Potential Flow and Method of Images with

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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
Maths Speed Dating: • Maths Speed Dating wit...
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....
For more maths content check out Tom's website tomrocksmaths....
You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths.
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Опубликовано:

26 сен 2024

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Комментарии : 285

@TomRocksMaths 3 года назад

Grant also taught me something - check out the 'Power Tower' video here: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-dnZ3xlif9VA.html

@leif1075 3 года назад

What does z represent at 6:15..you didnt say

@jaredjones6570 2 года назад

@@leif1075 z is a complex-valued number - z = x+iy. Think of this as the "coordinate" of the complex plane.

@danielkron2513 3 года назад

Ah, yes, two sexiest mathematicians in one video

@LeoStaley 3 года назад

Grant is so attractive he is the only man who could seduce me.

@davidgjam7600 3 года назад

I'm glad this is the first comment, cause I wanted them to kiss from the moment I looked at the thumbnail

@joshuajoshua46 3 года назад

BONK

@Arbmosal 3 года назад

you must be unaware of Ed Frenkel :D

@klutchboi3266 3 года назад

@@davidgjam7600 👀

@Nick08352 3 года назад

im taking a break from learning maths with a maths Video, weird isn´t it ^^

@dee8163 3 года назад

somehow the college experience is just about getting distracted from doing maths by another different kind of maths

@zetsubou1108 3 года назад

yeah i m watching this video instead of practising for my circuit theory test.

@Hi_Brien 3 года назад

I have a class in 4 minutes

@okhan5087 5 месяцев назад

Same!

@franteryda4730 3 года назад

Just in time for my fluid dynamics exam! Haven't seen the video yet but man, this collab is epic

@sudheerthunga2155 3 года назад

Hehe ikr!!

@Eyes_On_America 3 года назад

Good luck :D

@franteryda4730 3 года назад

@@aiyopasta lots of maths, not so easy but super interesting!

@tapuwachitiga2547 3 года назад

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

@Stream_Function 3 года назад

@@aiyopasta depends on your lecturer In general it should be easy

@ManojKumar-cj7oj 3 года назад

I was looking for a video on method of images of electrostatics but ended up watching this amazing fluid video ❤️

@TomRocksMaths 3 года назад

Awesome! Thank you!

@amaarquadri 3 года назад

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).

@marcocecchi9853 10 месяцев назад

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

@arnosuess9020 10 месяцев назад

reading this comment felt really good @@marcocecchi9853

@MrWhiteVzla 3 года назад

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!

@TomRocksMaths 3 года назад

It's good to know it does its job sometimes :)

@luckyw4ss4bi 3 года назад

Greatest math video ever created. I am in awe at how amazing this journey how perfect the format is.

@TomRocksMaths 3 года назад

Thanks Connor :)

@PapaFlammy69 3 года назад

@fidgetgadget3475 3 года назад

hello papa flemmy

@googleit1370 3 года назад

The method of images is so amazing that it deserves a background music of its own: "Mirror on the wall, here we are again...."

@thunder852za 3 года назад

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!

@TomRocksMaths 3 года назад

@gandalftolkien2879 3 года назад

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!

@jmdr48 3 года назад

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.

@TomRocksMaths 3 года назад

@EmilyMGin 3 года назад

Yay! More Grant and Tom collabs!

@sebastianmorales9787 3 года назад

Next: Laminar flows with Dustin, the epic fluid collab, and it doesnt get any better than that

@TomRocksMaths 3 года назад

I'm down

@leif1075 3 года назад

@@TomRocksMaths Are those walls supposed to be solid walls or walls of stationary fluid or something?

@tylercrowley2559 3 года назад

@@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

@theverner Год назад

I needed this for electrodynamics

@likithstochastic 3 года назад

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.

@nickwisely2581 3 года назад

"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.

