WATCH THE WHOLE SET... Playlist: bit.ly/NavierPlaylist Part 1 (Navier-Stokes): th-cam.com/video/ERBVFcutl3M/w-d-xo.html Part 2 (Reynolds Number): th-cam.com/video/wtIhVwPruwY/w-d-xo.html Part 3 (River Water): th-cam.com/video/5mGh0r3zC6Y/w-d-xo.html
how fascinating! as a kid in Melbourne, my friends and i would always swim on the right of the stormwater drains that emptied into the bay, because the water was always cleaner on that side -- and now, 30 years later, this explains why!
This trilogy of videos is SUPERB. Tom succeeds in making viewers understand and have a feeling for fluid physics. Such a “trouvaille” as the French say, what a discovery for this channel. Tom could probably explain many other areas of physics.
Just wrote a report on T. J. Crawford's thesis, gotta say I did not do it nearly the justice it deserves. If any viewers are watching this, I highly encourage you to give his work a read if you are in the field of mathematics or engineering or just find it neat. It was super cool but a 10 pager doesn't even come close to being long enough to discuss a 200+ page doctoral thesis. Very cool vid, and very inspiring work from a (hopefully) future mechanical engineer
Awesome video as always. Please feature Tom more in more of your videos. His subject matter is super interesting and he does a phenomenally good job explaining such complex matters.
This is a really fascinating topic and a lot of research is being done in this area. I’m a first year student in physical oceanography and while I have yet to dive into research as I am still taking classes how this volume flux (angular spreading) of a river entering the ocean affect the longshore currents and the creation of sandbars which affects the surf would be interesting to learn more about. Great stuff dude!
Started watching the video because I was drawn by the title and my curiosity to know what it was about. My interest picked up once you stated the real life application.
This gives me the following questions: 1) When working in this, did you look at the process and effects of delta formation at the river mouths? 2) Wouldn't it be easier to clean the water while it's in the river rather than once it's in the ocean/sea? That's a neat dissertation topic. I love these sorts of head-slappers where you realize no one has ever studied a phenomenon; it seems obvious someone else must have already asked that question.
One thing that's interesting to me about mathematics is that it's very easy to work out most of the factors or parameters that contribute to something like river flow. Math and physics would be similar to the quantitative difficulty of political science, writing, or art if that were a decent chunk of the work. But it seems so much more difficult to work out the precise interactions between some set of parameters than it is to qualitatively list them out. The difference is just enormous, and I think that's really the thing that makes math and physics so much more difficult than other fields. It's just extraordinarily hard sometimes to nail down the precise influence things have on one other.
Wow, this means we can expect coastal currents from left to right everywhere in the Northern Hemisphere. I did not expect that there is such regularity! Something learnt, thank you.
hey there, amazing work please upload more videos like this, where mathematics is applied in complex situations and simplifying them. The work was really creative and so was the idea of representing in such a simple way that it could be understood by anyone with very basic knowledge about of topic.
@rchandraonline, as @Silvar55x alludes to, flux depends on direction. However, it is not a vector. Rather, flux is a measure of how much stuff flows "directly" through a given surface. That is, it measures how much stuff flows through a surface in the direction perpendicular to the surface. In the case of volume flux for a river, the surface we consider is the boundary between the river and the ocean. The volume flux will indeed have units of [length]³ [time]⁻¹. For example, consider the case (call it case A), where water is flowing out of the mouth of the river directly (i.e. in a straight line, perpendicular to the coastline), then the flux is just the volume of water flowing out of the river per unit time. However, consider the case (case B) where that same amount of water was flowing out of the river mouth in a direction angled 60° away from the straight line perpendicular to the river mouth, then the volume flux in case B is only half that of case A, since cos(60°) = 1/2. *_EDIT:_* The above paragraph is wrong; volume flux is indeed equal to the amount of water displaced; the correct description of case B should be "water is flowing out of the river with the same speed (magnitude of velocity) as in case A, but in a direction 60° away from the surface normal". Then, indeed, the volume flux in case B is half that of case A.
The discharge is the net flow through the system. This rate is the same regardless of how you draw your boundary. Flow perpendicular to the boundary is faster but narrower. Flow at an angle is wider but slower. I think flux refers to flow per unit width. Technically it means flow per unit area, but for river flows the height dimension is ignored.
