard to go into more details without getting into the specifics of various functional spaces. That's a shame because these are the fields I study and it sometimes kind of feel like they're unappreciated by "pure" mathematicians. If you want to talk more about this kind of stuff, I think it would be really cool to have a video on distribution theory. I feel like the concept might be general enough to fit into one video, more so than sobolev spaces for example.

[@jerrysmcnuggets]

Not trying to be nitpicky and please correct me if I'm mistaken, but it's not strictly correct to say the equation you had up at 0:11 describes the flow of everything in the universe, is it? Isn't that equation the N-S for incompressible, Newtonian fluids (which would exclude honey, for example)?

[@apexmcboob5161]

You are a superb teacher. I've subscribed and as soon as I finish typing this I'll search for all your other videos. Thank you!

[@austinkmhi]

I absolutely have no knowledge on physics but I understood this...kudos man

[@AlexMoophty]

You are simply amazing! Pls keep going with your content🔥❤️

[@rohlay00]

Came here to procrastinate on my upcoming fluid mechanics exam. Wasn't expecting such a philosophical ending.

[@maxs5022]

best math youtuber i know!

[@atil4]

A part from being among the top top channel of math in all KZbin, there is something special about yours the way you simplify the key milestones to really understand a hard concept. Without the need of big animations, because it relays on these key Simplifications. As you side in other video; you are simplifying knowledge for us, thank you for sharing this digested math wisdom. I loved all your videos so is silly to suggest but I really calculus since is the language of Nature. Understanding the nature of calculus is understanding nature from Math point of view. I would love to see more related video to differential equations e.g. Laplace transformation. Or the relation between different fields of math. One subject that always fascinate me is the conical curves :) This channel can only grow, thank you for your efforts.

[@starstuff11]

I am so glad I found this channel!

[@finnbraaten3264]

I think we should be a bit careful. Navier Stokes does not describe everything that flows. It only describes flow of even viscosity, isotropic media. It would not describe nematics, polymer flow, or that of active matter/odd viscosity media.

[@LostAlienOnEarth]

Best math channel ever

[@Chill----]

Yep I also agree that your channel is gonna boom. I liked the channel after finishing my very first video from this channel.

[@mauriciougaz]

Beautiful and outstanding job man!

[@vijaysinghchauhan7079]

Today I discovered a great channel dedicated to math majors.

[@gustavoexel5569]

There's somrthing off with this equation, unless in your equation p represents specific pressure energy, it should be grad(p) / rho, where p is the pressure, and rho is the specific mass.

[@khangafar8505]

The efforts of the making this video is awesome..

[@sergiolucas38]

Great video, so much quality :)

[@albertora4391]

Congratulations! Nice and complete video!

[@PinakiSwain]

Thanks for sharing this. A very accessible presentation of a complicated equation.

[@gauravs3566]

Really love your work bro! Keep it going

[@IshanBanerjee]

Undoubtedly one of the most beautiful equations.

[@Aleph0]

Agreed!!

[@99bits46]

It's bullshit. It's not true. You are tearing me apart.

[@Godakuri]

@@99bits46 What

[@DragonRazor9283]

@@99bits46 ??

[@hopkinsyb4306]

Great vid hope you do one for every million dollar question

[@tantzer6113]

What is meant by a “solution”? You mean getting a *closed-form* solution? Many equations have solutions that do not have a closed form and can be approximated arbitrarily closely using numerical methods. Are you saying we don’t even know whether non-closed- form solutions exist?

[@pierrecurie]

Special cases aside, closed form solutions are clearly not possible. The millennium prize is asking for the _existence_ of "smooth" (and prob other conditions as well) solutions. Smooth is probably asking for finite L2 norm or some such - eg when you spill a glass of water, the energy doesn't suddenly all concentrate into heating some tiny region to 9999C. There's also a constraint on the assumptions - you start with "smooth" initial conditions. Otherwise, if you start with singular conditions, it shouldn't be a surprise you to encounter singular conditions later.

[@parkashsingh9044]

wow.. one of the best.. keep it up bro.. 👍👍

[@gowzahr]

I'm of the opinion that the solution of the Navier-Stokes Equation, if it exists, would be so complex that it will have little effect on computational fluid dynamics, beyond perhaps, deriving better turbulence models.

