Derivation of the Navier-Stokes Equations

🧠 Science Academy: courses.jousefmurad.com/
🌎 Website: jousefmurad.com
In this video, we will derive the famous Navier-Stokes Equations by having a look at a simple Control Volume (CV). A small introduction to classical mechanics at given in the beginning. Comment, Like & Subscribe!
► Part 1 - Derivation of the Mass Conservation Equation: • Derivation of the Mass...
► Part 3 - Derivation of the Energy Equation: • Derivation of the Ener...
► PDF of the presentation: / derivation-of-34090956
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🎵 Joachim Heinrich - Words
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#NavierStokes
#Physics
#Engineering
Time Stamps
----------------------
0:00 - 1:38 : Intro to Classical Mechanics
1:39 - 3:13 : History of the Navier-Stokes Equations
3:14 - 4:24 : Recap - Fundamental Equations
4:25 - 6:26 : Fundamental Equations of Fluid Mechanics
6:27 - 7:10 : What is Missing? - Normal & Shear Stresses
7:11 - 7:30 : Body Forces
7:31 - 8:04 : Normal & Shear Stresses - Visualization
8:05 - 8:45 : Assembling of the Equations
8:46 - 8:58 : Simplify the Equations
8:59 - 9:27 : Questions that need to be answered
9:28 - 10:46 : The Stress Tensor
10:47 - 11:07 : Pressure
11:08 - 11:34 : Separate Stress Tensor
11:35 - 11:40: Preliminary Equations
11:41 - 12:10: Stokes Hypothesis
12:11 - 12:49 : Product Rule for RHS
12:50 - 14:20: Final Form of the NSE
14:21 - 15:12 : Substantial Derivative
15:13 - 15:49 : Lagrangian vs. Eulerian Frame of Reference
15:50 - 16:12 : The Navier-Stokes Equation (Newton's 2nd Law of Motion)
16:13 - End : Outro

[@JousefM]

Thanks for watching! Let me know in the comments what you liked the most - keep engineering your mind! :) 🎓 My Science Courses - courses.jousefmurad.com/ ✍️ Latest blog posts: jousefmurad.com 📥 My Newsletter - jousef.substack.com/ Time Stamps ---------------------- 0:00 - 1:38 : Intro to Classical Mechanics 1:39 - 3:13 : History of the Navier-Stokes Equations 3:14 - 4:24 : Recap - Fundamental Equations 4:25 - 6:26 : Fundamental Equations of Fluid Mechanics 6:27 - 7:10 : What is Missing? --> Normal & Shear Stresses 7:11 - 7:30 : Body Forces 7:31 - 8:04 : Normal & Shear Stresses - Visualization 8:05 - 8:45 : Assembling of the Equations 8:46 - 8:58 : Simplify the Equations 8:59 - 9:27 : Questions that need to be answered 9:28 - 10:46 : The Stress Tensor 10:47 - 11:07 : Pressure 11:08 - 11:34 : Separate Stress Tensor 11:35 - 11:40: Preliminary Equations 11:41 - 12:10: Stokes Hypothesis 12:11 - 12:49 : Product Rule for RHS 12:50 - 14:20: Final Form of the NSE 14:21 - 15:12 : Substantial Derivative 15:13 - 15:49 : Lagrangian vs. Eulerian Frame of Reference 15:50 - 16:12 : The Navier-Stokes Equation (Newton's 2nd Law of Motion) 16:13 - End : Outro

[@ahsanhasnain29]

I need help in NS problems. I need help its urgent

[@jurivolodin1061]

One of the best explanations I've seen about Navier-Stokes!

[@JousefM]

Thanks a lot!! 🙂

[@user-bz4ez5mn7w]

Great job. Very explicit approach in explaining such a complex topic. Thank you so much

[@verygood6625]

This is the best video on the N-S equation, well documented, explained and aptly presented. Thanks for the great work, it helped a lot.

[@JousefM]

Very flattering, thanks a ton! :) Feel free to suggest other topics that interest you!

