The Navier—Stokes problem and the three atom universe.

What is the ocean, but a multitude of drops?

Okay, I previously studied materials science (which is a cross between engineering, chemistry, physics and maths). Maths was never my strong point. It’s nice to see people speaking english here. I am hopeless at maths. Really hopeless. I am more of a visual/graphical person.

I am having trouble even VISUALISING the problem here. Can someone show me *what the problem is* in a more GRAPHICAL format than showing equations?

Are you asking for a mathematical solution so you can predict where laminar to turbulent flow will occur in any theoretical system? Or are you asking what causes turbulence, where does it originate from?

If you are trying to understand and model where turbulence comes from in the real world, then I think you need to understand the real world at both large and small scales. Turbulence happens at all scales, from gases to galaxies, so it is a universal constant. So I’ll try to explain it like this:

On large scales, interactions between one atom and quintillions of millions of other atoms do not matter (I wouldn’t say ‘nothing’ but not very much anyway). Why not? I think that should be obvious, but it’s because minor fluctiations in an atoms’ position do not change bulk properties very much. It’s like adding a drop of liquid to a cubic metre of water, no big relative change. Well okay, “a drop in the ocean” has even become a common expression in the English language.

And what is the difference between large and small scales except just looking ever-closer at your X, Y and Z coordinates?

If the entire universe consisted of just one atom, the universe would be easy to predict. This one atom would just sit there, possibly vibrating and rotating around itself… and life would be pretty boring.

Now, if there were two atoms in the entire universe, they would probably orbit around eachother in a very predictable, ‘linear’ way. By ‘linear’ I mean “not progressive”; of course the atoms would move in circular orbits around each other. Or they would simply collide into one another leading to something I would called “the little crunch”. It would all be very easy to model. The point is, still nothing much would change with time. Even though in motion, it would be a fairly ‘static’ rather than a ‘dynamic’ system

However. If the entire universe consisted of only three atoms, nothing else. Things get very complicated, very quickly. Because, first of all, there exists in physics something known as the “n body problem“. Given three initial starting vectors, apparently it’s very difficult to predict the exact position and momentum where three measly atoms will end up.

Why is this so? Firstly, all of the atoms are attracted to one another because of Van der Waals forces. Also, each atom technically imparts an infinitesimally small gravitational pull on the other two. And any time gravity is involved, well, Newtonian mathematics doesn’t work too well. But there are magnetic forces from the nuclei as well. And when all of those forces are acting upon each other in three dimensions, it gets tricky. Because of the inverse-square law, the forces that interact for each of the atoms upon on the other two will be different at each point in time and space  (based on their proximity).

So on smaller physical scales, however, we know that a drop of water contains roughly 10^21 molecules. And those molecules are vibrating all over the place. They have all sorts of vibrational modes. They rotate. They spin. They vibrate at different frequencies. Nothing at all stays still. If it did, matter would soon annihilate itself. The electrons do not just orbit the nucleus in a circular fashion, but in complex dumbell shapes. The orbitals will even be different according to the type of atom, there are s, p, d and f shaped oribitals. We don’t even know with certainty where the electrons are orbiting due to the Heisenberg’s uncertainty principle (which takes effect whenever you actually try to measure anything at the atomic or subatomic scale). And all of the subatomic particles may interact with each other as well. Who knows? We don’t know.

So even when looking at the behaviour of a few atoms, as compared to a whole ocean, their position and momentum starts to become significant with respect to each other. Let’s look at what I’m talking about. A few people have tried to estimate the number of water molecules in all of this world’s oceans. And the number we get is from around 5 x 10^46 to 5 x 10^47. Yes you’ll note that those two numbers are out by a factor of ten. That is one order of magnitude margin of error just in the counting. Yes it’s that inaccurate. Because this is the real world and it’s just a ‘guestimate’ (close enough for the purposes of argument). Now keep in mind that they are very approximate figures, we certainly would never know the exact number, let alone the position and momentum of all that lot!

So for example, if I added one more molecule of water to an entire ocean, how much do you think all of those existing water molecules affect one single atom. The answer is “quite a bit”, because the number of them completely overwhelms that one molecule. But what about if I think about it the other way around? Does our additional lone water molecule impart much change to the other ten or a hundred ‘quattuordecillion’? How much influence does this extra molecule provide. It’s not ‘nothing’. It’s never ‘nothing’. Most people would say it is ‘negligable’. It is insignificant. Right? It wouldn’t change too many of the others’ motion paths. Or would it? Who can say? It’s like putting one more person in a stadium. Almost all of the other 100,000 people probably won’t notice the extra person. But if I add one extra atom to a two atom universe —if I get a divorce from the only person I know in the universe— then yes it does become very significant for both atoms.

One guy even mentioned on Quora that his instinct was that quantum (subatomic) interactions cause turbulence. If that is the case, then it truly would get ‘complicated’. Furthermore, physicists and cosmologists are saying that space and time itself might be ‘granular’ and not smooth. And then you are assuming that fluids are incompressible. Are they? Perhaps not 100%, no. I personally think there are no such things as fractions…

“I do not see two halves of an apple. I see quadrillions of whole atoms on each side.” Leslie Dean Brown

There are just that many variables. People like to quote god here. I don’t believe in god, because it isn’t necessary for me to understand the way the world works. For those people that insist on quoting god at this point, I’d say it like this: “not even god knows; nothing could or would know what is going on with all atoms and interactions at one instant in time, because EVERYTHING is interacting with EVERYTHING at that point in time”.

If you ask me, laminar flow is where maths and theory works. Turbulent flow is where maths breaks down. Does that sound like a grey answer? It is meant to be. I don’t think there is an answer. By the time you have done the calculations, the atoms will have moved elsewhere. It should be a trillion dollar problem. It’s going to be that complicated to try and figure out.