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

This video dives into Gauss's Divergence theorem to derive the partial differential equation (PDE) for mass conservation, known as the continuity equation. This is one of the most fundamental equations in fluid mechanics. Specifically, for incompressible flows, the mass continuity equation reduces to a condition that the velocity field must be divergence free everywhere.
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This video was produced at the University of Washington
%%% CHAPTERS %%%
0:00 Introduction & Overview
1:38 Mass Continuity Recap
3:21 Control Volumes and Death Stars
6:30 Smoothness Conditions and Shockwaves
8:56 Incompressible Flows
10:55 Math
14:16 Incompressible Fluid Flows
16:15 Divergence Free Condition

[@arvindp551]

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[@Eigensteve]

That is awesome to hear!!

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​@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

[@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!

[@user-ls2oq6yf5y]

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

[@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.

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You are the very rare thing, a good teacher.

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

[@fredericoamigo]

Another brilliant vid! Thank you so much for this!

[@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

[@andreacomparini9381]

Simply a Great Teacher!!

[@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.

[@curtpiazza1688]

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

[@Infinium]

This is a really nice video, thanks for sharing!

[@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.

[@zinanurmi6157]

Great explanation! Thank you!!

[@Eigensteve]

You're welcome! Thanks for watching :)

[@hdheuejhzbsnnaj]

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

[@ramakantgadhewal7042]

Thank you for the nice lecture.

[@innfdtfjord3340]

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

[@papawhiskeybravo]

Brilliant. Thanks!

[@issamn.e9032]

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

[@jexyl8071]

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

[@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.

[@seslocrit9365]

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

[@Eigensteve]

I hope so =)

[@metluplast]

Thanks

[@reyes09071962]

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

[@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?

[@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.

[@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

[@vidyasagarramalingam7067]

Awesome

[@muzaffergecim1933]

Thanks.

[@radouanelimane3468]

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

[@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.

[@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.

[@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

[@phoenixorder1]

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

[@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.

[@sbsarkavri5436]

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”

[@mariuspopescu7543]

correct name is Gaus-Green Theorem

[@takomamadashvili360]

🙌😍😍

[@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)

[@radouanelimane3468]

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