Potential Flow and Method of Images with

[@TomRocksMaths]

Grant also taught me something - check out the 'Power Tower' video here: kzbin.info/www/bejne/mp-9ZKuinstsjKM

[@leif1075]

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

[@jaredjones6570]

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

[@danielkron2513]

Ah, yes, two sexiest mathematicians in one video

[@LeoStaley]

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

[@davidgjam7600]

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

[@joshuajoshua46]

BONK

[@Arbmosal]

you must be unaware of Ed Frenkel :D

[@klutchboi3266]

@@davidgjam7600 👀

[@alexwolffe7805]

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

[@Nick08352]

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

[@dee8163]

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

[@zetsubou1108]

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

[@Hi_Brien]

I have a class in 4 minutes

[@okhan5087]

Same!

[@franteryda4730]

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

[@sudheerthunga2155]

Hehe ikr!!

[@Eyes_On_America]

Good luck :D

[@franteryda4730]

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

[@tapuwachitiga2547]

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]

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

[@PapaFlammy69]

:v

[@fidgetgadget3475]

hello papa flemmy

[@amaarquadri]

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]

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]

reading this comment felt really good @@marcocecchi9853

[@luckyw4ss4bi]

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

[@TomRocksMaths]

Thanks Connor :)

[@googleit1370]

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]

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]

[@MrWhiteVzla]

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]

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

[@nickwisely2581]

"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]

agreed :)

[@gandalftolkien2879]

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]

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]

[@theverner]

I needed this for electrodynamics

[@VibratorDefibrilator]

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!

[@EmilyMGin]

Yay! More Grant and Tom collabs!

[@sebastianmorales9787]

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

[@TomRocksMaths]

I'm down

[@leif1075]

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

[@tylercrowley2559]

@@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

[@likithstochastic]

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.

[@ashoulle8953]

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

[@asklar]

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

[@momolover3606]

HENCE PROVED TOM ROCKED

[@TomRocksMaths]

[@brad5387]

I think that’s so cool how they differentiate the sum to to get it into a harmonic series for that converges and then they just integrate it back once it’s neat and tidy for the potential they need.

[@aero33888]

Not just maths but also chemistry!! ❤️

[@benwinstanleymusic]

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

[@gaeb-hd4lf]

Channel is growing my man, awesome!

[@TomRocksMaths]

[@mith873]

i see tom i see 3b1b i click

[@luorisluo3634]

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

[@TomRocksMaths]

Glad it was helpful!

[@firusclad]

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

[@ManojKumar-cj7oj]

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

[@ShaunJW1]

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

[@TomRocksMaths]

[@koketsomohale8596]

This is the best video on the internet

[@TomRocksMaths]

A bold claim, but I'm not complaining - thank you

[@prdoyle]

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

[@mharbol]

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).

[@wingpandora6381]

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

[@muhammadsaid4654]

Kutta, blasius, zhoukovsy etc they were all extremely gifted

[@muhammadqaisarali]

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.

[@mudkip_btw]

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

[@mudkip_btw]

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

[@baoboumusic]

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

[@TomRocksMaths]

Welcome :)

[@arl5564]

"The wall is a mirror" love it!

[@vetrubio13]

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

[@TomRocksMaths]

Happy to help :)

[@Medellinish]

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]

Reminded me of the same thing

[@TomRocksMaths]

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

[@magtazeum4071]

Two legends.. love both of them

[@InvisibleThinker]

This is what happened when mathematicians go with the flow!

[@TomRocksMaths]

I laughed.

[@aftermath__7060]

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

[@TomRocksMaths]

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

[@berryzhang7263]

Yesss we need more Tom and grant collabs

[@TomRocksMaths]

I second this comment.

[@christianorlandosilvaforer3451]

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

[@TomRocksMaths]

Hello in Colombia!

[@wingpandora6381]

Youre a really good teacher wow

[@TomRocksMaths]

[@abhinanda2967]

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]

Glad you enjoyed it!

[@maurice22ravel]

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

[@domc3743]

brilliant content as usual, thank you

[@TomRocksMaths]

[@richcole157]

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

[@EpicMathTime]

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

[@LeoStaley]

Grant is large.

[@fburton8]

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

[@IrshaadAdatia]

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

[@lloydgush]

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

[@pappaflammyboi5799]

Is the word "seems" intentionally misspelled?

[@20031bibi]

@@lloydgush not me explain pls lol

[@lloydgush]

@@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]

@@lloydgush LMAOOOOOO

[@BobBeatski71]

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

[@TomRocksMaths]

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

[@johnboard407]

Eye candy, brain candy. Also happpy 2021 to the both of you!

[@TomRocksMaths]

Happy New Year John!

[@johnboard407]

@@TomRocksMaths Thanks Tom 😊

[@mu.makbarzadeh2831]

You both are incredible! Thanks for this video!

[@TomRocksMaths]

Glad you enjoyed it!

