Description and Derivation of the Navier-Stokes Equations

The equations of motion and Navier-Stokes equations are derived and explained conceptually using Newton's Second Law (F = ma).
Made by faculty at the University of Colorado Boulder, College of Engineering & Applied Science.
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[@mertgunes9854]

so this is what it feels like to be TAUGHT something *gratitude*

[@erickgomez7775]

There should be an organization that gives awards for best educational youtube videos each year. This one deserves an award.

[@ArpitAnand-yd7tr]

The amount of intuition conveyed by the instructor to the audience is absolutely unbelievable Thank you

[@AravindUkrd]

That's among the most meaningful 11 minutes I have ever spent

[@guilhermehambrusch7064]

Oh my god, what an incredible explanation. Congratulations.

[@adamchelchowski]

If you can explain someone the idea of something, it means you really do understand the matter of that topic. This video is a true example of how a really difficult topic may be presented to students and others in an extremely accessible way. Thank you for those 11 minutes.

[@fabiovezzari2895]

Most professors should start this way, before obtaining the formulas from the model and using them...make the overall concept clear and simple

[@diliprockz24]

That was really simple and easy to understand. Thank you!

[@sayyedahmadkhadem3321]

Truly enjoyed the way the instructor explained this phenomenal PDE! This shows how deep he understands the concept!

[@GaiaKnight11]

OH MY GOD. Sir, you have just OPENED MY EYES. Thank you!

[@smitkapadia8080]

Thank you for this amazing easy-to-understand breakdown of the Navier-Stokes Equations! Been trying to understand them since a while and this makes matters so simple :))

[@Junhyun-Park]

thank you so much!! It was the most beautiful explanation that I ever heard.

[@kacelamal4257]

the best explanation I have ever seen. You really made it easier to understand for us. Thanks a lot

[@noelgomariz3038]

This was great, now i challenge you to make a video on how to find a general solution 😂😂

[@adamfattal468]

Lol

[@siddharthkumar9349]

😂😂

[@sagarpatel3043]

😂😂😂

[@Nayan133]

Even original founders of these equations was unable to find a general solution. 😂

[@marcusrosales3344]

@@Nayan133 It's a Millennium prize problem so yeah.

[@thathsaranidiliniherathher2364]

Thank you so much for the wonderful explanations. Finally found a channel worth subscribing.

[@Eta_Carinae__]

Jesus, this was good. Been meaning to get into this for a while.

[@fslurrehman]

The best explanation of what Navier Stokes Equation is and how to derive it. Also a good correlation with a venturimeter.

[@MathsatBondiBeach]

Not bad for 11:17 mins. Euler's original work on fluids (in French - see the Euler Archive) follows this approach in substance.

[@ahmedhamadto8756]

Amazingly done, I have to say, I look forward to exploring your channel

[@marcelohurtadocastillo3982]

The best explanation of the Navier Stokes's ecuation that I have seen until now

[@tayyabrazashah2924]

same for me

[@arianaaraujofalcon7474]

Just beautiful, thank you so much! Perfect explanation!

[@durbhaganapati7227]

This channel has got surprisingly less number of subscribers The way of making things easier... just felt awesome

[@marc627]

One more comment of gratitude and praise for your teaching skills :)

[@yihe2806]

so cool, bro, it's a good method to show the explanation of these complicate functions

[@ProfeARios]

Great explanation!!! Thank you so much for sharing!

[@joshuachang4564]

beautiful explanation. thank you so much!

[@nicolaswiles3236]

You do a great job of explaining it, thanks!

[@AbDulRaHMaN-ox2yq]

Dude you just covered a whole unit withing 11 minutes, 😍 thanks man,,

[@anirudhsingh1729]

Really Sir, thankyou for such a good concise explanation!

[@joaopedrobarbosacoelho455]

Simple, clear, well done!

[@teachingengineering4205]

Very well done! Bravo!

[@cipaisone]

Excellent. I went through a couple of books trying to get what that all meant, beaver able to get it. You just explained so simple even the donkey I am could understand it!

[@melvingeorgechittilappilly1625]

Amazing video!!! You taught what college profs couldnt in an hour at least

[@007aha1]

WOOOOOOOW THIS WAS TOO GOOD!

[@chenjane4814]

really good explanation and everything! THX a lot !!!

[@kanakaraju1825]

thank you for this wonderful easy explanation this sure saves me for my final exam

[@tivnator]

Could you please explain how the constitutive equations for newtonian fluid are built? Please. You made an amazing explanation!!!

[@manuabasto6325]

This is the best explanation ever

[@akshatjhawer9155]

you really have a very nice way of teaching compared to my professors 😊😊

[@tayyabrazashah2924]

WOW, an amazing way to teach.

[@AdityaMotla-yt]

very well explained!

[@pawankhanal8472]

very nice explanation!!

[@SeppyDawg]

Beautiful, thank you

[@KanishkRajput-cm5lg]

really awesome explanation!