@@thumper8684, yes, it seems I had incorrectly described my example, but I've edited my comment now to correct this. In the case of volume flux, it is in fact equal to the amount of stuff (i.e. volume of water) that passes through the boundary, but this is in general only the case when we're measuring the flux generated by a _velocity field_ associated with some substance. The reason that this simple "volume of water displaced" quantity is commonly referred to using the term "flux" is because this quantity can easily be calculated for a given boundary surface S by F = *u* • *n* A where F is the volume flux over S, *u* is the velocity of the fluid over S, *n* is the unit vector perpendicular to S pointing towards the ocean (else F will be negative), and A is the surface area of S. Of course, this is for constant velocity fields and flat surfaces, so for a general velocity field *u* and general surface S, we instead have F = integral of [ *u* • *n* dA ], for all infinitesimal elements dS of S, each of which has area dA. These expressions are the prototypical expressions for what is, in physics as a whole, referred to as flux. Compare with magnetic flux through a surface, electric flux through a surface, etc.
Flux is flow and it comes from a definite volume of the river, so Volume flux is totally justified and time has nothing to do with it as a river never stops. So, flow of a particular volume = Volume flux.
that prevents new pollution (but where would you do that exactly? You can never be entirely sure there isn't another source of pollution further downstream), but we're looking for where the bulk of the existing pollution is, I assume...
One of the assumptions was that the water was basically infinitely deep for the purpose of the equation, but if the river water tends to follow very closely to the coast then that doesnt seem appropriate. The water is shallowest along the coast, from a certain perspective it's infinitely shallow! The inverse of that assumption!
Later on the video, it states that the width of these flows are usually of the order of 10km, so the section near the shore could be considered to be negligibly small. Plus there's the shoaling effect near the coast, which further complicates the picture. With the other strong assumptions being used here, the error caused by assuming deep water throughout likely doesn't cause too many issues.
That seems reasonable. I had it in my mind that the plumes were one or two orders of magnitude smaller than that. I was also was imagining bay water and not sea coasts; the beach/coast that I'm most familiar with seems to stay shallower than 10m for quite a ways out, but it being in a bay is probably a factor.
The whole of the North Sea is very shallow. Even 10 to 20 km out from the coast it's about 30 metres deep, so it seems his assumptions are not quite right. Then again, you do have to make lots of assumptions when working in fluid mechanics and if you're starting a theory you need to start somewhere
Yeah, of course there is probably error from the assumptions. But they said in the video that they checked the theoretical results against actual measurements, and they seem to line up pretty well, so the error can't be too high. But like a different commentor said, you have to start somewhere. So you make big assumptions, and then you find the error. And if that error is too big, you go back and try and redo it with less assumptions. With fluid mechanics and really all of engineering, you're never going to get it completely right. It's about knowing how big your error is, and deciding whether or not that's too big.
Hey there, I have a quick question: Assuming that the cross section of the flow is rectangular, the multiplication of all 3 equations should be equal to the volumetric flow rate Q, however when multiplied together they come to 3Q/2. Is this because the initial rectangular cross section assumption is incorrect, or due to the other assumptions in the video?
I'm assuming the equations in the video are supposed to give a rough approximation of what is going on. I wouldn't expect things to work out exactly. Also 3Q/2 isn't too far from Q for an approximation. I also think the cross section of the flow would be better approximated by a very stretched half-ellipse, maybe that explains the difference.
Ive had another thought. Getting 3/2Q suggests the formula for the cross section is 2/3*u0*w0*h0 doesn't it? So whatever shape that's the formula for might have been the cross section shape he assumed it was.
This is an excellent check on the results. I bet the actual vertical and horizontal profiles are not constant within this box, but rather h and w are scales for a varying functional form.
@warumbraucheichfüryoutubekommentareeinescheissgooglepluspagefragezeichen Depends on what you mean by “before it flows into the ocean.” According to Google, the Rhine river is ~1232 km long. It would be insane to clean the river anywhere along its length. It would make more sense to clean the water near the river’s mouth. As for why they let the dirty water enter the ocean instead of cleaning directly at the mouth, I would guess it’s because most of the cleaning vessels are somewhat large and need ocean depth to move. Alternatively, rivers usually have commercial and private boats driving up and down the river; maybe they don’t want to interfere with commercial flow.
If you apply _logic_ to the problem you'll find cleaning up the river before it meets the sea is far more effective than chasing chemical unbalance 100 km out to sea. Filtering a river without disturbing the flow is perhaps a problem for _topologists_ . This needs the counsil of mathematicians.
Alistair Shaw he means before it hits the sea at all, in essence questioning why you need to know the measurements for the water once irbid in the sea, which is what all the maths was for
I got confused about it turning to the right in the northern hemisphere.... does it not matter if the river flows east or west (ie: with or against the earths rotation?) or north or south?
Indeed it doesn't. The Coriolis effect does not care what your heading is, it just turns you in a given direction dependent only on your distance from and side of the equator.
The very amazon is a funky example as well because its flow reverses for a while during spring equinox (where sun and moon are well aligned to combine their tidal forces on the equator)... But these are not what that tried to model anyway. A decent approximation just has to work well within a given scope, and "This applies decently to most rivers out there" is already a nice result.