[@BobBeatski71]

Ahhh, that's what PotentialFOAM does.

[@carlosciudad-real2602]

This video deserves way more views

[@bryanbischof4351]

This was really great.

[@TomRocksMaths]

[@MatesMike]

Epic colab!

[@adayah2933]

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]

I really love this type of content

[@maxm1947]

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?

[@atrumluminarium]

That was so beautiful ❤️ I miss fluid dynamics

[@TomRocksMaths]

it's awesome isn't it?

[@timotay22]

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]

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

[@Ghost____Rider]

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

[@euclidselements9522]

Yes i love these team ups

[@TomRocksMaths]

you and me both :)

[@flirkami]

Could someone explain the differentiation and simplificatiom step? I don't even really know what he has written down there ..

[@felicote]

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).

[@scott_the_engineer]

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

[@TomRocksMaths]

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

[@Abhinav-ib2er]

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

[@TomRocksMaths]

Absolutely - the concepts introduced here are incredibly useful!

[@Aquadolphin314]

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]

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.

[@Upsallauniversity123]

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

[@TomRocksMaths]

Added to the video idea list - thanks!

[@pourushsood]

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?

[@asklar]

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]

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

[@adrianhimmelreich3911]

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]

I recommend 'Elementary Fluid Dynamics' by David Acheson

[@luisbreva6122]

Just in time for my EM exam lol thank u

[@alanx4121]

I want to learn about longitudinal potential flow with compression and decompression.

[@Erin-ks4jp]

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

[@ws_zilch]

U speaking the gods language hooman..

[@TomRocksMaths]

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

[@zubin8010]

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

[@TomRocksMaths]

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

[@zubin8010]

@@TomRocksMaths Nice! I had missed that

[@MA-nx3xj]

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

[@lloydgush]

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

[@johnchessant3012]

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]

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

[@TomRocksMaths]

I was thinking contour integrals...

[@mitchellsteindler]

Cool concepts

[@NachoSotoBustos]

This was amazing 👏🏻

[@TomRocksMaths]

Glad you enjoyed it Nacho!

[@squareroot1697]

Just found your channel!

[@TomRocksMaths]

Welcome :)

[@erinhopper6568]

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"

[@Michael-il8ls]

why complex values? why not x y?

[@johnchessant3012]

That was awesome! Especially the infinite reflection one

[@TomRocksMaths]

Glad you enjoyed it John :)

[@Celthiccness]

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

[@amandeep9930]

Hey Grant, make a series on Manim library

[@nopo_b3645]

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?

[@babajani3569]

Hello sir, I just found out that you are/were a proffessor at Oxford. I am currently doing my A levels and really want to get into cambridge. Could you plz recommend some books or give some tips on how to do better in the STEP exam (granted that in oxford, they do MAT) or the cambridge interview and what kind of problems I can expect to find in it? Thank you.

[@TomRocksMaths]

I don't know about any specific books, but my main advice would just be to practice solving these types of maths problems. The questions on the STEP and MAT are similar in style, and are generally very different to what you see at A-level, so the more of these you are able to work through the more familiar you will be with that particular type of thinking. This is the same type of logical thinking that we are testing in the interviews too!

[@babajani3569]

@@TomRocksMaths ok thx but for some reason I just cant seem to be improving at all. But thx anyways.

[@harrisoncentner6876]

My physics professor mentioned using this method for magnetic fields. Is that the same thing?

[@GuruPrasad-qu4vg]

Yes it is,you are solving the Laplace equation in both cases

[@tameemmabruk8089]

@@GuruPrasad-qu4vg ^ I concur with that.

[@khajiit92]

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?

[@rat_king-]

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?

[@7head7metal7]

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]

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

[@timanb2491]

hi Tom, is there any book or online course that can be the best intro of fluid dynamics?

[@TomRocksMaths]

G K Batchelor - an introduction to fluid dynamics

[@alexgu8745]

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]

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

[@NEONHYPERTURTLE]

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]

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

[@murilloprestesvilla9539]

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]

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.

[@drdca8263]

If instead of the boundary being a straight line, you instead have some wacky curve (that still extends out to infinity, not closing back on itself, so that the region we are dealing with is simply connected), can you do the same thing by using the Riemann mapping theorem to map the whole space to, I guess the half space, and doing it there? Would that work? edit: looked it up : it appears that conformal maps (which the Riemann mapping theorem gives) do preserve these things, and so my impression is that the answer is yes, that should work.

[@TomRocksMaths]

Yes, conformal maps are incredibly useful tools for these kinds of problems!

[@drdca8263]

Tom Rocks Maths Thanks!

[@matrics192]

what is linear exactly? so that we can sum up the potentials

[@TomRocksMaths]

the potentials come form solving Laplace's equation which is linear, meaning you can add solutions together and still get a valid solution. You do need to be careful of changes in the boundary conditions though.