[@jawad9757]

Breaking news: weather prediction becomes slightly more accurate

[@davidwilkie9551]

Supposing we accept Professor Susskind's lectures on Black Holes, Singularity function in pure-math Black-body Reciproction-recirculation connection of hyperfluid zero-infinity i-reflection entangled containment inside-outside inclusion-exclusion e-Pi-i sync-duration resonance timing-phase positioning .., and assume that the temporal superposition identification terminology is relevant to the Navier-Stokes Equations such that conic-cyclonic coherence-cohesion objectives condensed in/of QM-TIME Superspin-spiral Superposition Totality here-now-forever, then aprt from establishing a context for continuous creation connection cause-effect Logarithmic Time AM-FM Communication Perspective Principle, WYSIWYG, apparently all-ways all-at-once pseudo random cycles of fractal phase-locked coherence-cohesion objective-aspects of ONE-INFINITY.., there's not much to say..? In this case the Holographic Principle Imagery Actuality, is infinitely more informative than possible probability positioning floating point coordination objectives. Ie of no fixed solutions other than the 1-0 i-reflection containment roots of e-Pi-i omnidirectional-dimensional logarithmic interference resonances in zero-infinity.., aka "Renormalisation". Check with 3BLUE 1BROWN channel for ideas, and the Chain Rule has Factorial Quantum Operator Divisors under the term sequence that correspond to the 1-0 zero-infinity logarithmic sync-duration resonance quantization of dimensional orthogonality time-timing containment states of Partitioning Number Theory relationships in log-antilog superposition identification condensation. This is the real-time pseudo random containment cycles "symptom" of hyperfluid vertices in vortices nodal-vibrational emitter-receiver manifestation. IMHO, please ask a real Mathematician for a more formal argument. Fluxion-Integral Temporal superposition Calculus @.dt instantaneously throughout the Holographic Universe is the proof-disproof by observation of WYSIWYG Mathematical QM-TIME Completeness cause-effect Actuality.., hyper-hypo temporal fluidity. "Universe", ->it turns together; in/of ONE-INFINITY Singularity Eternity-now zero-infinity Interval. (Navier-Stokes formatting is amazing)

[@harryh5666]

Awesome vid. Would be great to have some more Millennials explained simply like that

[@michaelcollins7192]

Brilliant short video, well done! I'd like longer ones too.

[@DragonRazor9283]

Holy shit.... this made me think and perceive things differently

[@brunoolas]

Ah yes, the 5 elements: Air, Water, Ketchup, Gas and Smoke!

[@yash1152]

lololol 0:12

[@yash1152]

ohkay, u meant 5:25

[@brunoolas]

@@yash1152 both work

[@nelsonpereira9306]

I really appreciate your videos! Hope you keep on.

[@markattila9835]

No need to go as far as fluids to find impossible to solve equations, try the 3 body problem or the simple dual/triple pendulum. Such a complex thing as a fluid will go chaotic very quickly. the main problem seems to be that equations have only two sides, however in reality inside a fluid there are infinite many equations that "equal" one another simultaneously. This sort of "dependency" causes chaotic evolution very quickly. Chaotic only because in reality the evolution of systems is always chaotic except in the quantum world. Only in our equations does it seem that things are deterministic. Our equations model reality in the simplest cases very well. You have the best videos on these subjects, great work!

[@Ottmar555]

The equation presented here only works for incompressible fluids. Transport phenomena is one of my favourite topics.

[@cbbc711]

Yeah! You could also couple the equations with the Maxwell equations to study electromagnetic interactions. Also, the problem could be made even more difficult by coupling the N-S with the energy equation to solve the temperature distribution inside the fluid. Transport phenomena really is the best part of physics and I’ve been studying that for 2 years, just love it too much

[@Ottmar555]

@@cbbc711 What becomes interesting when including the Maxwell-Heaviside equations is that you get in problems with relativity when considering changes in frames of reference. Though understanding the Poynting vector as the flux of EM energy ties things in profound and interesting ways. But that is, IMO the biggest problem of the NS equation, it is essentially a linear dissipative themodynamics approximation, so information travels infinitely fast. I'm interested in multiphase TF and in carrying chemical thermodynamics into the continuum, as chemical energy is usually not properly defined in TF.