@TomRocksMaths 3 года назад

agreed :)

@ashoulle8953 3 года назад

i heard "sauce" when Tom says "source" and honestly that didn't asked myself any question before finding out it wasn't sauce

@asklar 3 года назад

These equations can describe sauce flow too... For the right choice of sauce (incompressible, inviscid, irrotational sauce)

@aero33888 3 года назад

Not just maths but also chemistry!! ❤️

@benwinstanleymusic Год назад

Thank you Tom and Grant! Just in time for my Fluids exam

@christianorlandosilvaforer3451 2 года назад

i never saw this aproach in fluids, since i just knew it from EM topic ... epic greetings from colombia

@TomRocksMaths Год назад

Hello in Colombia!

@VibratorDefibrilator 3 года назад

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!

@prdoyle 3 года назад

Wow, amazing. That's one of those concepts you never forget once you've seen it.

@mharbol 3 года назад

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).

@mith873 3 года назад

i see tom i see 3b1b i click

@momolover3606 3 года назад

HENCE PROVED TOM ROCKED

@TomRocksMaths 3 года назад

@luorisluo3634 3 года назад

i am doing a fluid mechanics master degree and this really brainstorming, thanks so much for sharing.

@TomRocksMaths 3 года назад

Glad it was helpful!

@ShaunJW1 3 года назад

Both of you are assisting me with my physics maths degree, final year student ❤️

@TomRocksMaths 3 года назад

@vetrubio13 3 года назад

Finally I got the images method! 👏🏼👏🏼👏🏼

@TomRocksMaths 3 года назад

Happy to help :)

@baoboumusic 3 года назад

Are you kidding me? I just found your channel and you have a collab with Grant? Christmas came really early this year :)

@TomRocksMaths 3 года назад

Welcome :)

@mudkip_btw 3 года назад

Never seen the source in a channel before. Thanks Tom great video!

@mudkip_btw 3 года назад

I now see why you chose potential flow and the method of images to show to Grant, you had a brilliant example :D

@firusclad 3 года назад

Great video! The method of images is very useful when dealing with phenomena that can be treated as linear, e.g. in (linear) acoustics.

@BobBeatski71 3 года назад

I understand the concept, but the math leaves me in the dust !

@TomRocksMaths 3 года назад

This is 2nd year maths undergraduate level so don't feel bad!

@arl5564 3 года назад

"The wall is a mirror" love it!

@wingpandora6381 3 года назад

Youre a really good teacher wow

@TomRocksMaths 3 года назад

@berryzhang7263 3 года назад

Yesss we need more Tom and grant collabs

@TomRocksMaths 3 года назад

I second this comment.

@magtazeum4071 3 года назад

Two legends.. love both of them

@wingpandora6381 3 года назад

Also a question how do people even think of this abstract idea it feels magical

@muhammadsaid4654 3 года назад

Kutta, blasius, zhoukovsy etc they were all extremely gifted

@muhammadqaisarali 3 года назад

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.

@EpicMathTime 3 года назад

Is Grant really huge or is Tom really small? Or am I just bad at understanding camera angles?

@LeoStaley 3 года назад

Grant is large.

@fburton8 3 года назад

Why, man, he doth bestride the mathematical world. Like a Colossus.

@abhinanda2967 3 года назад

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 :)

@TomRocksMaths 3 года назад

Glad you enjoyed it!

@MatesMike 3 года назад

Epic colab!

@IrshaadAdatia 3 года назад

Oh goodness.... Math bursting at the seems, all we need now is an onlyfans. Hahahahaha.... Kidding not kidding. Fluid flow, for sure.

@lloydgush 3 года назад

Tom is a good example why there isn't a nobel for math. I hope you got the joke.

@pappaflammyboi5799 3 года назад

Is the word "seems" intentionally misspelled?

@20031bibi 3 года назад

@@lloydgush not me explain pls lol

@lloydgush 3 года назад

@@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.

@20031bibi 3 года назад

@@lloydgush LMAOOOOOO

@alexwolffe7805 3 года назад

Maths, fluid dynamics, Tom and Grant. Simply amazing.

@InvisibleThinker 3 года назад

This is what happened when mathematicians go with the flow!

@TomRocksMaths 3 года назад

I laughed.