What effect does tidal action have? Many rivers in the US Pacific northwest have large estuaries. Does Coriolis force effect the tidal bore? If the water is mixed in the estuary does it still behave like you describe? I find this very interesting.
I use to live in Churchill Manitoba, we could see the water separation based on density and it made a clear line of fresh and salt water keeping separated as it flowed to the right, this is why the polar bears are there because the river flows into a natural basin and the fresh water cools and freezes faster pooled in the corner. That is why it freezes there first.
Great video, like all great maths applications it presents more questions than it answers. Like, what’s the impact of different contaminant densities (compared to the water, plastics or chemical solutions can have dramatically different densities), what’s the impact of lateral flows and pressures (like waves and tides), how do currents and sea streams interact?
You can see it with your own eyes, if you find a river that carries a lot of sediment so it'll be a darker colour and if there is a hill near by you can see the path it takes, you can see it from lower down you just get a smaller snapshot. A place I know where you can see it is Aberystwyth in Wales.
At 6:58 Tom assumes that the ocean is infinitely deep. I understand that this may hold true when we are looking at a point very far from the coast. However, I don't understand how the ocean can be so deep near the coastline. Doesn't the ocean depth decrease when we move further from the coastline (i.e. shallowest near the coast)? So is the assumption that the ocean is infinitely deep (near the coast) correct?
I love this channel. It changes me every time I watch a video. From henceforth, I will never look at a river's ocean outflow the same way again...these equations will forever flash through my mind as I watch the waters meet, and I will see the numbers come to life
Very interesting video... but I was going to ask the same question! XD I guess it's because the speed of the water in the river it's much higher and then it's more difficult to do it "in real time"... How could you filter 2000m^3/s? But actually I have no clue ':)
that's a solution a much larger scale though. most of the times the people accountable for what goes on the rivers are not the ones in charge of the coast
Cleaning trash is relatively* easy. Cleaning chemical or microparticle pollution is a completely different story. You would need to build a water treatment plant at the mouth of the river. But, in doing so, you can destroy the ecosystem of the river and maybe even the ecosystem at the mouth of the river.
There are a few probable answers. 1) You can't always deploy gigantic equipment in the river, both because of its depth and the constructions alongside it 2) Preventing pollution won't remove pollution that is already there. (and you will never 0 pollution output, that's impossible)
Very cool! I live in the midwestern United States, and the first thought I had was if this would apply to the Great Lakes? With fresh water into freshwater, would the densities be too similar? Are the lakes too small to experience the coriolis forces? A quick view at some satellite images looks like some of the plumes of larger rivers go counter-clockwise, so maybe it is strong enough? Very interesting, Thanks!!
From experience with applied mathematics I strongly suspect dimensional analysis was used to get the form of these results prior to the fancy footwork deriving them. Tip: math is easier if you start knowing what the answer ought to look like. Need a video on using dimensional analysis to amaze people.
What if there is archipelago? How Islands, plants and uneven shallow water affects the flow? (like some places in Baltic sea) I guess easiest is to either use numeric approximation with a super computer or install probes to measure flows.
I sail on the severn estuary which has strong tides. There is a reputation that the sea is particularly rough in wind over tide conditions (ie the wind going in the opposite direction to the tide). Naturally the tide will effectively add (or subtract) a few knots (maybe 5) to the relative wind, but the effect seems more pronounced than that. I have looked for an explanation but cant find one. Unless it is purely phsycological, navier stokes must be the reason. I wondered if it was something you had ever thought of, if you know of any research into it, or if you are curious. I would love to see a video on it.
How does Ocean currents impact the flow? The Ocean current in California flows north to south but according to your maths the river current flows south to north. Does it not mix and have two bodies of water on top of each other flowing in different directions?
I like how silly assumptions like an infinitely deep ocean can make sense in fluid dynamics, it always amazes me how much information can be found by making a few assumptions.
Would gradient of density be a variable? Like I can’t imagine none of this water is mixing with sea water to some extent. Is this negligible in modeling?
Here's a question/comment/concern: based on the numbers/estimates he gives, the volume flux doesn't quite add up. Typically for a fluid flow, the volume flux is equal to the product of the flow's width, height, and flow speed. But here are his numbers for the Rhine: width: 10-20 km height: 10-20 m speed: ~1 m/s volume flux: ~2000 m^3/sec BUT, if you multiply the flow's width, height, and flow speed, you get: 10,000m * 10m * 1m/s = 100,000 m^3/second on the low end, and 400,000 m^3/sec on the high end. What's the reason for this discrepancy?
Amazing! Nice one! I can say I understood one of your videos at least! Anyway it way great. BTW I'm studying the effect of the turbulence in a following formula car during a race. Navier Stokes is the way to go!