[@guidedmissile4154]

thank you for the explanation

[@AJ-et3vf]

Great video! Thank you!

[@aidanschulze9799]

bless your soul for this video

[@magicm8ball558]

Guys this looks really easy I’ll solve it and keep you guys updated!

[@oliversadek2016]

Thank you so much. it is simply illustrated

[@moonout478]

It helps! Thank you!

[@AnindyaMahajan]

Neat explanation, but I think you should have went into the derivation for the relationship between stress and velocity profile as well. Just for the sake of completeness.

[@allandavis6116]

That's the hard part. It follows from Newton's Law of Viscosity but it's a mess. That is, you have an expression for shear stress, and the fact that the fluid is isotrophic, and you have to 'derive' the expressions for the normal stresses from that alone.

[@dipsomania0]

I like this video!

[@wyattb3138]

I’m in high school and you helped me understand this. Thank you.

[@ib_aaravrakesh6830]

Nice seeing another high schooler interested in advanced math, cheers dude, just finished calc 2, doing some calc 3 stuff currently and exploring these concepts.

[@noobstudios7707]

Am also a high school student. To be honest I'm a tad confused by a few concepts since I never took physics (self-studied), but hey. That's why we're here, eh?

[@m.nahidulislam2498]

Thanks a lot .

[@brilliant1863]

you're a LEGEND!

[@atillaus9168]

great explenation

[@iitian123]

Very nyc saved a much time

[@Gealamusic]

Thank you so much :)

[@beshoyemad6587]

Good job tho for such explanation in dat short time and i loved that skip in writing btw Which software are u using for such work?

[@ALIRaza-MCE-]

Thank you Sir ❣️💖❤️❤️❤️❤️

[@JamesVestal-dz5qm]

That's why I taught me freshman to lable axes and units, because the more graphs you know the easier it is to learn new graphs.

[@__h.a.r.s.h.a__]

Wonderfull tq very much

[@pstark4]

Thank you for showing the stresses on all 6 faces, usually descriptions of tensors only have three faces(9 terms, 3 faces), which has confused me. Do you have a link to derivation of stress equations for a newtonian fluid at 9:22? Question: my understanding is that we use the material derivative as an equation between Lagrangian and Eulerian descriptions. If so, does the volume element move, do we follow it through the narrow part, or does it remain in a fixed place? sorry if the question isn't well stated.

[@vedhaswalke4406]

The material derivative portion is the right hand side of this equation(excluding rho). It describes the total acceleration of a fluid particle(Lagrangian description) using the velocity field(Eulerian description).

[@arslansadiqe6615]

when putting the value of sigma_xx into the equation we got for density time acceleration in x direction,there is a factor of 2 with (mu*del^2u/del_x^2). In the next step that factor if gone. Can you please explain, how did you factor that out?

[@user-yd7rc3cc9n]

Thank you

[@nidalal-sayed4019]

The beeeeest explanation

[@pradyumnchiwhane846]

Amazing

[@JamesVestal-dz5qm]

I was always in the top 3 percent of standardized math testing growing up in minnesota.

[@erfuniti3570]

Thank You. How could I learn about derive Navier-Stokes equation in cylindrical coordinates from Cartesian ? Is there any book explained it?

[@poisonpotato1]

Does this account for summing the moments on the element?

[@rinzan]

Nice explanation! Shouldn't the velocity at steady state at the middle of constriction reaches its maximum. I mean points 1 and 3 have the same velocity and point 2 have the maximum with left and right of it being the acceleration and deceleration respectively. I may be incorrect.

[@deborahgardner6786]

Important to note: This only works under the assumption that density fluctuates very little or not at all ( rho = rho_0 + rho' with rho'/rho_0

[@kylenemeth6252]

Well f*@%#%! done, thank you.

[@gustavocortico1681]

Please please please talk about the constitutive relations!

[@marianodeanquin]

estaria bueno que pusieran esto en cilindrica y polar...gracias colorado

[@JamesVestal-dz5qm]

You could memorize shear stress and substript definitions by linking them with dot and cross product from physics right hand rule.

[@ShadowZZZ]

Great explanation but I have 2 questions: 1. How exactly is the chain rule at 2:14 applied at u(x,y,z,t,)/dt in order to get that differentiative term? 2. Where do these terms at 9:15 come from? I know you said you won't cover it in this video, but I'm still curious because I want to understand every aspect of it.

[@TufanKirmaci]

+1

[@muhammadsaid4654]

Newtons law of viscosity

[@RumOwnesT]

4:50 I didn't understand the low, how did you get that?? Please replay, how can be m*gx = ro(dxdydz)*gx

[@mrcannotfindaname]

(dxdydz) is simply the volume of this infinitesimally small cube of length dx by dy by dz. So density times volume equals mass.

