The million dollar equation (Navier-Stokes equations)

PLEASE READ PINNED COMMENT
In this video, I introduce the Navier-Stokes equations and talk a little bit about its chaotic behavior. Make sure to like and subscribe!
What is a newtonian fluid? physics.stackexchange.com/que...
Divergence and Curl by 3b1b: • Divergence and curl: ...
Source code for the animations: github.com/vivek3141/videos
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Sources: docs.google.com/document/d/1l...

[@vcubingx]

Hey all, I removed a part of the video that had some misinformation, hence the "jump" from one section to another. I made a large error in what I was conveying, so here's a correction by viewer Jay Raut: From what I understand (and don't quote me, its been a while since I've dealt with fluid dynamics), the problem with the Navier-Stokes equations is the issue of them being ill-conditioned. By that I mean that a small change in the input does not result in a small change in the outcome. This is important since with any system, a small input change should always yield a small output change, otherwise the reliability of the solver is questionable (the results should be reproducible, and near infinitesimal changes should not result in drastically different answers). Now while the the equations are basically glorified F=ma equations, which means that they are most likely the correct DE that describe the underlying physics, the problem lies in the fact that we simply don't understand or appreciate them enough. Also, remember that the real underlying physics is much more complicated. We can break down the problem to its core where we consider all the fundamental forces of the universe and the quantum effects between each particle in the fluid. But, this is meaningless because we want a meaningful compressed description of the physics, similar to how Newton's laws of gravity are a simpler version of Einstein's. I've solved the Navier-Stokes equations by hand in undergraduate classes for simple problems, and in these cases the equations are very well behaved. The solving process is actually very logical to the point where you realise that all you are doing is Solving F=ma. The problem comes down to turbulence, and the fact that the simple Navier-Stokes model do not capture this phenomenon at all. There have been very complicated proposals to the NS equations which take turbulence into account, but these are loosely based on analytical physics and more empirical solutions. Introducing this does not only create a more accurate solution, but employing some numerical trick also make the solution very stable. Also, there is also the problem of the DE itself. Its not simple to solve, and the numerical methods we usually employ to obtain approximation, are exactly that: approximations. So if you read the problem statement more carefully, you will realise that there is no straight forward problem that has to be solved. It's like the people didn't know what to set as the problem itself, which has become the problem. To essentially solve the millennium problem, you would need to come up with some form of proof that the NS equations are truly the underlying physics of a fluid (or not). Remember I mentioned the problem of ill-conditioning? Well even if that is true, that does not mean that the NS is BS, and the turbulence modelling tricks can make the solution very stable. However, these tricks are sometimes based on nothing more than: 'it works'. This is not progressive work and that is what the millennium prize tries to address. So answering the question in terms of your words, we don't know if the solution (real) is smooth. because of which we don't know if using tricks to make our modeled solutions smooth is the correct thing to do to obtain meaningful answers either. And upon finding out whether or not it is, we'd also like to know why? Essentially: solve turbulence, because nobody knows wtf is going on. A second mistake is that isothermal refers to no loss or gain in TEMPERATURE and not heat. Sorry about that, I definitely got a bit confused when typing up the script. I'm considering making a follow-up video as to what was wrong with the video and explain what we are actually solving.

[@prometheus7387]

I tried joining your server, but it says that I have been banned or something. Could you see to it? Discord tag is Napoleon Bonaparte#1729

[@roberttelarket4934]

vcubingx:A cite like this for math, physics, chemistry is not the place to discuss politics including this "b.l.m."!!!

[@JivanPal]

@@roberttelarket4934, why not? His channel, his rules.

[@roberttelarket4934]

@@JivanPal: It may be his channel but it's MY RULE!

[@JivanPal]

@@roberttelarket4934, and thus, your rule is one that no-one is obliged to follow. It's also utterly daft.

[@PapaFlammy69]

Very nice Vivek

[@vcubingx]

Thanks Jens

[@HKCREATESUNIQUE]

🙄

[@parameshwarhazra2725]

Pappa is eager to solve this and win a millennium prize

[@floatingpointerror55]

Can you solve it papa?

[@yanry7152]

Papa Flammy Because of Your guidance I know theory of everything Now " [Universe in a Nutshell] = 42 "

[@raresmircea]

Kids today that have a natural inclination for maths live in the golden age of learning

[@AndT101]

Im a med student. Wish i could understand maths easily. Seems much more interesting than human biology. It's very hard for me to grasp these concepts but im not giving up.