Though I like me some Navier stokes, why would you try to clean the water right after it flowed into an ocean/sea and not before, like in the delta area? 🤔
I love the video. But what about tidal currents. The current must affect the vortex of water at the river mouth let alone the direction of water exiting and entering the river mouth?
Sir thank you for giving us knowledge. I published two bachelor Research paper in computer (infinite instructions time complexity by help of Ramanujan infinite sum and Reimann hypothesis ) by getting knowledge from your youtube videos
WATCH THE WHOLE SET...
Playlist: bit.ly/NavierPlaylist
Part 1 (Navier-Stokes): th-cam.com/video/ERBVFcutl3M/w-d-xo.html
Part 2 (Reynolds Number): th-cam.com/video/wtIhVwPruwY/w-d-xo.html
Part 3 (River Water): th-cam.com/video/5mGh0r3zC6Y/w-d-xo.html
Numberphile Y
Great
Where can we get his paper?
Thank you, this satisfies my October minimum requirements for a Navier-Stokes vid. This is a brilliant series.
how fascinating! as a kid in Melbourne, my friends and i would always swim on the right of the stormwater drains that emptied into the bay, because the water was always cleaner on that side -- and now, 30 years later, this explains why!
I just watched a guy talking about river water for 15 minutes. No regrets.
you can take any action and be reductive about it and make it sound trivial.
Stiggy Vanderkskeen you arent so bright are you
??
Vorticity: noun
1. How swirly a current is
adjective?
@@gabrielshaw2140 vortical
how about vorticious
Something to do with vortex I assume
@@pianojay5146 prefix -vert, means "turn." Vert is pretty close to vort, so I assume it's the same thing
Just met Tom Crawford today, he is an amazing fella, very friendly and energetic, awesome to listen and talk to !
I'd love to read the steps for deriving these equations from Navier-Stokes with those given assumptions.
Yes please
Upvoting this. Great idea.
Agree!
It would be cool if these equations match the equations we get from Dimensional Analysis.
It's the topic of Tom's PhD thesis. I would assume the steps wouldn't be easily understandable.
This trilogy of videos is SUPERB. Tom succeeds in making viewers understand and have a feeling for fluid physics. Such a “trouvaille” as the French say, what a discovery for this channel. Tom could probably explain many other areas of physics.
Thanks Luis - I'm glad you enjoyed the series!
You should make a live feed video of yourself talking about rivers. You know, stream it.
Most underrated comment of the year
Get out! 😂
??
1:51 - ah yes, my favourite Windows Media Player visualiser effect
Feeling nostalgic
I love this videos with the Water Professor! I hope there'll be more videos with him!
Just wrote a report on T. J. Crawford's thesis, gotta say I did not do it nearly the justice it deserves. If any viewers are watching this, I highly encourage you to give his work a read if you are in the field of mathematics or engineering or just find it neat. It was super cool but a 10 pager doesn't even come close to being long enough to discuss a 200+ page doctoral thesis. Very cool vid, and very inspiring work from a (hopefully) future mechanical engineer
Never considered the way a river flows out to sea before. Very interesting!
Holy moly, this guy is so clever! It’s so amazing that formulas as simple as these would be derived from the NS eqn’s!
Haven't watched Numberphile in a while, now excited to watch more
I was really hoping for this third one. Please, don't hesitate on doing more content with Tom Crawford.
Awesome video as always. Please feature Tom more in more of your videos. His subject matter is super interesting and he does a phenomenally good job explaining such complex matters.
This is a really fascinating topic and a lot of research is being done in this area. I’m a first year student in physical oceanography and while I have yet to dive into research as I am still taking classes how this volume flux (angular spreading) of a river entering the ocean affect the longshore currents and the creation of sandbars which affects the surf would be interesting to learn more about. Great stuff dude!
0:30 Wow, I never actually realized that the Amazon River is pretty much EXACTLY over the Equator
This was definitely the best of the group of his videos you've done so far.
Started watching the video because I was drawn by the title and my curiosity to know what it was about. My interest picked up once you stated the real life application.
The title is an anti-clickbait. I first was like 'duhhh to the sea, everyone knows that', but it was brilliant.
This gives me the following questions:
1) When working in this, did you look at the process and effects of delta formation at the river mouths?
2) Wouldn't it be easier to clean the water while it's in the river rather than once it's in the ocean/sea?
That's a neat dissertation topic. I love these sorts of head-slappers where you realize no one has ever studied a phenomenon; it seems obvious someone else must have already asked that question.
I like this guy, I'd like to hear more from him.
One thing that's interesting to me about mathematics is that it's very easy to work out most of the factors or parameters that contribute to something like river flow.
Math and physics would be similar to the quantitative difficulty of political science, writing, or art if that were a decent chunk of the work.