[@apriliasaktikusumalestari4132]

Thanks a lot, loud and clear

[@EddickC2D]

Great job man! Would be nice to see in future the explanation of DNS approach, what is this, how many equations do we have, why there is no turbulence modeling, what is Kolmogorov scale and hence LES, DES associated. May the stokes be with you!

[@JousefM]

Will come for sure!! :) I will mix FEM and CFD from time to time and put in some other entertaining videos to reach a broader audience. Suggestions are always welcome and the points you mentioned are already on my to-do list! ;)

[@villagefood6335]

Brilliant, clear natural presentation step by step, thank u so much man...please keep up.

[@JousefM]

Thanks mate! More coming over the holidays hopefully :)

[@Youtb8888]

Great video! Hats off to you!

[@hardiksharda9673]

Amazing! Thank You!

[@FranzBond007]

Great Video, I really appreciate this, a good contribution for Science. Thank you!!

[@JousefM]

Appreciate your comment Franz! Encouraging words :)

[@sriti_hikari]

Chill music. Love it.

[@JousefM]

[@tassioleno808]

Great!! Many thanks from Brazil!

[@JousefM]

Obrigado Tassio! Hope I wrote it correctly 😄

[@nitinnegi6797]

Amazing explanation of such complicated equations.

[@JousefM]

Thanks mate!

[@JamesVestal-dz5qm]

I love the soothing music at the start of the video!

[@JousefM]

Thanks! 🙂

[@MrDJPAPAYA]

Thank you very much!

[@alotte_of_anton]

Really helpful breakdown of the equation, thank you!

[@JousefM]

Thanks mate!

[@sadiaamin2918]

Clear description on this topics. Thanks. Wants more videos.

[@JousefM]

More coming soon :) Magnus Effect Video going live tomorrow!

[@ninadmehta3654]

Thank you. This video was very lucid in explaining NSE.

[@JousefM]

Thanks 🙂

[@vinaykumardaivajna7276]

Ohh god that was so elegant!!

[@rohitsuryawanshi2229]

You are on fire 🔥❣ amazing work again ✌🙌

[@JousefM]

Thanks as always my friend 🙂

[@QMCESAR]

Thanks for your amazing explanation

[@JousefM]

Thanks for watching mate! 🙂

[@tamtse6131]

really love how you start it

[@JousefM]

Thanks 🙂

[@albertoconde5404]

Equation of Navier Stokes aplicación heat exchanger in Chemical Engineering. Very good

[@parvathinarayanan6348]

It is amazing how the creation and it's function has been going on forever. And how the gross and subtle energy and forces work together. Including we humans born and live and think and understand. Whether we understand or not it still operat s beautifully. From an ordinary person. God bless you

[@bosonglin7462]

Good job! Very helpful

[@JousefM]

Thanks a lot :)

[@sabkabro2322]

I do Linear Stability analysis, Navier Stokes equation is my bread and butter

[@JamesVestal-dz5qm]

I took century college classes while working at papa John's and that motivated me to learn more about food chemistry and mathematics.

[@KapilVetrivel08]

Great Video :). It would be great if you can also upload a video on using the Kronecker's delta to derive the Stokes Hypothesis.

[@JousefM]

Thanks mate! Might come in the future - currently working on my turbulence video. 😉

[@riza.siregar]

interesting, that's easy to understand..thank you

[@JousefM]

Glad you like it Riza :) Feel free to share it with your friends!

[@prof.onkarsalavi7143]

very good explanation

[@JousefM]

Thanks 🙂👊

[@sheejuchandran8345]

excellent video.. thank you very much..

[@JousefM]

Cheers! 🙂

[@carmelpule1]

