The Continuity Equation: A PDE for Mass Conservation, from Gauss's Divergence Theorem

[@AsherZhu]

I've already made fluidsim systems for games and I'm learning to actually understand them. This series of math lessons are the best I've ever had!

[@arvindp551]

Having a great time learning with you Professor! I can see you are having it too teaching us passionately.

[@Eigensteve]

That is awesome to hear!!

[@juniorcyans2988]

The best professor I've ever found! I found this channel last semester to help me do complex analysis homework, now I'm in the mathematical physics class, still finding this channel is super helpful. Thank you so much!

[@Eigensteve]

Thank you so much!

[@pitsielias483]

​@Eigensteve I enrolled to study computational fluid dynamics graduate level because of your channel. I'm hoping to tackle turbulent modeling which you also teach very well

[@권현우-u3w]

I took two gap years and my math became a little rusty. Thank you for making such videos. Its helping me survive my fluid dynamics class

[@wp4297]

Nice. It's great you're using comments as a feedback, like a university class. But now things start to become tricky to be explained in short videos: lots of (not so) little details make the huge difference between a consistent learning of the subject and incredible mess. Keep going! I'll keep follow in my spare time because I'm so curious

[@Eigensteve]

Thanks! And good point about things getting tricky/subtle.

[@hdheuejhzbsnnaj]

You are the very rare thing, a good teacher.

[@sakeofrumour4679]

Thanks for your work. It is nice to map these ideas onto existing knowledge like how the GT can be applied to derive Maxwell's equations in electromagnetism

[@seslocrit9365]

Seconded. I'd love to see how we can derive equations like the heat, Navier-Stokes and Schrodinger's equations too

[@joe33444]

These are such easy to understand lectures, thanks! - And do you have to write backwards as you do these? Its a very cool setup how you can write, and be behind the text

[@andreacomparini9381]

Simply a Great Teacher!!

[@AliMohamed-jx1lk]

you are great!! please Prof. Brunton keep making these videos.

[@diveintoengineering6089]

Thank you again from Chile Prof. Brunton. Such a beautiful explanation. I´m on my first approach to PDEs and this lectures are of great help.

[@fredericoamigo]

Another brilliant vid! Thank you so much for this!

[@shanqiupang2011]

Really like the videos. One thing I am interested to learn is what is curl, div, grad under different choice of origin/coordinates. Because when we are talking about physics, the rules should apply with any choice of coordinates.

[@jexyl8071]

I can see why so many people go ME now - these ideas are Fascinating, intuitive and widely applicable

[@ganeshramamurthi9663]

Since divergence can be defined point to point and there can be local bubbles and cavitations having low static pressures which can become local divergent points upon breaking, can we say divergence always occur in opposite pairs ?

[@nagarajprashanth763]

Hello Mr Steve, Is there a video of yours explaining the derivation of conservation of momentum ? Sorry i was unable to find the video.

[@mathjitsuteacher]

Hi Steve, I have a question. Since the gradient vector is the vector of partial differential equations and rho is a function of 4 variables should't it's gradient be a four dimensional vector? Why do we ignore this when we calculate the gradient of rho F? Is it just an abuse of notation? Or you you are thinking that the equation is true for each fixed t?

[@Eigensteve]

Good question. It is kindof an abuse of notation where we fix t for this gradient. The idea is to separate out the time dependence from spatial dependence.

[@mathjitsuteacher]

@@Eigensteve Thanks, Steve.

[@mathjitsuteacher]

@@Eigensteve I am pure mathematician so I never really cared much for Physics and applications. Now I am trying to learn more about PDEs and how they are used in applications. I like really like your lectures.

[@curtpiazza1688]

Wow! This is GREAT! It's a little over my head....but I'm slowly catchin on! 😊

[@thomashofer6011]

Thanks for the great video. Actually, for all of them. Just a quick question from my side. The end result for the incompressible fluid is only strictly true for a thermostatic volume/no temperature gradient (same of salinity or other parameter that influence the density in space and/or time) as this could change the density of the fluid, regardless of its incompressibility, right?

