Potential Flow and Method of Images with
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- เผยแพร่เมื่อ 5 ม.ค. 2021
- Grant Sanderson of 3Blue1Brown asked me to teach him some Fluid Dynamics during his visit to Oxford last year (Feb 2020) - here's what we got up to...
We look at potential flow for a 2D incompressible, irrotational, inviscid fluid, beginning with the flow fields for the most common situations such as uniform flow, stagnation point flow and a line source. We then derive the potentials for each of these flows and use the method of images to construct the velocity field for a source next to a wall, a source next to a corner and finally for an infinite channel.
Find Grant's amazing channel 3Blue1Brown here: / @3blue1brown
More videos with Tom and Grant
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Power Tower: • Power Tower with @3blu...
Produced by Dr Tom Crawford at the University of Oxford. Tom is an Early-Career Teaching and Outreach Fellow in Mathematics at St Edmund Hall: www.seh.ox.ac.uk/people/tom-c...
For more maths content check out Tom's website tomrocksmaths.com/
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Grant also taught me something - check out the 'Power Tower' video here: th-cam.com/video/dnZ3xlif9VA/w-d-xo.html
What does z represent at 6:15..you didnt say
@@leif1075 z is a complex-valued number - z = x+iy. Think of this as the "coordinate" of the complex plane.
Ah, yes, two sexiest mathematicians in one video
Grant is so attractive he is the only man who could seduce me.
I'm glad this is the first comment, cause I wanted them to kiss from the moment I looked at the thumbnail
BONK
you must be unaware of Ed Frenkel :D
@@davidgjam7600 👀
im taking a break from learning maths with a maths Video, weird isn´t it ^^
somehow the college experience is just about getting distracted from doing maths by another different kind of maths
yeah i m watching this video instead of practising for my circuit theory test.
I have a class in 4 minutes
Same!
Maths, fluid dynamics, Tom and Grant. Simply amazing.
Just in time for my fluid dynamics exam! Haven't seen the video yet but man, this collab is epic
Hehe ikr!!
Good luck :D
@@aiyopasta lots of maths, not so easy but super interesting!
I had my exam like 4 weeks ago... We had a question on method of images and it was AWFUL, i just think this video was suggested to mock me
@@aiyopasta depends on your lecturer
In general it should be easy
The method of images is so amazing that it deserves a background music of its own: "Mirror on the wall, here we are again...."
I didn't know Tom had his own channel! I just saw the drag equation video on the Numberphile channel and the recommended video was this one. Thank you algorithm!
It's good to know it does its job sometimes :)
I was looking for a video on method of images of electrostatics but ended up watching this amazing fluid video ❤️
Awesome! Thank you!
:v
hello papa flemmy
Definitely one of the coolest ideas in fluids! One of my favorites is if you have a source at (1, 0) and 2 walls leaving from the origin at slopes of +30 degrees and -30 degrees, then you can replace it with 6 sources at the 6 roots of unity (hexagonal symmetry).
Your comment made me start thinking. Have you realized you can generalize this method? If you have a similar setup but with an angle of 2pi/2k between upper wall and x axis you can construct a solution with k charges on the vertixes of a poligon with k edges.
And if k goes to infinity? I think in some sense it converges to the infinite channel example
reading this comment felt really good @@marcocecchi9853
Greatest math video ever created. I am in awe at how amazing this journey how perfect the format is.
Thanks Connor :)
The thing that strikes me - is the simplicity with which it all has to be approached. If nothing else this shows how to make maths accessible; or that even some of the best minds in math, still when introduced to a new topic, take it from the most basic forms and build on that. Sublime!
"It's like you're looking at the mirror and then you give him a high five. Of course, it will stop there"
That's a very good analogy for a method of images.
agreed :)
Maybe you guys haven't seen it or it has been a while, but I would check out Feynman's Lectures on Physics. In volume 2 there is a neat section on the method of images for electrostatic potentials!
Could not asked for a better new year gift. I am working with these for past 2 years and it still amazes me how beautiful the math is.
