I have a old peach schnapps bottle that’s been repurposed as a meditation bottle. It has glitter and black, white, and clear plastic balls inside. You shake it and think deep thoughts while you watch the contents swirl and settle. I imagine that I could shake that thing until Heat Death, and I would never see the same pattern repeated.
The bottle is 12" tall, but 3" is unfilled neck, and it is 3" in diameter.
It’s got 20 black, 20 white, and 20 clear balls, each the size of a marble. They all eventually settle, but it takes a while. There are 1000 (let’s say) pieces of glitter ranging from fat flakes to tiny little specks. Most of the glitter settles, but some small amount floats at the top.
What kinds of modeling would mathematicians use to describe or predict the state of the bottle’s contents? My math background includes college Calc, but I’ve forgotten most of that, so be kind.
OR you need to more drastic simplifications of the contents and what you consider a “state” in which case you have your choice of level of complexity from high school math to the aforementioned model of all the content.
Depending on the definition of “same” you could alter the order of magnitude of the answer by nearly as much as you want. Pick a specific “resolution” to which you will evaluate the total travel paths of each object and an upper and lower limit for how little, or how much you want to agitate before beginning measurement. Number of particles to the power of the the number of discriminated paths, because of the limits is less than “until heat death” for a match.
The number of paths without resolution limits is transfinite, for each particle. (Aleph sub 2, I think?) The amount of agitation only matters with finite solutions. Number of particles is a much smaller variable. (necessarily finite.) However, long before the heat death of the universe, successive iterations of agitating will have ceased.
Horace Lamb, one of the mathematical greats, was quoted thus: “I am an old man now, and when I die and go to heaven there are two matters on which I hope for enlightenment. One is quantum electrodynamics, and the other is the turbulent motion of fluids. And about the former I am rather optimistic.”
To address the OP’s question a little more literally than others have thus far, this sort of problem is generally solved via partial differential equations. The Navier-Stokes equations are particularly relevant to fluid dynamics. And as others have suggested, this sort of thing would be solved on a computer, most likely using the finite difference method.
And reaching college calculus is a lot better than most Americans do—there’s no shame in that whatsoever!
I used to lead a team of CFD (Computational Fluid Dynamics) engineers in the past. On a scale of 1 to 10, where 10 represents the CFD model’s capability to predict the system fully, I would give current Commercial (and research) models about a 3 where two phase motion like the above is concerned.
For relevance, particles interacting with fluids is common in rocket nozzles, liquid fuels combustion in gas turbines, many chemical reactors, etc etc. Most of these designs are done empirically or with data from smaller scale experiments. Dipping dots ice cream is a result of one such experimental test.
In the past Eulerian - Lagrangian models were used for CFD which failed whenever the particle concentration go to about 5%. Eulerian -Eulerian models introduced in the last few years have improved representativeness but there’s still a long way to go.
Thanks, everybody. I won’t pretend that I can discuss the matter intelligently, but I have enjoyed looking up the various concepts mentioned here. I really like the transfinites and alephs. I will say that I have a better understanding now (keeping in mind the quote from Sir Lamb, of course).