What I am admiring about this video is not the derivation of the Navier-Stokes Equations themselves, but how does an education system prepare children and a good proportion of the people in a nation, to follow and understand the states and the mathematical symbols and what their operations actually mean in real life. Most people I know, tend to be led by their emotions and not their processing powers, and I feel they miss out a lot in life, not being conscious of the unseen and silent phenomena around them. I am old enough to remember World War 2, where many young pilots at the age of 18 were put in a cockpit of a fighter aircraft after only 12 hours of flying tuition time, and they themselves were so proud to think that they were able to fly, but little did they know that flying a fighter aircraft, to try and hit a target, moving at an oblique angle, took more than knowing how to fly, as knowing how to detect the relative location of heavy objects with mass (P), the relative first-rate of change P (dP.dt) and the second rate of change of P ( d2P/dt2). ( at least that information) Now when that applies to three-dimension, those operations will increase to nine operations and if the process needed an integral as in the case of existing side wind on an aircraft, then the human pilot needed to know and operate with higher rates of changes rather than using two separate eyes to locate the distance of a target ahead of him. Unfortunately, most young inexperienced pilots in WW2 handled the location of the target with respect to their own position and they corrected the aircraft or fired their guns when the target was in line with their sight, forgetting the first and higher rate of changes of masses involved in the state of affairs they were in. Many pilots died because they knew how to fly but were not conscious of how many rates of changes and integrals one need to observe and process to solve the problem at hand, and many emotional people, including artists, and religious people and many teachers of social subjects, live in a world of locations and distances, rather than the rate of changes, and Gradients and Divergences and Curls. Many people are not conscious of the following. 1. If one looks at a distant Patch of the area with the same colour (F(xy)) all over, then within the patch there is an intensity of colour, but there are no rate of changes with respect to the horizontal sweep nor the vertical sweep in the patch, dF/dx = dF/dy=0 2. When it comes to, say, a distant patch that has two colours or four different colours as in a flag, Then one needs to be conscious that there exists a rate of change with respect to the vertical and horizontal and what is more, the first-rate of change, dF/dx could have a rate of change with respect to the same axis x or the other axis y. as d2F/dx/dy. or f2F/dx2. 3. In fact, using such higher derivatives, one can come to a situation where THE CENTRE OF ATTRACTION OF THE SCENARIO IS located by such differentiation with respect to the reference axis, which in two dimensions is not too difficult but in three dimensions, it is not so easy. 4. Such operations with the detection and processing of a higher rate of change, on a work of art, could result in locating the centre of interest of the art piece being looked at. where the rate of change could be in the intensity of adjacent hews of the same colour or the difference of adjacent colours, or both. Even contours and trajectories of streamlines describing statues, or vector line drawings using the same colour can be processed to find their higher rates of changes as a vector, and within their contents, there would be an emotional interest to a human being rather than a mechanical acceleration vector!. Architects and fashion designers, interior decorators, car designers, and musicians and religious people do not really know that what they do is merely to select in their products, the right rate of changes at the various locations and time sequence they provide to the public whether in visual static or dynamic movement, or an oration or music in time. With Navier-Stokes equations, it is not merely the rate of changes of location of the mass particles and their first-rate of change ( velocity) and the second rate of change( accelerations) but they include a characteristic of separation of particles, had the rate of change of position be too fierce. From this point of view, it is better to handle Maxwell's equation to deduce what goes on in waveguides and antennas. I would be so proud to be able to handle the cavitation or boiling of water in immersed or surface water propellers, but I can only admire such work as your work. Anyway, coming back to how much, the people in all professions are conscious of THE HIGHER RATE OF CHANGES and INTEGRALS, they need to detect, combine and process to solve such a problem as a pilot flying in a heavy mass aircraft, which is just a complex distributed mass particle, located at a point and having velocity and acceleration, trying to hit a target having its own mass and location, velocity and acceleration, all having oblique vectors with respect to the first aircraft. Again many young pilots without a mathematical background just fired at the target, or when aiming to land a heavy aircraft at a distant runway, they corrected the error when the aircraft was in line with the target, forgetting that the mass of the aircraft has a drift velocity. acceleration and yaw vectors, and so it will drift past the target to oscillate and yaw about the line of sight. What is so remarkable about Navier and Stokes and Maxwell, is that they need to have the ability to process the higher rate of changes with respect to all linear dimensions and time. and this was 200 years ago when their technology had only paper and pencils to write their very meaningful mathematical symbols which are only images and shadows of the real deeper process of real-life processes. Their mental powers were remarkable. Jousef Murad, Sir, in your voice, I detect tones that you are a young man, and you are so conscious of the higher operations that go around you, and unlike many people, who are only conscious of locations of distant particles in a portrait or in a statue, or a static word in a dictionary, you can handle in your mind the contribution of the dynamics of the higher rates of changes and the integrals at different locations in a MATRIX of MASS PARTICLES uniquely connected to act like close membranes with distributed masses, and if the rate of change at any location is any higher than a certain value, then the membrane's adjacent connections will fail and tear up to make the particles in a zone keep, their mass, but not their connections! After all dry sand particles would not behave like fluids. I hope you bear with me, as when I saw this video, I just wondered what we can do so that children and most people in any emotional and social professions, can appreciate the mind of the author who did all this. Sir, my hearty congratulations, and I wish you and your work would be instrumental so that all people in the future will appreciate deeper values in life, which can even save their life, as it would have done to those young pilots in WW2, who thought that they could fly, but did not know how to process the " knowing how to wait for at least three glances, spaced by the short duration of time, to detect three separate distances of a target, and from that, through double differentiation, they could calculate the velocity and acceleration of the target and that would have predicted where the target was going to be after a delay, rather than die in the process of just thinking that it was enough for them to know, and be so proud, of knowing how to fly a fighter aircraft. Thank you for your elegant presentation of the silent and unseen processes in a function that everyone can see daily, but not really understand, Again Congratulations. This situation shown in the following video is a case when the young pilot did not know how to handle the higher rate of changes with respect to space dimensions and time. There is more tacit content to this pilot's case than what the videos show in the preparation of the pilot. kzbin.info/www/bejne/Y524l51_bsiHb80 kzbin.info/www/bejne/eWmsaWlracyogZo