Yay! More Grant and Tom collabs!
i heard "sauce" when Tom says "source" and honestly that didn't asked myself any question before finding out it wasn't sauce
These equations can describe sauce flow too... For the right choice of sauce (incompressible, inviscid, irrotational sauce)
Wow, amazing. That's one of those concepts you never forget once you've seen it.
Thank you Tom and Grant! Just in time for my Fluids exam
Two legends.. love both of them
Next: Laminar flows with Dustin, the epic fluid collab, and it doesnt get any better than that
I'm down
@@TomRocksMaths Are those walls supposed to be solid walls or walls of stationary fluid or something?
@@leif1075 those would be work with any walls along which the potential flow is 0 so perfectly stationary fluid walls with infinite mass would definitely work and solid walls would as well. Not sure about other stationary fluids with finite mass
Thoroughly enjoy these collaborations with Grant. I think the visuals with barriers and reflections would make a great 3Blue1Brown video (like a followup to the Maxwell's equations video).
Great video! The method of images is very useful when dealing with phenomena that can be treated as linear, e.g. in (linear) acoustics.
Channel is growing my man, awesome!
I needed this for electrodynamics
This was nice! Fluid dynamics is very similar to electric field theory we did in physics. The source is like a positive charge, sink being a negative charge and the velocity vectors are like electric field vectors. We do use potentials in electrostatics but I don't remember using complex potentials. In that way electrostatics might be a bit simpler.
The mirror method is elegant indeed! Visualizing images of source in mirrors and doing the calculations. In electrostatics the wall is in fact a conducting surface. The infinite channel example was particularly enlightening.
When I saw the first example with the source and the wall - 12:41 - I thought about an additional step: to imagine every point of the wall as a kind of source itself, but a linear one in particular direction alpha, depending of its position relative to the source of the flow. But, wait a minute!... this is the definition of the mirror, and as we already know, we can imagine the second source behind the wall, placed at its special spot as it was shown in the video. (By the way, the method of mirror images is also used in the field of electrodynamics, which is my speciality - so, you see, I was taught to think in this manner.)
How clever it is! What mathematical wonders are hidden in Fluid Dynamics... I can only guess!
Ah, and the last example was also very elegant! What am I talking about - all they are!
These things must be popularised and the host of this channel is doing great, I admire his efforts... as with the same favour for mathematics that 3Blue1Brown is doing with his magnificent visualisations... big fan!
Never seen the source in a channel before. Thanks Tom great video!
I now see why you chose potential flow and the method of images to show to Grant, you had a brilliant example :D
You both are incredible! Thanks for this video!
Glad you enjoyed it!
"The wall is a mirror" love it!
brilliant content as usual, thank you
i am doing a fluid mechanics master degree and this really brainstorming, thanks so much for sharing.
Glad it was helpful!
Not just maths but also chemistry!! ❤️
i see tom i see 3b1b i click
I really love this type of content
Are you kidding me? I just found your channel and you have a collab with Grant? Christmas came really early this year :)
Welcome :)
This video deserves way more views
Both of you are assisting me with my physics maths degree, final year student ❤️
Yesss we need more Tom and grant collabs
I second this comment.
Epic colab!
Finally I got the images method! 👏🏼👏🏼👏🏼
Happy to help :)
i never saw this aproach in fluids, since i just knew it from EM topic ... epic greetings from colombia
Hello in Colombia!
The series at the end actually doesn't converge. But since a potential is only defined up to a constant, we can subtract from each term an appropriate constant just so that it converges. This way we can use the Weierstrass product formula for sin(z) and get the same result.
I needed a quick revision on this topic for my PhD Quals and there you are Grant. Awesome collab Tom xD.
Also, the series should start from n=1,inf after taking the derivative. Sorry it had to be done :)
Glad you enjoyed it!
HENCE PROVED TOM ROCKED
This was really great.
Youre a really good teacher wow
That was so beautiful ❤️
I miss fluid dynamics
it's awesome isn't it?