@atrumluminarium 3 года назад

That was so beautiful ❤️ I miss fluid dynamics

@TomRocksMaths 3 года назад

it's awesome isn't it?

@maurice22ravel 3 года назад

Next video: "Grant and I are a couple now! #MathLove"

@mu.makbarzadeh2831 3 года назад

You both are incredible! Thanks for this video!

@TomRocksMaths 3 года назад

Glad you enjoyed it!

@carlosciudad-real2602 3 года назад

This video deserves way more views

@Medellinish 3 года назад

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.

@sebf9847 3 года назад

Reminded me of the same thing

@TomRocksMaths 3 года назад

The same method is indeed used in electrostatics - well remembered :)

@bryanbischof4351 3 года назад

This was really great.

@TomRocksMaths 3 года назад

@adayah2933 3 месяца назад

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.

@martinibarra4903 3 года назад

I really love this type of content

@Ghost____Rider 2 года назад

Where was this video when I was doing fluid mechanics last year 😭

@Upsallauniversity123 3 года назад

Please makes videos on streamline, streakline, pathline and stream functions etc 🙏 please 🙏.

@TomRocksMaths 3 года назад

Added to the video idea list - thanks!

@pourushsood Год назад

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?

@Aquadolphin314 3 года назад

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?

@TomRocksMaths 3 года назад

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.

@luisbreva6122 3 года назад

Just in time for my EM exam lol thank u

@NachoSotoBustos 3 года назад

This was amazing 👏🏻

@TomRocksMaths 3 года назад

Glad you enjoyed it Nacho!

@MA-nx3xj 3 года назад

As Richard Feynman put it - the flow of "dry water"...

@johnchessant3012 3 года назад

That was awesome! Especially the infinite reflection one

@TomRocksMaths 3 года назад

Glad you enjoyed it John :)

@richcole157 11 месяцев назад

What about a wall with two holes in it and does it generalize to n dimensions.

@Abhinav-ib2er 3 года назад

Heyyy, you can also describe flow around rotating circle in uniform flow which replicate flow around airfoil as used by earlier aeronautical scientists

@TomRocksMaths 3 года назад

Absolutely - the concepts introduced here are incredibly useful!

@timotay22 3 года назад

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!

@TomRocksMaths 3 года назад

Oh there were far too many good ones to include them all...

@BobBeatski71 9 месяцев назад

Ahhh, that's what PotentialFOAM does.

@kartikkalia01 3 года назад

Cool nerdy stuff

@TomRocksMaths 3 года назад

It's what we do :)

@ws_zilch 3 года назад

14:39 plot twist 🔥😂

@vwcanter Год назад

It's not clear why Tom says there is no divergence in the field with a source. There would obviously be a divergence, wouldn't there? Isn't that what a source is? I also don't understand the complex factor z that is added to the expression for w. It appears to already have a magnitude and a direction everywhere, without z. And it just appears out of nowhere. Why is there a z in there?

@rat_king- Год назад

Hold on.... doesnt this assume that the source is positionally static. as in, not migrating in space due to its changes upon the flow field... What occurs for a migratory source?

@lloydgush 3 года назад

Well, tom shows us the reason why we don't have a nobel for math... lol!

@Michael-il8ls 2 года назад

why complex values? why not x y?

@johnchessant3012 3 года назад

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?

@johnchessant3012 3 года назад

On second thought, I realized that you can do it with a contour integral, which is a bit less tedious.

@TomRocksMaths 3 года назад

I was thinking contour integrals...

@zubin8010 3 года назад

18:14 Grant is thinking: "did you see that throw? nailed it!" but, unluckily, Tom is focused on the graph :(

@TomRocksMaths 3 года назад

Haha I think I said a quiet 'nice' before immediately getting back to the maths...

@zubin8010 3 года назад

@@TomRocksMaths Nice! I had missed that

@Celthiccness 3 года назад

Now I'm remembering fluid dynamics... Oh no... the screams. The terrible screams.

@phenixorbitall3917 2 года назад

Great :)

@TomRocksMaths 2 года назад

Thanks!