[@mikethe1wheelnut]

this is interesting. I can't quite tell whether it's narrated by an ai or not. I think it is because it's just too perfect. I have trouble believing a human could be that perfect.. perfection, in this particular case, is a good thing.. 🙂

[@theInternet633]

I think that the equations look a lot nicer and more concise in vector notation but otherwise solid video

[@Ottmar555]

Yeah. Besides this is the incompressible and constant physical properties version, which wasn't specified in the video. But to really get there, you usually need to get into tensor calculus, which I think may be outside the scope of this video.

[@67-abdulrehmanshahzad20]

Solve this in terms of cylindrical coordinate system

[@MSuriyaPrakaashJL]

Good explanation sir but I couldn't understand those equation for Newtonisn fluids which came out of nowhere and helped us to prove it

[@jordifrias8829]

Amazing work. I am glad I found this. But I have a remaining question: Do you have any reference on how to get the sigmas and taus in 9:24?

[@holmessherlock5106]

hello, did you find any reference for your question? it happens that i have the same remark, thank's

[@jordifrias8829]

@@holmessherlock5106 No, Sorry. It was not very important for me and I did not made the effort.

[@holmessherlock5106]

@@jordifrias8829 thank you for your answer, I've just found a lot of references. For info, all we need to do is to search in google using the key words: "constitutive relation for a newtonian fluid"....

[@bh-rf9dd]

You guys should definitely look up the derivations for the constitutive relations to stresses. The derivation is quite complex but is also quite beautiful. It deals with tensors up to the fourth order I believe.

[@ethanmartin2781]

2:10 I thought you use the chain rule not just because velocity is also a function of x y z, but because velocity is a function of x(t), y(t), z(t), namely, u(x(t), y(t), z(t), t)?

[@patrickamstad5091]

Yes that is true, because remember, we are describing here how the momentum changes of fluid particle that travels with the flow; therefore the coordinates x, y and z of the fluid particle are functions of the time t; and that is the reason why we have to take the chain rule. What he said in the video is incorrect

[@benjaminsmus8553]

@@patrickamstad5091 Thank you

[@gergodenes6360]

For differentiating multivariable functions, it's useful to always expand everything out. Assume there was some other variable h in there, that we don't even know if it depends on time or not. If we use the chain rule, and it turns out that h is independent of time, dh/dt will just be 0 and we're good to go, we're at the same conclusion. So really, he's right: It has multiple variables, so we use the chain rule.

[@atmonotes]

if you didn't apply the mass continuity, would the equation directly become the compressible navier stokes?

[@hanafieuwais3110]

Anyone knows any video for the derivation of the last part?

[@sabarirajansingaravelu886]

how did we get that expression for sigma xx = -p + 2(u)(du/dx) ?

[@trik3002]

Taught in adv fluid mech in m tech. Theres a porf R J Garde I guess, u can read his books of afm for explaination. This is a very long derivation, we used to get this for at least 10 marks to derivd in our exams

[@ntorres1152]

goated video

[@rahulmenon4003]

I have only one thing to say... Thank You

[@hendemad8757]

please can you explain how to slove problems

[@honkhonk8009]

I hope I get to the point in life where I actually get to understand this shit lol. Im rn in highschool, but about to head off to college. My grades were basically fucked by the pandemic, so im gonna hopefully be able to transfer off community college if possible.

[@siddharth_20]

Notations for shear forces are a bit confusing... Does the subscrits following the pattern τyx meaning y face & along x direction...

[@jakee6275]

Little proviso this derivation is true for liquids but not necessarily gases as the second viscosity term is generally assumed to be zero for a liquid, loved the explanation though

[@Karim-qb6uj]

Nice !

[@LouisLiuMusic]

still don't get it. why is g on the x direction?

[@gabrieldigiuseppe3106]

It´s just a generalization, you don´t actually have a gravity force in that direction, so when you do the math, you consider it to be 0.

[@pranavsateesh3002]

Take a case when the control volume is at an angle to the x axis, then there would be a component of g acting on the x axis.

[@itswhoppertime]

Imagine you had fluid flowing down an inclined plane, and you chose your coordinate system to that the x-axis coincides with the incline and the y-axis is normal to the incline, then there will be components of gravity in the x and y directions.

[@aleenalaique6618]

Is this the Conservative form or non-Conservative form of Navier stokes equation?

[@stex3849]

tag a friend

[@pablofts1569]

This is the best fakin explanation! yAAAAA!

[@veritythomas5682]

Anyone got a derivation or link to one in the y and z direction please?

[@deborahgardner6786]

He provided general expressions for each. It's the same process.

[@pochinapeddisaibhargav3637]

Lot more simpler than using Reynold's transport theorem

[@moonman239]

Why does the function u depend on 3 coordinates, and not just one?

[@alexandertownsend3291]

It is because the Navier Stokes equations describe the flow of fluids like water or honey in 3 dimensional space. There is one coordinate for each of the 3 coordinates of space.