[@Ottmar555]

@@cbbc711 I'm pleasently surprised, though, as unfortunately my experience has been that physicists tend to overlook TF and continuum mechanics, often dismissing it as an engineering discipline. It's interesting to note that many Physics curricula don't even include a fluid mechanics course. Where did you study?

[@cbbc711]

@@Ottmar555 as you almost guessed, I am a nuclear engineer student and I do research in the marvelous field of nuclear fusion. That is the main reason why I’m studying so much Fluid Dynamics, Thermal Fluid Dynamics, Magneto Hydro Dynamics and so on. Even tho I properly am, I would not describe my profession as an engineer, since the nuclear fusion field is still very theoretical!

[@Ottmar555]

@@cbbc711 Yes, I can imagine. How much QM do you study? I have the impression that the transport theory of radiation is still in need for further development, but I haven't studied it sufficiently to have a definite conclusion. I'm also interested in studying nuclear physics, any book recommendations?

[@ClemoVernandez]

Amazing video! you deserve more subs

[@Aleph0]

Aw thanks! That's really sweet :)

[@willm6598]

lol learned more about this equation in 7 min then I have learned in my entire time in engineering school

[@nathanbarnard7896]

I've just started working on fluid mechanics, it's nice to see where it's leading, even if it's leading somewhere unsolvable :))

[@arduous222]

2:02 "If we just deleted the bad term" Oh, the horrible mindset of mathematicians. In reality, the convection term is the very core of the fluid which should never be removed in any cases, and the one term that could be ignored is diffusion term (Euler's eq.). Of course I'm not complaining. It's just that these kinds of approachs are almost heretic, that I'm glad to see this differences between people with different backgrounds. :) I mean, for example, small-scale turbulences will be dissipated very quickly (and transferred to heat) thus there exists a lower limit to sizes of "physical" turbulences. So, these "making convection irrelevant" mindset is impractical to physical applications. Nevertheless, it's good to have a good foundation...

[@Edwin-wn3ss]

Similar to homogenous and non homogenous ODEs i guess, gotta start simple then go general

[@arduous222]

​@@Edwin-wn3ss Of course, of course. But without that term, it's just a heat dissipation equation, which has been studied quite well, with additional terms (which can be regarded as "going general"). Also the physical background for viscosity term is far weaker in comparison to the advection/convection term, which is embedded in the very core of the laws of motion. On the other hand, the advection/convection term would change the class of the equation, which seems far more difficult approach. Of course, for example, if one has limited the problem so that for ANY phenomenon to be LARGER than a certain scale along all directions, you can transform to the frame of reference to make v~0, but this has a very limited application in terms of numerical calculation, since 1. This transform is pointwise while we need to solve for every point in space, 2. all "interesting" phenomena of fluid dynamics arises when v is large (or Reynolds number is high). And frankly, one can find the term (v).( dv/dx ) everywhere in different fields of physics, so, solving Navier-Stokes is good, but giving a recipe to solve ANY equation involving that specific term is far better and a thing I'd like to see soon.

[@nikhilyewale2639]

@@arduous222 Neglecting convection term is something of routine for stokes flow regime(also known as creeping flow regime). Field of microhydrodynamics thrives on this assumption. But otherwise I agree that its the most essential term in higher reynolds number regime

[@arduous222]

@@nikhilyewale2639 I guess we are defined by what we neglect :)

[@陳嘉傑-g8l]

How can this awesome video only get 1k views!!

[@Aleph0]

thanks!!

[@norlan718]

This is an excellent video. Keep it up.

[@sudheerv6941]

can you please make a video on curl, divergence and electromagnetic equations. intuition behind those concepts is elusive for me

[@neosokretes]

Those who make fun of weather predictions should watch this video!! It is like “reading the mind of God” 😃

[@homejonny9326]

Good point

[@yash1152]

0:51 wait, what does "that will last for all time" mean???

[@michaelmoccio2225]

Though I do love this video, I find it important to point out that ketchup, in particular, is not well described by Navier-Stokes since it is non-Newtonian (i.e. the constant viscosity mu does not apply since shear rate is dependent on the magnitude of shear stress in the fluid). You would need to use the more general Cauchy Momentum Equation for the ketchup case. I think your video does a great job explaining scaling arguements and I think it's a great resource so please don't take it as disliking the video, keep it up!