@asklar 3 года назад

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?

@TomRocksMaths 3 года назад

Yes, this will work for any potentials. The definition I use for a 'potential' is something that can be written as a gradient.

@aftermath__7060 2 года назад

could you explain how did you differentiate and apply the limits at 21:08

@TomRocksMaths 2 года назад

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

@balajisriram6363 3 года назад

Love eeeeeeeeeeeeeetttttttttt!!!!

@Erin-ks4jp 3 года назад

Are the conditions of incompressibility and irrotation equivalent to a meromorphic potential? - it looks like they should be.

@ws_zilch 3 года назад

U speaking the gods language hooman..

@TomRocksMaths 3 года назад

There is a lot of crossover with Complex Analysis which is where the 'complex potentials' idea comes from

@cheasify 3 года назад

Cool just like electrostatics.

@TomRocksMaths 3 года назад

Yes, so I've been told!

@NEONHYPERTURTLE 2 года назад

Im a little confused, why are we able to just add another potential flow when using the method of images? won't that equation be different than the 1 source case (This is for the infinite plane case)

@TomRocksMaths 2 года назад

adding in the other sources acts to create the 'wall'. the flow pattern is in fact the same for two symmetrical sources and one with a wall - and you can verify this with experiments

@adrianhimmelreich3911 2 года назад

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

@TomRocksMaths Год назад

I recommend 'Elementary Fluid Dynamics' by David Acheson

@erinhopper6568 3 года назад

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"

@maxm1947 3 года назад

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?

@scott_the_engineer 3 года назад

Amazing video. How would you calculate the flow with a curved surface instead of a flat plane?

@TomRocksMaths 3 года назад

Ah, now that requires a completely different theory... this only works for 2D flows.

@squareroot1697 3 года назад

Just found your channel!

@TomRocksMaths 3 года назад

Welcome :)

@7head7metal7 3 года назад

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 😁

@TomRocksMaths 3 года назад

Ah yes, the infinitely-sided polygon that is the sphere...

@kristianwichmann9996 3 года назад

I'm getting flashbacks from electrostatics class.

@saqarkhaleefah6159 3 года назад

I just failed an exam on partial differential equations 🧚🏼‍♀️ method of images contributed to that failure

@TomRocksMaths 3 года назад

Hopefully this helps :)

@khajiit92 3 года назад

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?

@murilloprestesvilla9539 3 года назад

Great video! I have some doubts regarding pressure drag, and I thought this could be a good place to post it xD. Maybe someone could help me :D Starting from the beginning: why potential flows circulating a body (like a sphere) produces zero drag? I ready this is the D'Alembert paradox, and I can see that there is no pressure difference between the front and the rear of the sphere, so there is no resulting drag. But it is very non-intuitive to me in the sense that in the front, the flow is going towards the sphere, hitting it, and in the back, it does not hit it. It simply moves away. How does conservation of momentum work here? Second: is D'Alembert, right? Does pressure drag for flows circulating a body exists only for fluid with non-zero viscosity? If so, why is pressure drag always independent of viscosity? (at least it is what I read in various places). What is the explanation for the D'Alembert paradox? I ready that Prandtl said it was because of the separation of the boundary layer, but apparently, this was not right.

@TomRocksMaths 3 года назад

The paradox can be addressed by including viscous effects. In this video - and potential flow in general - we assume the flow is inviscid, and so viscous effects are ignored to help to simplify the governing equations. Whilst this approximation holds true in some cases, it of course will not hold true in all cases. Hence the paradox.

@sschmachtel8963 3 года назад

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?

@alexgu8745 3 года назад

Wait. Why don't you use a mass balance? Or use simply the eqns of continuity and momentum? Wouldn't it be more intuitive than just throwing model eqns

@TomRocksMaths 3 года назад

In this situation, the method of images is much easier than solving the full NSEs...

@timotejbernat462 3 года назад

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

@TomRocksMaths 3 года назад

Yes z is a complex number, so z= x + yi where x and y are real.