Derivation of the Navier-Stokes Equations

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!

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🎵 Joachim Heinrich - Words

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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! :)



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

[jurivolodin]

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

[verygood]

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.

[EddickCD]

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!

[parvathinarayanan]

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

[MartinsEdegbe]

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

[JamesVestal-dzqm]

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

[albertoconde]

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

[MarHerH]

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

[carmelpule]

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.

[JamesVestal-dzqm]

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

[RealLifeTops]

Thank you so much for your efforts in making this very informative video!

[deepaksharma]

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.... 😊

[SonTran-bhtt]

Thanks a lot for your sharing, very easy to understand and open new study

[KapilVetrivel]

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

[JITISH]

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?

[ayazfarooqui]

Please make an entire playlist on tutorials of CFD simulation

[erwinphillipcabanban]

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

[dakirmossadeq]

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.

[nitinnegi]

Amazing explanation of such complicated equations.