[@everab1209]

@@AndT101 Oh man, you must have a lot of free time. I am considering to let go med school to study physics. But ,regardless of what happens, doctors are always welcome. Good luck.

[@AndT101]

@@everab1209 No man i don't really have lot of free time it's because of covid im stuck at home and have more time, So instead of watching netflix and stuff i prefer learning new things. Thanks though :)

[@everab1209]

@@AndT101 It is good to see people interested in physics despite his main aims. Good luck man.

[@gabrielcarranza6139]

As a professor of Mathematics this comment is spot on. There is so much information for students at their disposal at any given time.

[@ra3ealawlh]

Isothermal refers to a constant temperature process. A process during which no heat escapes is known as adiabatic process.

[@UnfinishedEngineer]

if there is no heat escape or addition then temp constant only right

[@aaronrahman8751]

@@UnfinishedEngineer lets say a gaz is compressed by a piston, the temperature of the gaz will increase as we have a higher particle agitation due to high pressure, every variation of temperature is concerved as we dont have any heat transfer with anything. That is adiabatic. If we had colden down the gaz by letting out some of the heat in order to conserve the initial temperature, we would have a constant temperature variation yes but that is because we did a heat tranfer. That is isotherme. You can have an adiabatic isotherm process.

[@Mysteriousmachine1]

Iso means ‘equal’ like isometric means ‘equal measurements’. So isothermal means ‘equal temperature/heat’ Just pointing this out because I literally only found out recently what iso means and it was driving me crazy beforehand trying to remember the meaning of these names.

[@panosjretos]

The exchange occurs slowly for thermal equilibrium in an isothermal process.

[@ArthurShirinka]

Thermodynamics

[@andrewfischer-garbutt2867]

"In terms of divergance we have no divergance." - Gru

[@anilsharma-ev2my]

We have to face our karma curve at some point so divergent is something like diversity but we followed the same space so we got equilibrium at some point

[@tensorproduct3666]

@@anilsharma-ev2my I just went way over my head.

[@NovaWarrior77]

Nice one.

[@turolretar]

ugh millennials and their problems..

[@longdragon3]

lol

[@georgepp98]

Millennials and their million dollars

[@ravencoinspremiumadminasse6265]

kzbin.info/www/bejne/f3erq2Brn9h8abs

[@rafaellisboa8493]

I love the navier-stokes equations, I'd definitely watch a continuation of this. Good job man I like your channel very much

[@vcubingx]

Thank you!

[@shoam2103]

@@vcubingx I second this!

[@effexon]

Is this something I can argue I need Threadripper for my desktop pc to calculate these? So hard to find any game to actually do demanding calculation, other than synthetic Pi calculation.

[@user-nj1cg3ye8q]

I'd love to see more in-depth coverage of Navie Stokes and fluid mechanics. The video helped me understand so much. thank you

[@carlos24497]

This is the best overview of the Navier-Stokes equations that I have seen. The intuitive explanations were very helpful. Thanks!

[@vcubingx]

Thank you Carlos!

[@Sciencedoneright]

@@thealienrobotanthropologist was it really ???

[@pierrebaudemont4606]

Thank you for this video! I always have wanted some introduction to those equations and now it’s done in a nice and concise way 👍

[@vcubingx]

Glad it was helpful!

[@woodensheep]

Love the color scheme, keep it up with your videos!

[@vcubingx]

Thanks!

[@rzhang3927]

Nice animation and clear explanation! Good stuff!

[@vcubingx]

Thanks!

[@swastikbiswas8293]

This channel will be having 1M subscriber in 3-4 years .. I got this after solving Navier Stokes equation

[@prateekgupta2408]

Solve again correctly

[@swastikbiswas8293]

@@prateekgupta2408 that might lead to chaos!

[@borekcikz3173]

That was a great video for this topic .Thank you so much for sharing with us .

[@ashwinidixit5981]

Great work I always use RANS (Raynolds average navier Stokes equation) but never had this much clarity of it.

[@mikeock3164]

Thank you 😊 I learn a lot from your channel!

[@woofle4830]

Yes! Thank you so much for this video! I’ve been waiting for this for forever!

[@vcubingx]

You're welcome! Thanks for watching!

[@pkmath12345]

Pretty amazing video graphics! Good work!