Yes i love these team ups
you and me both :)
This is the best video on the internet
A bold claim, but I'm not complaining - thank you
Next video: "Grant and I are a couple now! #MathLove"
Also a question how do people even think of this abstract idea it feels magical
Kutta, blasius, zhoukovsy etc they were all extremely gifted
Its today's computer,, internet and technology, which paralyzed our mind and creativity. we are so much dependent on computers that we even don't try to imagine things, we search youtube for animations etc, which feels super easy to grab the things but in long term our brain gets lazy.
the time when there were no computer machines, all computations were supposed to be done in the brain, as a matter of fact, the more you use the brain the more it gets trained and powerful. and then curiosity will be developed for nature, and the ultimate result will be discoveries and inventions.
This was amazing 👏🏻
Glad you enjoyed it Nacho!
This is what happened when mathematicians go with the flow!
I laughed.
I understand the concept, but the math leaves me in the dust !
This is 2nd year maths undergraduate level so don't feel bad!
Cool concepts
Eye candy, brain candy. Also happpy 2021 to the both of you!
Happy New Year John!
@@TomRocksMaths Thanks Tom 😊
I studied physics and then went on with a not related degree. This video reminded me of when I used this mirror method for potentials in electrodynamics where there is e.g. some point charge (Punktladung in German) in a plane.
In such moments I dont know if I feel sad to have "abondend" the world of physics/maths and their methods.
Reminded me of the same thing
The same method is indeed used in electrostatics - well remembered :)
Oh goodness.... Math bursting at the seems, all we need now is an onlyfans. Hahahahaha.... Kidding not kidding.
Fluid flow, for sure.
Tom is a good example why there isn't a nobel for math.
I hope you got the joke.
Is the word "seems" intentionally misspelled?
@@lloydgush not me explain pls lol
@@20031bibi The joke is that nobel didn't made a nobel for math because a mathematician was fucking his wife.
Tom is heavily flirting with a married man.
Therefore, a joke.
But he flirts with everyone, after this christimas season I'd say he had an only fans, but who am I kidding, this is youtube, everyone has an onlyfans.
@@lloydgush LMAOOOOOO
Thanks for the great video! Really interesting and well-explained 😊
I just have one question that's been bothering me since the beginning of the video: why do you take the potential to correspond to u *minus* iv, and not u+iv?
Is there some physical or mathematical logic behing this choice?
We very briefly touched on this in the video, but the idea is so that when you calculate the derivative of the potential as dw/dz the velocities match up with the real and imaginary parts. If we instead define dw/dz as u + iv then the vertical velocity would be the negative of the imaginary part of the derivative.
That was awesome! Especially the infinite reflection one
Glad you enjoyed it John :)
it's probably bad that i saw the january timestamp and immediately went "oh well it's november now so i guess this is about 10 months old"
Heyyy, you can also describe flow around rotating circle in uniform flow which replicate flow around airfoil as used by earlier aeronautical scientists
Absolutely - the concepts introduced here are incredibly useful!
Tom missed the best one! Where you can put a source and a sink (negative source) infinitesimally close together to get a dipole. Add in a uniform flow, and you get flow around a cylinder!
Oh there were far too many good ones to include them all...
Just in time for my EM exam lol thank u
Great :)
Thanks!
Where was this video when I was doing fluid mechanics last year 😭
Love eeeeeeeeeeeeeetttttttttt!!!!
Is Grant really huge or is Tom really small? Or am I just bad at understanding camera angles?
Grant is large.
Why, man, he doth bestride the mathematical world. Like a Colossus.
Amazing video. How would you calculate the flow with a curved surface instead of a flat plane?
Ah, now that requires a completely different theory... this only works for 2D flows.
Please makes videos on streamline, streakline, pathline and stream functions etc 🙏 please 🙏.
Added to the video idea list - thanks!
Just found your channel!
Welcome :)
Hey Grant, make a series on Manim library
Very nice video, thanks for presenting it!
Can this be applied to Electromagnetism?
And there is the concept of Sink (opposite of Source)?
Or vice versa. I don't know which question should be the first XD
Yes I believe so. And yes sinks are the same as sources but the flow lines are going in rather than out.
Well, tom shows us the reason why we don't have a nobel for math...
lol!
instead of doing the whole, take the derivative to get coth then integrate to get ln(sinh), would it be possible to just use the taylor expansion for sinh somehow? it being complex is confusing me abit but it seemslike starting from log(infinite series) and ending with log(sinh) they should match? or is another infinite series that isn't the taylor series that also represents sinh?