[@JamesVestal-dz5qm]

What beautiful intro music!

[@JousefM]

Thanks! 🙂

[@MarHerH]

Thanks for sharing your knowledge and time. Please do OpenFOAM videos!

[@JousefM]

Planned to do a mini course on it once I have more subscribers as this will take ton of time. Please consider sharing my channel with your colleagues and friends and make them request the OF course then I see how much you need that beauty to come out 😄😄

[@YitzharVered]

Thank you! My textbook is great, but it has a very confused explanation of navier stokes. This is great!

[@JousefM]

Thanks :)

[@JITISH1]

This is a brilliant video man. Keep up the great work. Is there any chance of deriving the Shallow water equations from the Navier-Stokes equation?

[@JousefM]

Thanks buddy! I will put it on my to-do list and maybe implement it in Python/Matlab but I have so much stuff on my to-do list at the moment :D I will still track it :)

[@gabrielpisciottano6077]

You must include Coriolis Force as a body force in order to go to SW Eqtns .... .and ussualy simplify or disregard viscous force terms .... Saludos desde Montevideo, Uruguay, y drisfruten la Mecanica de Fluidos !!

[@geovanylopez2980]

Congratulations for your amazing video. I'm deeply interested in OpenFoam, I'd llike to know if there will be OpenFoam's beginner tutorials videos or something. Your videos have helped me in mi CFD classes, thank you. Another question, which book do you recommend for CFD Finite Volume Method?

[@JousefM]

Thank you very much Geovany! It is already planned in my head 😄😄 We will start from scratch how to install OF and how to manually create a domain that can be simulated :) I would tie this to a community goal. Let’s say if we are at 6.000 subs I am working on that and will also create PDFs for my Patreons that they can download as a OF guide - does that sound good? Will post some books in this thread later on :) Edit: I think I will just create a PDF for CFD & FEM each that you guys can download from my homepage. Give me some time to create it - should be done in less than a week!

[@jibeneyto91]

Very good video. Reminds me of my Fluid Mechanics course at the Technical University of Madrid, School of Aeronautical Engineering.

[@JousefM]

Thanks a lot, appreciate it!

[@erwinphillipcabanban5582]

Hi, can you do a vid on wall functions? Im still in the process of understanding cfd and so far your videos have been helpful

[@JousefM]

On my list Erwin :) have a deadline for my thesis around the end of March. After that I’ll have a bit more time.

[@dakirmossadeq1675]

Great video! I just have one question, I would like to know why an additional velocity is multiplied to the momentum equation? It’s the part you show at 5:36.

