🚀 go-pugleaf

I have not seen a reply to the following inquiry:

>Sat, 1 Aug 1992
>Noam Bernstein: Air bubbles in water
>We came across a problem a few days ago when a frosh here asked us about
>it.  First consider two spheres of water (in zero G), in an infinitely
>big container.  Because of gravity, they attract.  Now, consider two
>spherical air bubbles in an infinite volume of water (0 G, again).  Now,
>which way do the bubbles move (if at all) ?
>After a long discussion (which involved some of the participants switching
>sides several times), we've convinced ourselves of what the right answer is
>(we think), but we want to see what you people think about it.
>                                                        Noam
>                                            noamb@egg.gg.caltech.edu

Consider first the case of a *finite* volume of water; for simplicity
(& to avoid the expense of a container), let it be spherical.  In
general, there will be buoyant forces on the bubbles; to minimize
these, place the midpoint of the pair of bubbles at the center of the
sphere.  Then the buoyant forces are in opposite directions, and the
bubbles will separate.  The gravitational field, and therefore the
buoyant force, at each bubble is somewhat lessened by the presence of
the other; in that modest sense they may be said to attract each
other, but still they separate.  This will continue to be true as the
sphere is made larger.  The putative limiting case where the sphere
fills all space and the buoyant force is 0 by symmetry is unreal.  It
is of course just as legitimate to make the body of water go to
infinity in an asymmetrical manner; in that case the buoyant forces
may become arbitrarily large, but in no case are they negligible.

[In general, the limit of "an infinite volume" of matter is
notoriously unphysical & must be approached with caution.  In
particular, the gravitational field is indeterminate: one may apply
Gauss's law to a sphere of chosen size & orientation and get any
answer one pleases.  Likewise, in hydrodynamical problems the integral
for the momentum is often not absolutely convergent, so that its value
depends on how one sweeps thru space during the integration.  It is
especially dangerous to combine spatial infinity with
incompressibility (as is tacitly done here), because that makes the
speed of sound infinite and allows momentum to disappear from the
visible universe instantaneously.]

I conclude that this is a nonsense question.  Nevertheless, I think
the frosh who asked it should have been named.

			Joe (Caltech '58, Ph, F)
			jcf@world.std.com