But it seems so much more difficult to work out the precise interactions between some set of parameters than it is to qualitatively list them out. The difference is just enormous, and I think that's really the thing that makes math and physics so much more difficult than other fields. It's just extraordinarily hard sometimes to nail down the precise influence things have on one other.
humble tip: clean it while it's in the river
*before it's in the river
What if you don’t have any access to the river that’s polluting your sea!!!
@@numberphile sue them big time
@@kosmar I don't think China cares about lawsuits topkek
...or maybe just stop polluting the rivers?!?!
Wow, this means we can expect coastal currents from left to right everywhere in the Northern Hemisphere. I did not expect that there is such regularity! Something learnt, thank you.
hey there, amazing work please upload more videos like this, where mathematics is applied in complex situations and simplifying them. The work was really creative and so was the idea of representing in such a simple way that it could be understood by anyone with very basic knowledge about of topic.
These three videos were fing excellent! Thank you, and more power to Tom Crawford.
That must be the coolest looking PhD guy of all times.
@@琥珀-u3o okay boomer
"Volume flux" ...fair enough term, although that seems quite the same as "flow rate"...like [liters or cubic meters] per [minute], or something.
Flux has direction. It's a vector (or vector field/function), whereas "flow rate" is a scalar.
@rchandraonline, as @Silvar55x alludes to, flux depends on direction. However, it is not a vector. Rather, flux is a measure of how much stuff flows "directly" through a given surface. That is, it measures how much stuff flows through a surface in the direction perpendicular to the surface. In the case of volume flux for a river, the surface we consider is the boundary between the river and the ocean. The volume flux will indeed have units of [length]³ [time]⁻¹.
For example, consider the case (call it case A), where water is flowing out of the mouth of the river directly (i.e. in a straight line, perpendicular to the coastline), then the flux is just the volume of water flowing out of the river per unit time.
However, consider the case (case B) where that same amount of water was flowing out of the river mouth in a direction angled 60° away from the straight line perpendicular to the river mouth, then the volume flux in case B is only half that of case A, since cos(60°) = 1/2.
*_EDIT:_* The above paragraph is wrong; volume flux is indeed equal to the amount of water displaced; the correct description of case B should be "water is flowing out of the river with the same speed (magnitude of velocity) as in case A, but in a direction 60° away from the surface normal". Then, indeed, the volume flux in case B is half that of case A.
The discharge is the net flow through the system. This rate is the same regardless of how you draw your boundary. Flow perpendicular to the boundary is faster but narrower. Flow at an angle is wider but slower.
I think flux refers to flow per unit width. Technically it means flow per unit area, but for river flows the height dimension is ignored.
@@thumper8684, yes, it seems I had incorrectly described my example, but I've edited my comment now to correct this. In the case of volume flux, it is in fact equal to the amount of stuff (i.e. volume of water) that passes through the boundary, but this is in general only the case when we're measuring the flux generated by a _velocity field_ associated with some substance. The reason that this simple "volume of water displaced" quantity is commonly referred to using the term "flux" is because this quantity can easily be calculated for a given boundary surface S by
F = *u* • *n* A
where F is the volume flux over S, *u* is the velocity of the fluid over S, *n* is the unit vector perpendicular to S pointing towards the ocean (else F will be negative), and A is the surface area of S.
Of course, this is for constant velocity fields and flat surfaces, so for a general velocity field *u* and general surface S, we instead have
F = integral of [ *u* • *n* dA ],
for all infinitesimal elements dS of S, each of which has area dA.
These expressions are the prototypical expressions for what is, in physics as a whole, referred to as flux. Compare with magnetic flux through a surface, electric flux through a surface, etc.
Flux is flow and it comes from a definite volume of the river, so Volume flux is totally justified and time has nothing to do with it as a river never stops. So, flow of a particular volume = Volume flux.
14:42 - subtitles on point
Loving these fluid mechanics videos, so interesting to see...
As a chemical engineer I really enjoyed this fluid mechanics series
Why not clean the river water.... BEFORE it gets to the Oceans...
Easier to calculate the flow, depth and capacity in the River.... IMO
Hmm, just thinking out loud... what if the river is in a different jurisdiction and you have no control over cleaning it?
You are correct, but it would have to be a combination of both that as well as cleaning up the already polluted ocean.
that prevents new pollution (but where would you do that exactly? You can never be entirely sure there isn't another source of pollution further downstream),
but we're looking for where the bulk of the existing pollution is, I assume...
Agreed... The oceans still need to be cleaned. Stopping the original pollution at the source will make the oceans cleanup less overwhelming.
It might be inconvenient to boats that go up and down the river, if you do it on the coast, it's much less problematic for this.
Absolutely fascinating video. Expertly explained, and extremely relevant. Thank you for producing these videos!
Love this and the whole trilogy!
Wow... today I realised how fluid mechanics helps to improve our environment!!