[@vcubingx]

Thank you!

[@gaeb-hd4lf]

Awesome videos bro, hope the channel keeps growing!

[@vcubingx]

Appreciate it!

[@luis5d6b]

Great video, clear and deep at once, loved it, thanks for it

[@vcubingx]

Thanks!

[@KakoriGames]

Some already pointed out mistakes, some key information left out, but overall a nice video. Having tried myself, I know how difficult it is to make videos like these with Manim, so congrats. Also, nice to see more people doing videos on math subjects.

[@brijeshpr6543]

Is NS equation is applicable for laminar flow only or for turbulent as well?

[@KakoriGames]

​@@brijeshpr6543 It's applicable for any flow, both laminar and turbulent, but it's complicated, there's a lot of practical limitations regarding numerical simulations. Turbulent flows often require a very refined mesh for numerical analysis or some sort of turbulence modelling that are usually not derived from first principles. Computational Fluid Dynamics is a very interesting, but very complicated, area of study.

[@user-uq6pz1sf5z]

@@brijeshpr6543 for every type of flow, laminar, turbulent, compressible, incompressible, steady, unsteady. Just the form changes. For example, in this video mass equation is simplified to incompressible flow, and he pointed it out.

[@haimbenavraham1502]

I managed to flow through that quite smoothly. T.u.

[@zxaj]

Just subscribed. Thanks making such detailed informative video.

[@boukharroubamediane119]

Nicely explained. So I liked it and shared it. I am already a subscriber.👍❤️

[@metelicgunz146]

Navier-Stokes one of the best ways to scare prospective engineering students.

[@josemanuelmedeltorrero7622]

That is not the definition of smoothness, smoothness means that it is infinitely differentiable. (Whatever that means) It comes to the study of functions on smooth manifolds, hence smooth functions. The pendulum, for example, I’m not sure that it’s solution has a closed form, but Banach Fixed point theorem assures us that there is a solution!! And it is smooth!!! Now, you can ask then, what would it mean to not be smooth? Well for example the absolute value is not smooth since it is not differentiable at 0. But more than that, experiments on turbulence have shown that turbulence in fluids looks like a fractal!!! And let me tell you, fractals are not smooth in general!! In my opinion turbulence shows us that there is a loss of structure (again, whatever that means).

[@vcubingx]

Thanks for your comment Jose! I was typing up a correction to this at the same time as I saw you're comment. I got confused with a few concepts when typing up the script. I should pay more attention and run it by a few people next time.

[@josemanuelmedeltorrero7622]

Now since you read my comment, I hope you read this one too. Great video, you have a lot of talent and I encourage and celebrate it! So congratulations, and please keep doing videos!

[@josemanuelmedeltorrero7622]

vcubingx I am just a graduate student, but if you can contact experts to check the drafts for your videos, it may help to rise the quality of your work even higher! Again great job and thank you for your excellent effort! You can contact me and I can answer your questions if I can or even better, get you directly with the great professors from my university.

[@vcubingx]

@@josemanuelmedeltorrero7622 Thank you! I'll keep this in mind when I make my next video

[@ishworshrestha3559]

Ll

[@gershommwale7618]

brilliant work vivek

[@lukamitrovic7873]

So when I solve it, will it be navier - stonks?

[@pakiboi463]

U should go now lol

[@yagmurterzioglu1916]

this video helps a lot, thank you!!

[@trigon7015]

Oh my god I’ve been wanting to learn about this for so long

[@beyondlwm]

Thanks for explaining the fomular!

[@BoZhaoengineering]

Fabulous video on this topic. I am learning fluid mechanics this is very helpful

[@vcubingx]

Glad you enjoyed it!

[@NovaWarrior77]

Brilliant work my friend!

[@vcubingx]

Thanks a lot!

[@terryyoon1856]

I'm actually doing a research paper for the Navier-stokes equation!! Very complex but very fun to read!

[@vcubingx]

I agree, they're really fascinating!

[@diegocastillo6470]

Pretty onpoint use of Manim. Nice video

[@vcubingx]

Thanks!

[@aaronsmith6632]

Great explanation, thanks!

[@BTae9293]

Thank you so much for the explanation

[@nadiyayasmeen3928]

Ah yes. The beautiful Navier-Stokes equations

[@BinyaminTsadikBenMalka]

Hey man! This was a really great explanation. Thanks!

