There is fluid flow *without* many particles

This is the second part in a series about Computational Fluid Dynamics where we build a Fluid Simulator from scratch.
We derive the Macroscopic Perspective (Continuum) from the Microscopic Perspective (Molecules) covering: Collective Molecular Behavior, Local (Non-)Equilibria, Classical Statistical Mechanics, Rarefied Gas Dynamics, and Continuum Gas Dynamics.
The Macroscopic Perspective provides the ground for the next part where we make it all numerically accessible - the Discretization.
Timetable:
-----------------
00:00 - Why we need a Macroscopic Perspective
01:44 - Particles Collective Behavior
05:33 - Using Equilibria for Reduction
08:17 - Statistical Mechanics and Rarefied Gas Dynamics
12:00 - Continuum Gas Dynamics
16:09 - Building Macroscopic Quantities
23:05 - Linking Macroscopic Quantities
34:09 - Recap
Selected Papers and Learning Resources:
-------------------------------------------------------------------
Sorted by Topics:
01:44 - Local (Non-)Equilibrium; Necessity of Collisions; Fluctuations; Molecular Chaos; Initial Perturbation; Statistical Perspective; Statistical Ensemble Averaging; Time Reversibility (to be discussed), Equilibria of different Degrees-of-Freedom (to be discussed): [1,2,3]
08:17 - Rarefied Gas Dynamics: [1,4]
08:17 - Phase Space (one-particle vs. N-particle): [1,5,6]
08:17 - Boltzmann Equation derived via BBGKY Hierarchy from Liouville Theorem: [6]
12:00 - Continuum Gas Dynamics, Continuum Hypothesis/Assumption, Alternative Flow Regimes Classifications: [1]
12:00 - Local Knudsen Number and alternative Rarefaction Indicators: [1,4]
16:09 - Macroscopic Quantities: Density, Flow Velocity, Pressure, Temperature [1]
23:05 - Macroscopic Equations: Conservation Laws; Mass; Momentum; Energy [1]
27:20 - Navier-Stokes Equations; Densities of Forces; Pressure Gradient; Viscosity: [1,7]
27:41 - Time Derivatives along with flow; Lagrangian vs. Eulerian Formulation; Lagrangian vs. Eulerian Coordinate Systems: [8]
30:12 - Velocity and Temperature Profiles for Couette Flow: [9]
32:45 - Macroscopic Equations: Equations of State; Ideal Gas Law; Calorically Perfect Gas: [1]
Selected References:
[1] - Lecture Notes: from "volkov.eng.ua.edu/ME591_491_NE..." to "NEGD-06"
[2] - Paper: Maes, Christian, and Karel Netočný. "Time-reversal and entropy." Journal of statistical physics 110.1 (2003): 269-310.
[3] - Paper: "Parker, J. G. Rotational and vibrational relaxation in diatomic gases. The Physics of Fluids 2.4 (1959): 449-462."
[4] - Paper: Macrossan, M. N. "Scaling parameters for hypersonic flow: correlation of sphere drag data.", 2007.
[5] - Lecture Notes: "Cerfon, Antoine. Mechanics (Classical and Quantum). www.math.nyu.edu/~cerfon/mech..."
[6] - Lecture Notes: "Kenkre, V. M.. Statistical Mechanics. www.unm.edu/~aierides/505/" specifically ".../bbgky2.pdf" & ".../bbgky3.pdf"
[7] - Book: Anderson, John D. "Governing equations of fluid dynamics." Computational fluid dynamics. Springer, Berlin, Heidelberg, 1992. 15-51.
[8] - Essay: Price, James F. "Lagrangian and eulerian representations of fluid flow: Kinematics and the equations of motion." MIT OpenCourseWare, 2006.
[9] - Paper: Marques Jr, W., G. M. Kremer, and F. M. Sharipov. "Couette flow with slip and jump boundary conditions." Continuum Mechanics and Thermodynamics 12.6 (2000): 379-386.
Disclaimer:
------------------
This series focuses specifically on the aspect of information reduction in dynamical systems. For the sake of clarity, I had to omit many interesting aspects of the topics addressed in the video. So, the video itself is a reduction. :-)
Please note:
------------------
Watching this video with very low resolution produces the continua right away :-), thanks to the video compression!
I hope you enjoyed this little braintruffle!
If you like this series and want to support my work, you may consider subscribing to the channel. I would really appreciate it!
Thank you for watching and I hope to see you next time!

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I'm very happy you like it! Thank you a lot for your kind words!