[@wp4297]

In this video, there is a wrong definition of incompressible medium. Incompressibility says that the density of each material particle (and so the volume) does not change: you have to follow material particles in their motion, so that mathematically speaking, you have to take a look at the material derivative of density, defined as D rho/Dt = \partial rho/\partial t + Vel \dot grad rho You can recast the mass equation in the convective form D rho/ Dt = -rho div( Vel ) This way, you clearly see that the incompressibility constraint, div( Vel )=0, implies D rho/D t = 0, i.e. the density of each material particle (you can think of it as a point of the medium, moving with the medium itself) does not change in time. So, you can start with a fluid whose density is not uniform in space. If the flow is incompressible, it will carry material particles with constant density. Compressibility constraint is a kinematic constraint. And it doesn't tell you how to achieve it. You should take a look at the process you're studying to see if there is anything that makes the material particles change their density. If you think that your incompressibility constraint suits the model you're using to describe your process, you can replace mass continuity equation if it's convenient, as in the case of very-low Mach number flows. However when you enforce this kinematic constraint you're giving up something else, usually thermodynamics...anyway it's not true that the temperature (or whatever it is, since we're giving up thermodynamics) must be uniform in the flow. If you think at the Navier-Stokes equations for incompressible flows, you usually don't see any energy or temperature equation but not because temperature is constant, but because: - temperature equation can be solved after you solved for "pressure" and velocity the system of PDEs built with the momentum equation and the incompressibility constraint, since temperature equation is not coupled with the other two; - Temperature is just an approximation of thermodynamic temperature, since you rejected thermodynamics with the kinematic constraint Sorry for the long reply

[@hdheuejhzbsnnaj]

Just a thought: suggestions for textbooks for self-learners is always appreciated!

[@amjuarez6246]

I wonder if the derivation of divergence of F=0 is valid for incompressible and homogeneous density scalar fields. You may have an incompressible fluid with an inoogeneous density field ( example, planet earth, with different density at the core, the mantle..) then the gradient of the density (rho) does not have to be necesarily zero. Am I missing something here?

[@reyes09071962]

Would the use of atmospheric events to relate to the math of incompressible flow not be quite right?

[@doutormanhattan5680]

Why the Q term (source) is being integrated? The left side of the equation is just the rate of change of mass inside the volume. If Q represents mass being created, shouldn't it being differentiated instead? Or Q already represent the rate at wich mass is being created or destroyd? In any case, there would be not an integral.

[@charlesschmidt4272]

Great lecture, minor quibble. Instead of chain rule the product rule is being used to expand $\vec{ abla} \cdot ( ho \vec{F}$.

[@lemuelcarlosramosarzola5241]

Thanks for the great lecture. Comment in 11:47 min and 13:57 min, would it be "the product rule"?

[@wp4297]

yes. derivative of the product of 2 functions

[@issamn.e9032]

Thank you and god bless you ... You have to stay constant 😁

[@Infinium]

This is a really nice video, thanks for sharing!

[@sureshbbhatta513]

❤ Great explained

[@seslocrit9365]

Thank You so much for these. Could you do a video on shockwaves?

[@Eigensteve]

I hope so =)

[@edmald1978]

Hi thank you so much for the lessons really great explanation. Do you have some lesson where you derivate the Navier-Stocks equation?. Thanks in advance.

[@zinanurmi6157]

Great explanation! Thank you!!

[@Eigensteve]

You're welcome! Thanks for watching :)

[@innfdtfjord3340]

Thanks. I mean for all of us could be every interesting to hear about fractal derivative and practices using fractal derivative) Thanks.

[@ramakantgadhewal]

Thank you for the nice lecture.

[@papawhiskeybravo]

Brilliant. Thanks!

[@sawab.sarkavri]

Very nice video but, sorry isn’t finding an equation for the movement of fluids in space a mathematical problem which has a 1,000,000$ prize ? What’s the difference between this and that? “Navier-Stokes existence and smoothness”

[@metluplast]

Thanks

[@totoantibes]

couldn't the last "proof" of divergence free condition for incompressible fluids be achieved by just using the linearity property of the div operator in the mass continuity equation? (Since rho is a constant) Or another way to look at it is that Dr. Steve proved that Div is in fact linear.

[@radouanelimane3468]

Could you please make an explanation about the drawing of bifurcation diagram of fractional order chen system on matlab

[@kylegreen5600]

I'm sure you get this a lot, but are you right or left handed... or more specifically are you ridiculously good at writing backwards?

[@wp4297]

kzbin.info/www/bejne/j57Ze4mhrq-VgsU and if you're interested in Physical Chemistry, that is a very good channel

[@wp4297]

In this kind of videos, everything runs smooth, except for the orientation of space, because mirroring reverse it (all these things like right-hand rule and vector product...actors must pay attention only to those things, as an example showing right-hand rule with their left hands)

[@phoenixorder1]

chemical reactions can easily be sources for conservation of species...

[@vidyasagarramalingam7067]

Awesome

[@muzaffergecim1933]

Thanks.

[@mariuspopescu7543]

correct name is Gaus-Green Theorem

[@takomamadashvili360]

🙌😍😍

[@radouanelimane3468]

Could you please make an explanation about the drawing of bifurcation diagram of fractional order chen system on matlab