[@federicogottardo4869]

Awesome video. Keep the good work

[@1ucasvb]

Great stuff. Also, I commend your boldness on tackling fluid dynamics in an accessible way!

[@vcubingx]

Thanks Lucas!

[@nadiyayasmeen3928]

Are you the same Lucas I follow on Twitter. Similar profile picture

[@1ucasvb]

@@nadiyayasmeen3928 Yes, that's me.

[@AJ-et3vf]

Awesome video! Thank you!

[@piyushsingh6462]

Heads off to you bro Amazing explanation

[@abderrahmanenedjadi7475]

You are really amazing, go ahead, you gonna be our new 3b1b

[@ayberkduzenli5412]

Coolest presentation of the good old N-S Equations. Here , have my upvote .

[@alio.e.g.8881]

Excellent video, thank you !

[@vcubingx]

Glad you liked it!

[@danielmohoushmand5544]

These are not the Navier-Stokes equations but rather the initial startup of Hagen-Poiseuille equation. You have forgotten the nonlinear convective acceleration term u⦁∇u on the left hand side, which is what this price is all about in the first place. This term is responsible for turbulence and the white water you’re referring at in the beginning of this video. It should be like this: ρ(∂u/∂t + u⦁∇u) = ∇p + μ∆u + F Or with material derivative ρDu = ∇p + μ∆u + F Or more commonly ∂u/∂t + u⦁∇u = ∇p/ρ + ν∆u + F/ρ Where ν = μ/ρ is the kinematic viscosity. It’s a great video though. Time consuming or not, I would seriously change that, because significantly different equations, more than million dollars to say at least.

[@arkie87]

I noticed this too. Thank you!

[@royburson4579]

Dude doesn't know what he is doing

[@ayaanshaikh9908]

What is your insta can we connect ?

[@gavingavinchan]

I see someone has taken continuum mechanics for fluids in grad school.

[@vivekfaldu]

Very good explanation 👍

[@studentemail8067]

Nice explanation, thank you

[@rackydichminky1751]

That's great... ur video and the equation

[@freddiemeinertzhagen]

Brilliant explanation thankyou

[@karankapur2245]

Too good Nice representation of the equation!

[@vcubingx]

Thanks!

[@dronemonkey2038]

Great video, well done.

[@johnthatcher2014]

good introduction video. Well done

[@qqn4531]

damn thanks to you I finally understood why div(u)=0 when a fluid is incompressible. Thank you

[@x_gosie]

I will be happy if you make a series about the 7-millennium problems, with this kind of visual representation.💕😍

[@moonout478]

it's wonderful! thank you.

[@JousefM]

Very nice one! I have derived the whole NSE as well as the Mass & Energy conservation on my channel to actually grasp the concept of where these equations come from a bit better. You did a great job in explaining the main ideas and problems under 10 minutes, props! :D

[@vcubingx]

Thank you! I was a bit worried that I may have squeezed it in a bit too much, but I think it worked out well!

[@daan6693]

Good video, first explanation I can understand. Have you used the 3b1b library? It looks very familiar.

[@byronvega8298]

Excellent video!

[@vcubingx]

Glad you liked it!

[@alexcheng2498]

He never misses.

[@vcubingx]

yessir

[@massilaitabdeslam1316]

Good video. Nice job!

[@vcubingx]

Thanks!

[@kofitesla1566]

Nice presentation. Please may I know how u did the animations and which software tool you used. Thank you very much.

[@johnerlacher9911]

The equations you are showing represent the incompressible Navier Stokes equations, where flow density is assumed constant (Mach < 0.3). This is already a great simplification of the physics and this subset of the equations will not apply to flow over commercial airplanes (Mach > 0.3) and certainly not to rockets (Mach > 1). The full set is comprised of 5 PDE's, conservation of mass (1), conservation of momentum (3), and conservation of energy (1). Solving these equations numerically by marching them in time from an initial flow condition is relatively easy and straightforward, yet it requires significant computing power.

[@notlegal99]

its like the butterfly effect. a small change in the system adds up over time and makes something we can't predict easily.

[@vcubingx]

Precisely!

[@slikclips2966]

I'm pretty sure that's the chaos theory you just described, this on the other hand shows small changes do add up but don't drastically change the outcome. Please correct me if ive misunderstood

[@slikclips2966]

The million dollar question is why small changes don't result in drastic outcomes overtime. I think it might have something to do with the correlation between the area said newtonion fluids are operating in.