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👍 69 👎 *REPLY*

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[@braintruffle]

I am very grateful for your appreciation. It really means a lot to me, thank you! And also thank you for your support by sharing this video. :)

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[@braintruffle]

I'm very happy you like the overall approach and hope you like the upcoming parts as well! Thank you very much :-)

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Why doesn't this masterpiece have hundreds of thousands of views yet

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[@braintruffle]

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[@mennolubbers9196]

The quality of this video in both conveying the information as well as the visual esthetics is excellent. Thank you.

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[@braintruffle]

I really got to thank you for your motivating words! I am very happy that you like it and seeing your videos is a pleasure as well! You clearly have a sound theoretical background. I remember watching your wave function videos as an additional source of verification in preparation to my first video! Excellent visualizations. My background resides in non-linear dynamics, but pretty much everything that moves catches my attention. Since my studies, I just loved building my own simulations that really helped me understanding what is going on. My goal here is to share this experience of understanding by playing with simulations.

[@nou5440]

69th like

[@brunoritter5984]

@braintruffle simulating to understand is such a powerful idea. I'm just about to start my physics formal study, and I learned SO MUCH from the videos. It helped me to really understand a ton of other content that I've consumed during the last year. Lots of them left me with a taste of "I need a degree to really get it", but apparently I just needed to watch your work. Thank you and congrats.

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[@braintruffle]

Thank you soo much for your support and for leaving such a nice comment! I really appreciate it!

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[@braintruffle]

Thank you very much :-)

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[@braintruffle]

I'm so glad it motivates you! Enjoy every step of building your simulations :)

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this right here. it was amazing, congratulations on the work! together with the microscopic perspective, it was one of the most beautiful expositions I've ever seen. the quality of the animations and theoretical backgroud you provide is simply astounding. keep it up.

[@braintruffle]

Wow, just want to say thank you for your compliment. I'm so happy you like it! :)

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Great work. I love how you start at the quantum level and only then abstract upward. Never have seen that done in a contiguous way. I really highlights the levels of simplification needed

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[@braintruffle]

Thank you very much! The truffles are glad about your comment ;)

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Well done! I'm looking forward to following along with you over the next installments and hopefully building my own fluids simulation software.

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[@braintruffle]

Wow, it's great to hear that it could help a little bit in guiding your interest and I hope you have fun starting your journey in that direction.

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That animation of the two sphere guys at 4:30 is so cute

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[@braintruffle]

You perfectly describe my intention with this series and I'm so glad the actual message is resonating! Thank you so much for your feedback and also for sharing! I wish you and your friends the best of luck with your studies.

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Nice! I was looking forward to this!

[@braintruffle]

Thank you so much!

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[@braintruffle]

Thank you so much for your appreciation! :)

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A bit late to the ride, but just wanted to let you know that this series is truly amazing and extremely underrated. Videos like show just how beautiful physics and math can be even for people who are not that into it. Would love to see more!

[@braintruffle]

Thank you so much! I often feel stunned and excited when I dive into certain topics, and to hear that I was able to convey some of that beauty makes me very happy! Thank you so much for your comment!

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I just finished both videos on the channel so far. As I was watching, I had in my mind "Okay, this is a new amazing channel of someone who's been making science videos for many years." When I looked though, you don't have any other channels linked! This can't honestly be your first work!? This is incredibly good. If you have other work please link it so I can enjoy that as well.

[@braintruffle]

I'm so happy you want more. These here are my first videos ever, but I'm working hard on new content. I hope to see you then! :)

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[@braintruffle]

I'm happy you like it :)

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Wow. i can't wait for the next part.

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This video is simply amazing, thank you very much for your work :)

[@braintruffle]

I'm glad you like the approach here. Thank you very much!

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I'm definitely watching this again next weak, the pure amazingness needs rewatching to unpack it

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Daym bro, gonna help me a heck ton on my sim

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Truly wonderful work

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REALLY nice visuals!

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Thank you for this video!

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Awesome video! Thank you!

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and another fantastic high quality video 👍 i am rly impressed how much effort u put into your videos! cant wait for more 😊

[@braintruffle]

I'm glad you like this one as well! Thank you for your appreciation :)

[@ottolehikoinen6193]

I keep on flipping from the abstract consepts to practical consepts as I watch and listen. Maybe that's intended and represents the dual nature of things, static solid - fluid dynamic. Space could be defined by the amount of particle collisions, as could the generated vacuums on earth. Pressured gas behaves differently from unbound gas. Really good explanations, thank you.

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Amazing work!