[@techlead3743]

Damn, that's a good video!!

[@JousefM]

Thanks mate :)

[@taiwoadedeji3232]

Great job. I would like to know why the value 2 disappered.

[@JamesVestal-dz5qm]

I think Logan Casper would enjoy learning about the navier stokes equations applied to the kitchen sink.

[@nowfinnizam9918]

What are the 7 unknowns in viscous flow equation??

[@tharathepphiboon6207]

Dear Jousef, I'm interested in "Stokes Hypothesis". Could you make a video and tell more a detail about stokes hypothesis. I would like to know how they can derive those the eqs. to became the eqs. at 12:00 and put them into the momentum equations. Thank you

[@JousefM]

Thanks for letting me know Tharathep! Putting it on my to-do list and see what I have time for that :)

[@sktyt2012]

Thanks

[@JousefM]

Thanks for watching! Feel free to share with your peers who might benefit from it :)

[@user-cu7fg1nr8f]

at 13:57 at the end of the page for the vector version isnt it suppose to be w.d/dz instead of w.d/dx

[@JamesVestal-dz5qm]

I'm getting better at remembering pathlines and streamlines.

[@rat_king-]

small thing: Symmetry condition is technically ZX=-XZsine(theta) The crossproduct is anticommutative.

[@mesharyRD]

Man you are a legend, I really like what you are doing and enjoy every bit of your vedios. I'm really interested in compressible flows and thier behavior, if you can make vedios about that I will be very happy. Thanks

[@JousefM]

Thanks a ton for your kind words! :) Will come in the future in a bit more detail. For that we need to derive the energy equation which will hopefully be finished soon - then we have everything we need to work on harder problems! 😎

[@JamesVestal-dz5qm]

Logan should take some math classes from century college where I used to tutor math. He'd learn some interesting math that applies to every day subway life.

[@deepaksharma8337]

Hats off jos.... I understood some percentage but have to open my vectors calculas... To get it in apt way.... Suggestion from my side... Can you include graps Or any pictorial slide or animation so that we can understand it's better and also remember it... Fab work 💼.. Bro keep it up... Waiting for the series.... 😊

[@JousefM]

Thanks Deepak!! What I tried to do for this video is to give each term a color at the last slide to make it a bit easier to digest and remember it in terms of Newton‘s Second Law of motion. I believe if you remember it that way it’s very easy :) Also let me know what exactly is not clear and I will see how I can implement that in future videos!

[@deepaksharma8337]

@@JousefM working on my doubts first ,as suggested... If still I will face issues I will address it in a format... You concern is really appreciated😊

[@tentalks4287]

Wow....awesome explanation dear Pls upload Reynolds transport theorem

[@JousefM]

Thanks a lot! Hopefully in the near future 😌

[@JamesVestal-dz5qm]

Remember we have a moral duty as Americans to advance and teach knowledge!

[@JousefM]

Im german 🤓

[@richardjiang]

Hi, Jousef. Great video! After removing 2/3▽v because of imcompressive fluid (13'06''), how did you derive the △v term? Please could u elaborate a bit about it? Thanks!

[@MuitaMerdaAoVivo]

For example: for the x direction, after that you can apply the partial derivatives to the terms d(u)/d(n) where n is (x, y or z). You are left with d^2(u) /d(n)^2 for each n, and other derivatives like d(d(v)/d(x))d(y), since v is constant in x, this derivative equals 0. After that you can join everything in the vector form with △v.

[@wojciechangielczyk3846]

Hi Jousef. Thanks for the nice video. Question about the slide at 8:44 and the next one. In the first-mentioned slide, the time derivative contains "x-component" of the velocity (u). Then, in the next slide the eqs. for separate direction contains instead of u a proper velocity component. Thus, the time derivative in the first-mentioned slide should rather contain the velocity vector itself instead of its component u. Right?