[@slikclips2966]

We can predict streams via geography. Maybe aerospace is harder because of the vairing outside pressure and gravitational changes through a flight

[@notlegal99]

@@slikclips2966 where is the proof that small changes don't change the outcome drastically? i think the more time passes the more change will happen.

[@uzulim9234]

manim!!! thanks for this informative video. im a topology guy so it was a nice peek into pde world

[@vcubingx]

Glad it was helpful! I'd love to cover topology one day

[@uzulim9234]

yess maybe you could explain Hodge conjecture using simple geometric analogies about determining all shapes (~= homology classes) of algebraic varieties.

[@jeffgalef121]

That was fantastic. I wish the video was longer.

[@mattieohya]

When looking at Navier-Stokes the fundamental properties you are looking at are bulk properties and are impossible to define as a individual atoms. The infinitesimals are assuming a continuous fluid where there are no such things as particles. Think of density in the context of a particle, outside of the arbitrary area that defines that particle the density would be 0 and thus the system wouldn't be continuous. Rarefied gas dynamics is the feild of fluid mechanics where a gas is treated as a random assortment of molecules. And uses a variety of methods to figure out fluid flow when molecules are so far apart these bulk properties break down.

[@AdhyyanSekhsaria]

Amazing video!

[@vcubingx]

Thanks!

[@zenithalizesquads4873]

Fascinating

[@Doctor_Drew]

Oh most excellent video. I see you're using manim. I gotta learn it!!!

[@vcubingx]

Yes! Thank you!

[@hani8133]

great vid. thank you

[@prometheus7387]

Nice video!

[@vcubingx]

Thanks!

[@143mathematics]

very good. subscribed.

[@kanishka.s.jaeronauticalen8053]

Thankyou so much ❤️

[@GAPIntoTheGame]

2:05 Correct me if I’m wrong but an isothermal proces just means that the temperature remainins constant, not that there is no exchange in heat. In fact an isothermal process means there is no change in internal energy, which through the 1st law of thermodynamics entails that the work done by the system is equal to the heat gained by the system (I believe that was the correct phrasing of the first law given the change in internal energy is 0). So if there is work being done at a constant temperature there must be heat gained or lost.

[@vcubingx]

Yep you're right! I corrected myself in the pinned comment

[@GAPIntoTheGame]

vcubingx sorry didn’t see it.

[@redwoodenjoyer]

I've been working on this project since quarantine started and have made so much progress, so this video came a little bit later... but I was more interested in CFD and calculating things through code. Luckily, after soo many hours put into research, learning all this calculus stuff (currently in 10th grade so I had barely any experience with PDEs lmao) I finally got some C# code working with a Windows Form that allows me to specify the initial velocity, pressures for each cell and can tell me the next frame. Personally a great accomplishment. Something Ill definitely be putting on my college app for my projects during quarantine haha Thanks for making the video!

[@vcubingx]

Nice job!

[@joshuaallen8504]

Love this video

[@waynedeng9604]

you are a million dollar man, keep up the good work buddy

[@alexismisselyn3916]

When they say prove the solutions are smooth, does it mean that the solutions are smooth but we can’t prove it? As you said we can’t predict weather too many days ahead, so that means the solutions are chaotic but we haven’t proven that either? Can chaotic solutions be smooth?

[@gaeroot]