[@redpepper74]

This formula doesn’t apply to ketchup!! DISLIKED

[@AlexAnteroLammikko]

Im not a physicist at all, I don't even study physics, but work in healthcare and have a question on what an "initial condition" is and what "all eternity" means in the context of these equations. I assume this is undergrad level stuff (not the solving the equation, but those particular terms) so am sure some people in the comment can help. Basically, if I have a container of water like a bucket, and the water is settled, and thenI punch it, there is now an "initial condition". But any calculation of the resolving of that reaction can't, and won't, take into consideration other outside acts like me throwing in a brick or tipping the water over or rain occurring. Now lets take the Ocean, or the "air" (the various spheres the names of which I can't recall now but all of which interact with each other). All of these are complex systems with various dynamic entities in them. The Ocean isn't a closed system but even if it was and no water escaped and no additional creatures went in, the existence of animals that can "randomly" change directions would greatly interfere with the calculations as they create new flow. So is the equation assuming a perfect closed system and, if so, what value, if any, does solving the equation have for the "real world" (I have a friend in mathematics who hates the " real world" question but I am not in mathematics so ill ask it :p )

[@Shroomssss]

Really great video! But Navier stokes also has its limits! Even for fluids. You can only apply them to continuums where the Knudsen number is small enough.

[@inf0phreak]

Professor Terrance Tao wrote on his blog something to the effect of (I'm paraphrasing here): "It's an observed fact that fluids and gasses don't randomly spit out particles moving at near light speed". But of course whether that means that we should expect singularity-free solutions to the equation is another question entirely. As I recall, Tao's work on the equation has been mostly in the direction of proving divergence.

[@luizcarlosf2]

Very inspiring video... Wish I had a calculus teacher like that.

[@soulintent4129]

Amazing video ⭐❤

[@thunder852za]

Navier stokes in incompressible form hurts my eyes - detracts from the problem - and I mainly work with incompressible flows. In any case, great video and very well presented.

[@siquod]

I come straight from Paul Washers videos on proverbs and I think he would object: The mind of God has a lot more wisdom to offer than just the solution to Navier-Stokes.

[@KStarGamer_]

0:16 I think you missed the density term besides the acceleration derivative to represent a fluid’s mass?

[@Ottmar555]

It's in kinematic form. You divide by the density.

[@Shynobyn]

@@Ottmar555 Then the gradient of pressure should be divided by the density.

[@Ottmar555]

@@Shynobyn Everyone has their own notation. Both the pressure grandient and the external force should be divided by the density, but it is common to have a force per density term. Likewise here, I'm guessing the term represents pressure per density, which is almost hydraulic head. Most likely a mistake by the author, since he made more generalizations about the NS equations than they might merit.

[@kylevandeventer1037]

Wow… I just graduated with my math degree and I believe you’ve just convinced me to go back for phd… dang it

[@nirmalya_nanda]

The most amazing explanation and video on navier stokes Btw what and where do you study

[@72saurabhsen73]

Wonderful job 👍

[@phyarth8082]

Pour water into container and to find solution to Navier-Stokes equation that is a bit a challenge, but industrial pipes, tubes, channels and gravity driven rivers that is point for solution.

[@Sidionian]

You should do Yoneda Lemma soon. That's a big one.

[@KW-12]

Correction: If I remember well: The N-S equations are validONLY for Newtonian fluids, where the viscocity μ is related linerally to the deformtion rate, so it fails to describe the flow of fluids like honey, paint, ketchup, blood, toothpaste and many many others. *Correct me if I am wrong please We are still in diapers in fluid mechanics

[@AndrewPa]

Good video. As physicist I can say what NS means. Mathematicians see equation but do not understand and do not care about it smeaning. It is not about vorticity or non-linearity - it is about most dense packing of energy :-). So NS is closer to oranges packaging that to what god thinks ( who cares). When diffusion is much smaller than inertial terms - fluids require new regime to store excess of energy ( packed in combination of velocity or pressure). Greetings from Australia.