Two bubbles

[nancylebov]

If there were an infinite universe full of water, and there were two bubbles of air in it, would they move towards each other? Away from each other? Stay put?

If they moved close enough together to touch, would they coalesce into one bubble?

Would anything turn out differently if the universe were full of cold honey?

I don't have the answer--a couple of my friends who know more physics than I do kicked the initial question around. They and I have intuited the same answer, but no one has a proof.

Comments

[sturgeonslawyer.livejournal.com]

If there were an infinite universe of water and there were two bubbles of air in it, they would eventually dissolve in the water.

[juuro.livejournal.com]

Beautiful. Absolutely so.

[juuro.livejournal.com]

Cold honey -- same end result, slower process.

On touch, definitely coalesce. The total energy of one big bubble is lower than that of two smaller bubbles.

Initially, the bubbles would be drifting rather randomly. It is conceivable that when they drift "sufficiently close", they would be starting to "shade" each other from the random buffeting of the surrounding fluid, and be pressed closer together. What this "sufficiently close" means, I have no clue.

[brooksmoses]

On touch, not definitely coalesce; you've got to break the surface tension in the wall between them first. That's actually pretty hard to do, and in a universe without impurities it will be even harder. Most bubbles in our universe tend to simply bounce off each other.

(Soap bubbles are really bad for the intuition on this, because -- being film bubbles rather than bubbles in liquid -- they act quite differently. To begin with, they stick to each other and happily produce large flat regions of contact surface that are easy to break. Bubbles in liquid don't do that, unless they're physically pushed into each other.)

[brooksmoses]

(lilairen pointed me at this, presumably because I'm a grad student in geekery-of-bubbles-and-droplets, and because I enjoy this sort of thing muchly.)

Here's my take on the matter:

In an infinite universe full of water, you will have an infinitely large black hole in an amount of time that may or may not be infintesimally small; the gravitational forces will cause it to collapse. The bubbles will move towards each other in the process -- everything is moving towards everything else -- and then become annihilated like everything else.

So, to make this interesting, we need to postulate an "expansion" force in this universe that counteracts gravity and prevents the universe from collapsing. There are a couple of obvious simple ways to do that. The first is to simply "turn off" gravity completely; the second is to postulate a uniform expansion force everywhere that is tuned so that the universe neither flies apart nor collapses.

The answer turns out, as one might expect, to depend on what sort of additional expansion force one postulates.

First off, let's ignore any turbulent motion in the water. If there were such, it would die out from viscous effects, and then we'd be reduced to the problem where there weren't such.

Another thing to ignore is the possibility of charged bubbles. In our universe, most bubbles have some electrical charge, albeit miniscule, stored on the surface of the bubble. If the bubbles in this hypothetical universe are charged, they will either repel or attract each other according to the laws of electrostatics. If they attract each other, they will eventually touch, and at that point probably equalize their charges and start repelling each other again.

So, with that out of the way, we consider the interesting stuff.

If gravity is turned off, the motion of the bubbles will be directed by molecular forces -- the sorts of things that in our universe will be completely negligible in practical problems. After gravity (which fades at r^2, but has been turned off) and electric charge (which has also been declared zero), the next most long-distance effect will be electric dipole effects (which are, I think, r^4) that arise on a molecular scale. There will, as a property of the air-water interface, be some form of very small dipole layer at the surface of the bubbles. Since this is the same on both bubbles, it will be a repulsive force. A teeny, tiny, fading as r^4 repulsive force that is only relevant because there is nothing else going on in the universe anywhere. And so they will, at speed measured probably in nanometers per fortnight, move away from each other.

At those speeds, the only difference that honey instead of water would make is how strong the surface dipole layer is and how fast they move.

If instead we go for the uniform repulsive force to counteract gravitational collapse, then we can say with a bit more certainty what happens. Consider the forces on one of the bubbles. From any given bit of water, it receives a gravitational pull, and it also receives (from somewhere) an expansion force that exactly counteracts the gravitational pull. The net there is zero. Now, from the other bubble, it receives a very tiny fraction of that much gravitational pull, but it's still receiving an expansion force large enough to counteract the gravitational force that would be coming from water in that location. The expansion force is thereby much stronger than the gravitational attraction from the other bubble, and thus it (very slightly) pushes the first bubble away from it.

It is, of course, possible to postulate an expansion force that was carefully designed to push the bubbles together, or even to keep them at exactly some fixed distance from each other, but it would be an unnecessarily complicated one.

[nancylebov.livejournal.com]

Just wondering about that infinitely large black hole--obviously, there's no particular location where it would form. I suppose random motion could cause small increases in density (in water? maybe not) which would start a positive feedback process.

Still, what with infinity, I'm not sure that all the black holes would catch up with each other, and an infinite number of infinite black holes seems as likely.

[(Anonymous)]

Assuming there is no circulation or currents of any type in the water, and neglecting Brownian motion, they should drift apart.

The air bubbles are regions of lower-than-average density in the universe. As such, they will effectively be gravitationally repelled from each other; as far as one bubble is concerned, there's ever-so-slightly less mass in the direction of the other bubble than there is in every other direction.

This very small effect would of course be swamped by any circulation that may be present. I don't really want to work through the math to figure whether it
would be swamped by Brownian effects; the Brownian motions are certainly larger, but it's a random walk and so I suspect it would mean the change in separation distance would be non-monotonic, but still tending to separate.

I think they'd bounce if they touched, or stay stuck together without coalescing. Surface tension is pretty powerful.

The creator would have to be a bee.

[wolfdancer.livejournal.com]

Would anything turn out differently if the universe were full of cold honey?