Very nice 💓
3 over 2 to the nine-quarters...
Yer a wizard, Tom!
Oh it was but a trilogy? Well, I hope it's like star wars then... I can't wait for the second trilogy on the Navier-Stokes equations!
I'm looking forward to the prequel to Navier-Stokes. Preferably starting via Economic problems.
@@ElijsDima Meesa Nav-Nav Stokes! Yousa gonna be biiiiiig excited with prequels!
@@ElijsDima if we wanna go there, why not start from continuum mechanics?
One of the assumptions was that the water was basically infinitely deep for the purpose of the equation, but if the river water tends to follow very closely to the coast then that doesnt seem appropriate. The water is shallowest along the coast, from a certain perspective it's infinitely shallow! The inverse of that assumption!
I was thinking the same thing, but then he said W0 was 10 - 20 km. Which is hardly the coast.
Later on the video, it states that the width of these flows are usually of the order of 10km, so the section near the shore could be considered to be negligibly small. Plus there's the shoaling effect near the coast, which further complicates the picture. With the other strong assumptions being used here, the error caused by assuming deep water throughout likely doesn't cause too many issues.
That seems reasonable. I had it in my mind that the plumes were one or two orders of magnitude smaller than that. I was also was imagining bay water and not sea coasts; the beach/coast that I'm most familiar with seems to stay shallower than 10m for quite a ways out, but it being in a bay is probably a factor.
The whole of the North Sea is very shallow. Even 10 to 20 km out from the coast it's about 30 metres deep, so it seems his assumptions are not quite right. Then again, you do have to make lots of assumptions when working in fluid mechanics and if you're starting a theory you need to start somewhere
Yeah, of course there is probably error from the assumptions. But they said in the video that they checked the theoretical results against actual measurements, and they seem to line up pretty well, so the error can't be too high.
But like a different commentor said, you have to start somewhere. So you make big assumptions, and then you find the error. And if that error is too big, you go back and try and redo it with less assumptions.
With fluid mechanics and really all of engineering, you're never going to get it completely right. It's about knowing how big your error is, and deciding whether or not that's too big.
Tom Crawford - The new Numberphile Superstar xD
Wow. These formulas are so amazing. My mind is blown
I had no idea riverwater could be so interesting
as Richard Feynman once said "nearly everything is interesting, once you dig deep enough"
@@dle511 Don't go much deeper than 10-20m though, then it is all boring ocean waters :-)
@@vincentpelletier57 Or... you could find underwater volcanic vents with strange sea life living near it!
Wait till you see mixed compressible/incompressible fluid mechanics (gas/liquid mixed state) :|' *bangs head against mandelbrot fractal poster*
If you think that's interesting, you should look into marsh water!
Hey there, I have a quick question:
Assuming that the cross section of the flow is rectangular, the multiplication of all 3 equations should be equal to the volumetric flow rate Q, however when multiplied together they come to 3Q/2. Is this because the initial rectangular cross section assumption is incorrect, or due to the other assumptions in the video?
I'm assuming the equations in the video are supposed to give a rough approximation of what is going on. I wouldn't expect things to work out exactly. Also 3Q/2 isn't too far from Q for an approximation. I also think the cross section of the flow would be better approximated by a very stretched half-ellipse, maybe that explains the difference.
It does equal Q doesn't it? You get 1 power of Q when 1/2+1/4+1/4=1 from h0*w0*u0. How did you get 3/2?
Oh I see what you did now, nevermind.
Ive had another thought. Getting 3/2Q suggests the formula for the cross section is 2/3*u0*w0*h0 doesn't it? So whatever shape that's the formula for might have been the cross section shape he assumed it was.
This is an excellent check on the results. I bet the actual vertical and horizontal profiles are not constant within this box, but rather h and w are scales for a varying functional form.
Love this dude's look 0:11. Mad respect. Mad inspiration.
Wouldn't it make more sense to clean up the pollution from the river BEFORE it flows into the sea?
@ warumbraucheichfüryoutubekommentareeinescheissgogglepluspagefragezeichen genau deinen Namen habe ich mich auch gefragt xD
@warumbraucheichfüryoutubekommentareeinescheissgooglepluspagefragezeichen Depends on what you mean by “before it flows into the ocean.” According to Google, the Rhine river is ~1232 km long. It would be insane to clean the river anywhere along its length. It would make more sense to clean the water near the river’s mouth. As for why they let the dirty water enter the ocean instead of cleaning directly at the mouth, I would guess it’s because most of the cleaning vessels are somewhat large and need ocean depth to move. Alternatively, rivers usually have commercial and private boats driving up and down the river; maybe they don’t want to interfere with commercial flow.
It would make the most sense to not add pollution to the river. probably cheaper, too.
Would it make sense to close the door AFTER the horse has already left the barn?