[@JousefM]

Hi Wojciech and thanks for your question! The equations are correct as we will need to derive the equations for 3 directions. The slide at 8:44 is only for the x-direction to make it a bit easier to understand. The slide that follows is then for all three directions. Maybe this is confusing because in the general form of the NSE we have vec(v) as a general naming for all the three components but if we have a look at the equations separately we would have to specific which direction we mean. Is that somehow clear or did I miss your point here? Maybe there is a mistake that I do not see. Also checked this with other sources and the equations seem to be right :) Best, Jousef

[@wojciechangielczyk3846]

@@JousefM, Ah, OK! Now I see. Thank you for the fast answer.... but I will be annoying (sorry): 5:00--> you said that def. of mass is (∂ρ/∂t dx dy dz) but since (dx dy dz= dV) then (∂ρ/∂t dx dy dz = ∂mass/∂t) consecuently (∂ρv/∂t dx dy dz=∂momontum/∂t. Right?

[@JousefM]

@@wojciechangielczyk3846 That's cool, no worries about asking. I wish more people would have the courage to ask more :) So Density = Mass / Volume, thus Mass = Volume * Density - our Volume is dx*dy*dz and the density is indicated by rho! What's missing now is the velocity which we put into the brackets, so we are fine! Your focus might be on the time derivative if I am not misunderstanding it. You can see it as some sort of momentum change per time. We will also see in later episodes what simplifications we can make for the NSE so that in case of a steady-state system, time derivatives are not interesting for us anymore. All in all: You are technically correct! The slides (or better...I) should say "Momentum Change per Time!" because the definition of the momentum is still correct but my wording was probably incorrect! :) Thanks for that questions ;-)

[@wojciechangielczyk3846]

@@JousefM You're on fire! Thanks again for the fast answer. I have also issue with the stress tensor symmetry ... I've googled it and found that "if the continuum body is in static equilibrium (...) the stress tensor is symmetric". However, we consider the flow of the continuum body thus this condition is not satisfied... saw also some quite long articles on this subject (I leave it for later... maybe when retired). Thank you for your hard work and have a good one Jousef!

[@JousefM]

Getting back to you soon! That’s a very good question and is indeed not trivial :) will get back ASAP!

[@ahmetegesahin4761]

At 4:54 shouldn't be d(momentum)/dt = [d(gv)/dt ] .dx.dy.dz thus that equation is equals to force not momentum.

[@federicopagano6590]

5:14 momentum should be define as the change in time of ( the mass(density) times velocity)

[@JamesVestal-dz5qm]

Drinking soda through a straw is an example of navier stokes at work!

[@14mahreen]

At 5:08, could you please explain how does mass equal to derivative of density w.r.t to time.

[@Potatochips77105]

I have a problem at 6:11

[@ubaidmundewadi6368]

We want videos on turbulence modelling

[@JousefM]

Will come Ubaid! :) Invite your friends as well who might be interested in turbulence modeling.

[@ubaidmundewadi6368]

@@JousefM yeah sure sir

[@walterisraelmoscosozarate8768]

You are awesome

[@JousefM]

Thanks :)

[@tygodankers6526]

3.14k subscribers, coincidence, I think not!

[@JousefM]

LOL! What a good observation 😉 now it’s 3.143 and it used to be 3.141 2 minutes ago... noooooo! 😂

[@nourelislamlounis3328]

thx man long time ago and i want to study CFD from you but am really lazy these days i hope i can break the damn chain and study seriously again

[@JousefM]

You're welcome! :)

[@alejandroduque772]

I don't understand how you get from the eqs at 13:01 to 13:08 . Focussing on the x component equation. Why does the v_(xy) and w_(xz) disappear, and what happened with the 2 of the u_x term. Also, great video.

[@JousefM]

Are you talking about the derivatives? If you want you can send me the equation on paper via Instagram and mark the parts of the equation. There are some assumptions going into the simplifications like constant viscosity as well as the continuity equation which result in the simplified form of the NSE. For that a bit of knowledge about the so called "Stokes Hypothesis" can help which I have not explicitly derived here but can do that in a future video.

[@ag_008]

Thanks. By the way, which book did you refer?

[@JousefM]

My favorite one which also used to prepare for tutorials explaining my students how things work is the one from Ferziger and Peric. Have a video on books in CFD where I mention it :)

[@amrsayed6677]

للاسف فاتنى دراسة كثير من هذه العلوم الرائعة ...