From what I understand (and don't quote me, its been a while since I've dealt with fluid dynamics), the problem with the Navier-Stokes equations is the issue of them being ill-conditioned. By that I mean that a small change in the input does not result in a small change in the outcome. This is important since with any system, a small input change should always yield a small output change, otherwise the reliability of the solver is questionable (the results should be reproducible, and near infinitesimal changes should not result in drastically different answers). Now while the the equations are basically glorified F=ma equations, which means that they are most likely the correct DE that describe the underlying physics, the problem lies in the fact that we simply don't understand or appreciate them enough. Also, remember that the real underlying physics is much more complicated. We can break down the problem to its core where we consider all the fundamental forces of the universe and the quantum effects between each particle in the fluid. But, this is meaningless because we want a meaningful compressed description of the physics, similar to how Newton's laws of gravity are a simpler version of Einstein's. I've solved the Navier-Stokes equations by hand in undergraduate classes for simple problems, and in these cases the equations are very well behaved. The solving process is actually very logical to the point where you realise that all you are doing is Solving F=ma. The problem comes down to turbulence, and the fact that the simple Navier-Stokes model do not capture this phenomenon at all. There have been very complicated proposals to the NS equations which take turbulence into account, but these are loosely based on analytical physics and more empirical solutions. Introducing this does not only create a more accurate solution, but employing some numerical trick also make the solution very stable. Also, there is also the problem of the DE itself. Its not simple to solve, and the numerical methods we usually employ to obtain approximation, are exactly that: approximations. So if you read the problem statement more carefully, you will realise that there is no straight forward problem that has to be solved. It's like the people didn't know what to set as the problem itself, which has become the problem. To essentially solve the millennium problem, you would need to come up with some form of proof that the NS equations are truly the underlying physics of a fluid (or not). Remember I mentioned the problem of ill-conditioning? Well even if that is true, that does not mean that the NS is BS, and the turbulence modelling tricks can make the solution very stable. However, these tricks are sometimes based on nothing more than: 'it works'. This is not progressive work and that is what the millennium prize tries to address. So answering the question in terms of your words, we don't know if the solution (real) is smooth. because of which we don't know if using tricks to make our modeled solutions smooth is the correct thing to do to obtain meaningful answers either. And upon finding out whether or not it is, we'd also like to know why? Essentially: solve turbulence, because nobody knows wtf is going on. I think that last paragraph addresses the question you had about chaos?

[@vcubingx]

Awesome reply Jay! Thanks for this

[@ghader2375]

Jay Raut i understand it very well

[@cboniefbr]

Great video! My only complaint is that the sound could be a bit louder...

[@kaustoovmitra1260]

I hear fluid mechanics, I click like.

[@jacobjohnson3726]

You know the subject is unimaginably hard if there’s no tutorial from our lord and savior the organic chemistry tutor

[@hoodedR]

Good stuff!

[@vcubingx]

Glad you enjoyed it!

[@jujoropo]

Amazing bro

[@mrx42]

Brilliant! You should get paid by the ministry of education for that! All faculty teachers should use your videos to teach their students (same for 3blue) Cheers, A physics student

[@vcubingx]

Thank you so much! This made my day for sure :)

[@varunahlawat169]

What a good video!

[@gorantrkulja7053]

Source of scaled and shaped flows accumulates heat and tension, so we cannot describe, or solve it, but we can fell it...

[@dosomething3]

Excellent.

[@tozrimondher4250]

I think that this equation is used to simulate fluids in 3D software like Autodesk’ 3D Max and Blender

[@joseftrojan7664]

Love this

[@ryans7536]

Good video!!

[@vcubingx]

Thanks!

[@razterizer]

Shouldn’t smoothness depend on Reynolds number which determines the level of turbulence?

[@smiley_1000]

How exactly do the dimensions add up here? On the left side, we will have acceleration by mass over volume, but on the right side we just have acceleration by mass. Shouldn't we divide the right side by volume too?

[@MuitaMerdaAoVivo]

Na mate, you just substitute the m's for rho, since that's the "infinitesimal" mass (i forgot the actual name from fluids 1). You can see he uses rho x g to express the weight force, instead of m x g. If you want a more acurate derivation, this video shows a classic textbook analysis. kzbin.info/www/bejne/hJvSfqJ-apaIkMU

[@abraarsameer9521]

I saw the thumbnail and thought 3blue1brown just uploaded a new video

[@lexanris]

At 4:45, the second Navier stokes equation has the term F as the external force term. I’ve watched videos from Numberphile and searched up the equation but I keep finding the external force term being either ‘Fp’ (just take the p as rho, density’) or ‘gp’, etc. could someone please explain to me which is the actual right one? Cuz if you jsut divide F by V, you will get density x acceleration in reality.

[@themoroccanbee6720]

Can you please tell me what is the relationship between this equation and lattice gas model?

[@thunder852za]

6:13 What about surface tension, evaporation, general acceleration.

[@ryantran8800]

Hey! I have a question, what does the first variable in the friction section mean? The u with a line going down on the left looking thing.

[@franciscoreyes7370]

The greek letter mu. In this context it is the viscosity constant. Its usually not a variable, unless you want to complicate the model.

[@haroldhousen3876]

The name of the professor that solved the Navier-Stokes equation is Dr. Gabriel Oyibo

[@gabrielcarranza6139]

Nice