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Laplace's Equation and Potential Flow
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- Опубликовано: 5 авг 2024
- Potential flows are an important class of fluid flows that are incompressible and irrotational. They are found by solving Laplace's equation, which is one of the most important PDEs in all of mathematical physics.
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This video was produced at the University of Washington
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0:00 Introduction & Overview
3:48 Laplace's Equation
9:26 Example of Potential Flow, Phi(z) = z^2
12:03 Analytic Functions
![Div, Grad, and Curl: Vector Calculus Building Blocks for PDEs [Divergence, Gradient, and Curl]](http://i.ytimg.com/vi/lKXW7DRyyro/mqdefault.jpg)
The fact that this is freely available on youtube is really insanely incredible, I appreciate it so much.
No it’s not insane. It’s just math. You’re English is tacky. Unpolished.
I look forward to the day when mathematicians mention "flow fields" in a live lecture, and some future variant of Dall-E automatically creates an animated flow field on the screen. Dynamic systems deserve dynamic representation.
that would be wyyyld
The Professor writes out the subject material on the modern blackboard effectively in sync
with clear explanation, making the subject easy to learn. This is
what excellent teaching it is.
This channel is so damn good b/c Steve knows so damn much
Clear and concise. Couldn't ask for more.
Great video! I am extremely impressed that you can write so neatly reversed and backwards. Huzzah! That is a skill in and of itself.
the video is flipped during editing lol
Thank you for this deep theoretic lecture. I agree to your opinion of universal phenomenona.
God bless you!!!
Thank you, Doc.
This is GREAT! I'm learning a lot! New horizons for me! 😊
Thank you sir. It was so helpful
An important fact about harmonic functions (functions that satisfy Laplace equation), is that their sum also satisfies Laplace equation.
Del^2(f + g) = Del^2(f) + Del^2(g) = 0 + 0
Since the Laplace operator is a linear operator.
Sir, I want to pursue Ph. D. under you. Currently, I am in IIT Bombay, India. I am your big big fan!
Love it!
Question though, doesn’t the potential field still hold for Div * f x Grad V, where “f” is a scalar function? For example, in the standard wave equation, often derived/thought of as a fixed string under tension, the tension is constant. But, for say, a hanging chain, the tension in the chain varies with height and the governing equation involves something like Div * T(x) x Grad U.
Laplace’s equation and the Laplace operator are really just special cases of the former in a homogenous medium.
Also, I would LOVE a video about solving Laplace’s equation and the Helmholtz equation (really, finding the Eigenfunctions) on irregular domains. Triangles, or the square with one quadrant removed, etc.
Thanks for the video!
Nice video!
Thank you!
Prof. Steve i am new to all of this, I am becoming your FAN. I am studying fluid mechanics here in france as an international student and you are my beacon of hope. Allah has sent you into my path for that I am greatful.
Nice class.
Thanks a lot for these videos! How often are you publishing them?
Lgga di lgga di aag lgga di 👏
Thank you
Hello, i just watched the two last videos. I'd be great to deal with streamline function, i mean the scalar field which gradient is orthogonal to a given potential vector field. This is very useful to visualize streamlines.
@Eigensteve Great Series, Dr. Brunton. But why the flow has to be steady? Can't the potential exist at all times?
Steve, what exactly is your background? I can take a guess from the aero/ML content you’ve been pushing out lately, but loving it all nonetheless
Hi Steve, is it correct to assume that divergence free means, colloquially speaking, divergence AND convergence free? In other words, it appears that the term divergence accounts both for expanding and retracting systems?
Yes absolutely, that is a good way to think about it
Excellent. Many, many thanks for sharing this gift with your viewers!
What is counterintuitive with mathematics, is that to be more efficient, variables that we can not intuitively know, have to be used.
Here, we intuitively understand what is the velocity, but it's better to use another quantity, the potential, that is very abstract.
The same for the complex space. We don't know what it is, but it's easier tu use it rather than the real space.
So i have to ask. Steve, are you a wizard at writing backwards or do you just flip the video. I can't tell its bothering me lol.
Thank you for all that you share!
I think that he is left-handed and he flip the video in post production
For me its pretty much easier to do the curl in 3 dimensions for Cartesian coordinates .
Superbe
How the hell do you write backward and still manage to be readable?
I'm lost at complex potential...
That's why it's called complex
It's weird (complex analysis) after 2 semesters, I still have a very tenuous grasp.
I’m planning a little mini series on complex, so hang tight!
@@Eigensteve can't wait to see it!!
@@Eigensteve Lotsa demos, pictures, movies, mathmatica? The symbols fail to provide intuition.