Cool nerdy stuff
It's what we do :)
Great video as always,. Just to clarify though, “z” represents some arbitrary complex number in the plane and not the third coordinate axis as it is used traditionally, right? That caused me a bit of headache upon first watching
Yes z is a complex number, so z= x + yi where x and y are real.
Is there a book that explains the computation steps of the last problem a bit more in depth? Tried to do the computations on my own but failed :D
I recommend 'Elementary Fluid Dynamics' by David Acheson
Linear magick beauty :-) If only there would be similar roules for nonlinear stuff as well :-o imagine.
Isnt that kind of like the boundary knot method?
As Richard Feynman put it - the flow of "dry water"...
Won't we see multiple reflections even in the case of corners? When the boundaries are aligned at 90 degrees? Why did we consider only a single reflection there?
Could someone explain the differentiation and simplificatiom step? I don't even really know what he has written down there ..
You start of with the derivative being the sun from -inf to inf of 1/(z - 2nai) since the derivative of ln is 1/z. Then multiply top and bottom of each term of the summation by its complex conjugate. You get sum from -inf to inf of (z - 2nai)/(z^2 + 4n^2a^). Then extract the n = 0 term and group each of the rest with its corresponding negative term. You get 1/z + sum from n=1 to inf of (z - 2nai + z + 2nai)/(z^2 + 4a^2n^2). Cancelling out the 2nai terms and adding the z's and factoring them out of the sum you get the desired result. (Tom got it slightly wrong, the sum should start at 1 instead of 0. You could also include 0 in the sum but that would then negate the 1/z term we pulled out earlier).
What about a wall with two holes in it and does it generalize to n dimensions.
hi Tom, is there any book or online course that can be the best intro of fluid dynamics?
G K Batchelor - an introduction to fluid dynamics
could you explain how did you differentiate and apply the limits at 21:08
ln(f(x)) differentiates to f’(x) / f(x) and then I rationalise the denominator by multiplying the top and bottom by the complex conjugate
Also the source sink idea seems totally be evident in a level further maths discrete haha do mathematians like sources or. Sinks
18:14 Grant is thinking: "did you see that throw? nailed it!" but, unluckily, Tom is focused on the graph :(
Haha I think I said a quiet 'nice' before immediately getting back to the maths...
@@TomRocksMaths Nice! I had missed that
The method of images is used also for electric field potentials (e.g. what's the electric field when you have a point charge close to a sheet of metal). So i guess the method is valid for any sort of "potential"? Are there conditions that the potential most meet in order for the method to work?
Yes, this will work for any potentials. The definition I use for a 'potential' is something that can be written as a gradient.
I just failed an exam on partial differential equations 🧚🏼♀️ method of images contributed to that failure
Hopefully this helps :)
I'm getting flashbacks from electrostatics class.
Cool just like electrostatics.
Yes, so I've been told!
Early gang
The audio quality is still rough despite the clear effort you put in with the lapel equipment. Good content nevertheless, thanks
The issue here is that only I'm wearing the lapel mic. Though I do now have 2 so hopefully will be fixed for future collab videos.
Now I'm remembering fluid dynamics... Oh no... the screams. The terrible screams.
I kid you not, we covered this method of images in Theory of Electromagnetic Fields a few weeks ago, when talking about potentials and fields of charges. Mirroring against various walls was one of the examples we got.
If you want to step the fun up a bit more, try mirroring not against a wall, but a sphere. That gave me some head scratches 😁
Ah yes, the infinitely-sided polygon that is the sphere...
Ahhh, that's what PotentialFOAM does.
Mathematically, the point perpendicular to the mirror (15:00) is fine, but physically what would happen to the atoms and building up of the energy around that point?
I love how Grant just knows the answer was some cotangent thing. You prove it with Parseval's identity, correct? Or is there a more fun way?
On second thought, I realized that you can do it with a contour integral, which is a bit less tedious.
I was thinking contour integrals...
Are the conditions of incompressibility and irrotation equivalent to a meromorphic potential? - it looks like they should be.
U speaking the gods language hooman..
There is a lot of crossover with Complex Analysis which is where the 'complex potentials' idea comes from