[@JamesVestal-dz5qm]

Navier stokes quantum mechanics and quantum mechanics are the god in physics.

[@dennisestenson7820]

14:20, hmm. The laplacian is the divergence of the divergence, no? If the divergence of v is zero, doesn't the laplacian also have to be zero? The divergence of zero is zero.

[@kunchevijayalakshmi7771]

nice explanation sir can you send this in the form of ppt

[@JousefM]

Thanks! The slides can be downloaded from my Patreon page at a certain tier. Feel free to check out the links in the description - cheers :)

[@iansanderson6207]

Wow look at , that tanasity ✌

[@JousefM]

Appreciate your comment Ian!

[@gabrielpisciottano6077]

Lean las Notas de Mecanica de Fluidos del Profesor Julio Borghi de IMFIA, @ Facultad de Ingenieria, Universidad de la Republica (@Montevideo, Uruguay) y van a dntfnder todo esto de forma mas natural, perfecta y comprensible. Sus Notas han sido publicadas por el CEI (Ofic. de Publicaciones), o pidan las @ IMFIA. Si la usan citenlo.

[@muhammadsuleman-bf5oh]

Good

[@JousefM]

Appreciate it Muhammad!

[@faruqueazmiallahsubkoeakau7386]

One more linkedin connection added in your subscribers😊😊

[@JamesVestal-dz5qm]

Just like the momentum balance that defines your kitchen sink.

[@subramaniamprajesh]

I am sooooooooooo jelous on your knowledge... I'm sorry for that but you are genius

[@JousefM]

Thanks bro! Not a genius at all, also had to look up a lot of stuff to make sure that I explain it as easy as possible 🤓🙂

[@JamesVestal-dz5qm]

Your stomach is a control volume semibatch reactor if you eat slowly.

[@gokulup]

I need the pdf but i don't have enough money to pay for this.

[@A.Hisham86]

It's sigma xx for the compression cell. not tho xx

[@mba2ceo]

I want to see these equations solved using values NOT symbols

[@JousefM]

Good idea 🙂

[@roger_isaksson]

This problem seem to assume rigid body methodology (tensor notation). The fluid simply facilitates an energy transfer field. Thus by solving for energy transfer maxima, the fluid will reposition (move) as to maximize this. *The universe is maximally lazy*

[@JamesVestal-dz5qm]

It took me a while to realize navier stokes for couette flow between parallel plates is derived from hydrostatics.

[@JamesVestal-dz5qm]

My mom quit smoking because of statistics.

[@JamesVestal-dz5qm]

Taking notes is for undergrads.

[@leemontgomery7914]

Night crawler?!!

[@JousefM]

Why? :P

[@JamesVestal-dz5qm]

Real chem e grad students memorize.

[@jad_23]

Too easy bruh

[@Barsik1956]

Uhahaha... уравнения Эйлера-Коши (источник - сила давления от питерского Леонарда Эйлера, сток - тензор вязких напряжений из теории эфира от галла Августина Коши). Прапор Навье, конечно, великий гидромеханик...

[@JousefM]

Had to translate your comment. ;)

[@int16_t]

I like pretending I understand.

[@JousefM]

😄 what was the issue in understanding ? Please let me know.

[@int16_t]

@@JousefM No, actually the issue is me because my knowledge of math is all in crumble. I might as well learn calculus when I'm intrigued by it or have a reason to.

[@JamesVestal-dz5qm]

Cincinnati is more conservative than minneapolis. My youtube account is tolerated only because I'm book smart.

[@plekkchand]

I wish "Hey guys" , with all of its ingratiating, false, chummy informality could be dispensed with as an introductory phrase, but it has grown like a virus. It is more respectful of your listener to simply say "Hello."

[@JousefM]

Well it’s my channel and I can say whatever I want :) I’m not a big fan of it either that’s why I use „Hey friends“ most of the time. I’d suggest you do some videos on engineering so that we can learn from you.

[@azzteke]

Learn to pronounce "EULER" correctly!!!

[@Potatochips77105]

Oiler