That way ships can use the river as normal because we know the river water doesn’t assimilate that quick and we know exactly where it is
LOVE THIS! Physics is so much fun to learn!
I would LOVE to see Tom Crawford on a Numberphile2 podcast!!!
If you apply _logic_ to the problem you'll find cleaning up the river before it meets the sea is far more effective than chasing chemical unbalance 100 km out to sea. Filtering a river without disturbing the flow is perhaps a problem for _topologists_ . This needs the counsil of mathematicians.
Exactly what i was about to comment.
You havent said anything different to the video.
Alistair Shaw he means before it hits the sea at all, in essence questioning why you need to know the measurements for the water once irbid in the sea, which is what all the maths was for
I was thinking the same, like wtf, do it where the river flows out or just Be4...
I got the impression they're trying to clean up past damage, which has left the river a long time ago.
Beautiful. Thank you for this amazing series
This is brilliant indeed , thanks for such fascinating videos
Perfect! Thank you. Didn't know such simple things might be such interesting.
This is a GREAT video. Beautifully explained.
I got confused about it turning to the right in the northern hemisphere.... does it not matter if the river flows east or west (ie: with or against the earths rotation?) or north or south?
Indeed it doesn't. The Coriolis effect does not care what your heading is, it just turns you in a given direction dependent only on your distance from and side of the equator.
Coriolis effect is counterclockwise in the northern hemisphere and clockwise in the southern hemisphere.
So, is the river water layer still 10-20m deep when it is 10-20km out, or is the depth tapered as it gets further from the coast?
When you live on the Baltic coast and he video ends "none of this applies there"...
The very amazon is a funky example as well because its flow reverses for a while during spring equinox (where sun and moon are well aligned to combine their tidal forces on the equator)...
But these are not what that tried to model anyway. A decent approximation just has to work well within a given scope, and "This applies decently to most rivers out there" is already a nice result.
Same mate, same...
Exactly xD
But it explain why Baltic sea is more polluted.
@@ashishranjan4623 I believe, much of the pollution comes from its souther coast.
What effect does tidal action have? Many rivers in the US Pacific northwest have large estuaries. Does Coriolis force effect the tidal bore? If the water is mixed in the estuary does it still behave like you describe? I find this very interesting.
I use to live in Churchill Manitoba, we could see the water separation based on density and it made a clear line of fresh and salt water keeping separated as it flowed to the right, this is why the polar bears are there because the river flows into a natural basin and the fresh water cools and freezes faster pooled in the corner. That is why it freezes there first.
"Let's make it easy for ourselves"
-Dr. Crawford
💕 6:24
Keep up with the good work Numberphile team 🔥🔥🔥
There was a great experiment to test this in Fukushima. The water was marked with radioactive isotopes to make them easier to track.
This was the best video of the three, I felt like I understood all of it-and I did A level maths! (I am aware this is degree level)
Great video, like all great maths applications it presents more questions than it answers. Like, what’s the impact of different contaminant densities (compared to the water, plastics or chemical solutions can have dramatically different densities), what’s the impact of lateral flows and pressures (like waves and tides), how do currents and sea streams interact?
Does Tom have a video etc showing how he got from the naiver-stoke equates to the 3 parameter equations?
You can see it with your own eyes, if you find a river that carries a lot of sediment so it'll be a darker colour and if there is a hill near by you can see the path it takes, you can see it from lower down you just get a smaller snapshot. A place I know where you can see it is Aberystwyth in Wales.
Intresting, I never thought of that myself!
These can be striking from the air. A local example (western Canada) is where the Thompson River enters the Fraser River at Lytton, BC.
Great work, Dr. Oak.
This sounds as "physics teacher" as it gets. Love it :D
At 6:58 Tom assumes that the ocean is infinitely deep. I understand that this may hold true when we are looking at a point very far from the coast. However, I don't understand how the ocean can be so deep near the coastline. Doesn't the ocean depth decrease when we move further from the coastline (i.e. shallowest near the coast)? So is the assumption that the ocean is infinitely deep (near the coast) correct?
I love this channel. It changes me every time I watch a video. From henceforth, I will never look at a river's ocean outflow the same way again...these equations will forever flash through my mind as I watch the waters meet, and I will see the numbers come to life
Why not clean the river while its nicely contained bettween land, so while it is still a river?
Why not prevent polution in the first place? O.o
Very interesting video... but I was going to ask the same question! XD I guess it's because the speed of the water in the river it's much higher and then it's more difficult to do it "in real time"... How could you filter 2000m^3/s? But actually I have no clue ':)
that's a solution a much larger scale though. most of the times the people accountable for what goes on the rivers are not the ones in charge of the coast
Cleaning trash is relatively* easy. Cleaning chemical or microparticle pollution is a completely different story. You would need to build a water treatment plant at the mouth of the river. But, in doing so, you can destroy the ecosystem of the river and maybe even the ecosystem at the mouth of the river.
