Batchelor's law describes the process of mixing in turbulent fluids.
It has been around since 1959. It started out as a phenomenological
law, but a series of papers over the last 1.5 years has mostly proved
it.
I mention this because you might get asked about it. There has been
a fair amount of media buzz. Alas, most of the coverage has been
incomprehensible, wrong, or both. The primary sources (the research
papers) are not comprehensible to non-experts. HOWEVER, a remarkably
good discussion, aimed at a general audience, appeared about a week
ago, written by Kevin Hartnett: https://www.quantamagazine.org/mathematicians-prove-batchelors-law-of-turbulence-20200204/
I also mention it as an example of what current high-end research
looks like. Students sometimes get the impression that all of
physics was figured out in the 1600s or 1700s, but that's really
not true.
Even when you have a phenomenological law, proofs are important,
because they tell you the limits of validity. Otherwise you are
always walking on eggshells, wondering if the law is going to
betray you at the worst possible moment.
Mixing is a big deal. I mix stuff essentially every day of my life.
Besides ordinary kitchen mixing, consider swimming-pool chemicals.
I mix reagent A with some water in a vial and shake it. Then I add
indicator B and shake some more. Molecules are reeeeally tiny. Even
when there is a large excess of B, is it even possible in principle
for every (or nearly every) A-molecule to meet up with a B-molecule
on any practical time-scale?
And suppose I need to add acid to the pool. I buy concentrated acid
that would be insanely dangerous if not diluted. I built a fancy
mad-scientist apparatus that dilutes the acid by a huge factor and
feeds it in slowly over the course of an hour or two.
-- But even then, how long do I need to wait before it gets evenly
distributed throughout the pool? (Hours, unless people are in
the pool stirring things up.)
-- And then, how long before the reactions go to completion?
(Many hours for dissolved CO₂ or nitramines to come out of solution,
unless I sparge it with a powerful aerator.)
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I once asked Richard Feynman what he would work on if he couldn't do
particle physics. He immediately replied: Fluid dynamics. He explained
that it was really complicated, even more complicated than elementary
particles, and there were unsolved problems lying around all over the
place. Unsolved, but probably not unsolvable. And they were problems
people cared about, problems that had important real-world consequences.