There are a few probable answers.
1) You can't always deploy gigantic equipment in the river, both because of its depth and the constructions alongside it
2) Preventing pollution won't remove pollution that is already there. (and you will never 0 pollution output, that's impossible)
Maybe for pollution that is already in the sea... before you deploy that.
This little series is amazing and I absolutely love the way Tom explains its just great
Math is fecking awesome. Thanks for sharing this marvel.
Would be interesting to see what the outflow for a bay like the San Francisco Bay looks like.
No mention of rivers entering the ocean east or west (guess they only go north/south) or ocean currents traveling along the coast?
Very cool! I live in the midwestern United States, and the first thought I had was if this would apply to the Great Lakes? With fresh water into freshwater, would the densities be too similar? Are the lakes too small to experience the coriolis forces? A quick view at some satellite images looks like some of the plumes of larger rivers go counter-clockwise, so maybe it is strong enough? Very interesting, Thanks!!
From experience with applied mathematics I strongly suspect dimensional analysis was used to get the form of these results prior to the fancy footwork deriving them. Tip: math is easier if you start knowing what the answer ought to look like.
Need a video on using dimensional analysis to amaze people.
Wonderful. Absolutely Marvelous!
Now I want to see a video about the Baltic sea specifics...
Same
"the ocean is 100's metres deep"... Surely at the coast the sea water is only 20m-30m deep?
In fact, the whole of the North Sea is not much more than about 30 metres deep
What's interesting to me is ive never been past 2 meters deep swimming wise.
This video was super awesome
Tom mainly mentioned salinity but I guess it can matter to also encapsulate the temperature into Q because of thermal expansion? Great video!
What if there is archipelago? How Islands, plants and uneven shallow water affects the flow? (like some places in Baltic sea)
I guess easiest is to either use numeric approximation with a super computer or install probes to measure flows.
Will you make a video on yang mills equation?
Would love a link to Tom's dissertation...
Check the description
To Tom Crawford, I dedicate the song "Riverman" by Nick Drake, because he tells us about how his river flows.
15:00 Nice picture of the baltic sea. You can see clearly from that that eventually there will be a lake between Finland and Sweden.
I'm not sure why, but I have a feeling that... Could it be that he's a big fan of Pokémon?
?
@@ThePharphis He has a big Poké Ball tattoo on his left arm. It's a device for catching and storing Pokémon
@@Te4mRyouko ah, I didn't notice the tattoo
I want more videos of him!
I sail on the severn estuary which has strong tides. There is a reputation that the sea is particularly rough in wind over tide conditions (ie the wind going in the opposite direction to the tide). Naturally the tide will effectively add (or subtract) a few knots (maybe 5) to the relative wind, but the effect seems more pronounced than that. I have looked for an explanation but cant find one. Unless it is purely phsycological, navier stokes must be the reason. I wondered if it was something you had ever thought of, if you know of any research into it, or if you are curious. I would love to see a video on it.
How does Ocean currents impact the flow? The Ocean current in California flows north to south but according to your maths the river current flows south to north. Does it not mix and have two bodies of water on top of each other flowing in different directions?
I like how silly assumptions like an infinitely deep ocean can make sense in fluid dynamics, it always amazes me how much information can be found by making a few assumptions.
Would gradient of density be a variable? Like I can’t imagine none of this water is mixing with sea water to some extent. Is this negligible in modeling?
This guy has got impeccable mean of teaching
Here's a question/comment/concern: based on the numbers/estimates he gives, the volume flux doesn't quite add up. Typically for a fluid flow, the volume flux is equal to the product of the flow's width, height, and flow speed. But here are his numbers for the Rhine:
width: 10-20 km
height: 10-20 m
speed: ~1 m/s
volume flux: ~2000 m^3/sec
BUT, if you multiply the flow's width, height, and flow speed, you get:
10,000m * 10m * 1m/s = 100,000 m^3/second on the low end, and 400,000 m^3/sec on the high end. What's the reason for this discrepancy?
Amazing! Nice one! I can say I understood one of your videos at least! Anyway it way great. BTW I'm studying the effect of the turbulence in a following formula car during a race. Navier Stokes is the way to go!
first time something useful
Though I like me some Navier stokes, why would you try to clean the water right after it flowed into an ocean/sea and not before, like in the delta area? 🤔
I love the video. But what about tidal currents. The current must affect the vortex of water at the river mouth let alone the direction of water exiting and entering the river mouth?
Sir thank you for giving us knowledge. I published two bachelor Research paper in computer (infinite instructions time complexity by help of Ramanujan infinite sum and Reimann hypothesis ) by getting knowledge from your youtube videos
And love from India
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