From: pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt)
Subject: Re: bubble in container
Organization: Computer Science Department,  Stanford University.
Date: Mon, 28 Dec 1992 21:49:17 GMT

In article <1992Dec26.173915.16068@sfu.ca> Leigh Palmer <palmer@sfu.ca> writes:
>
>Second, note that the bubble now establishes a pressure in the fluid in 
>the horizontal plane of its location, since it is in hydrostatic 
                                                      ^^^^^^^^^^^
>equilibrium with the fluid. Only the position of the bubble is necessary 
 ^^^^^^^^^^^
>to specify the elevation of this horizontal isobar.

How do you justify hydrostatic reasoning in a hydrodynamics problem?
If the bubble is on its way up the system is not in equilibrium.

>The pressure increases everywhere in the container by an amount
>            delta P = rho g h,
>where 
>                rho = density of water
>                  g = acceleration of gravity
>                  h = the vertical distance the bubble rises

I'll buy this for the pressure change at the bottom, but the following
hydrodynamic argument would appear to invalidate your "everywhere."
Consider a *large* bubble starting at the bottom and rising to the top:

                         -------           -------
                        |~~~~~~~|         |       |
                        |~~H20~~|         |  AIR  |
                        |~~~~~~~|         |       |
                        |~~~~~~~|  --->   |       |
                        |       |         |~~~~~~~|
                        |  AIR  |         |~~~~~~~|
                        |       |         |~~H20~~|
                        |       |         |~~~~~~~|
                         -------           -------

Now neglect all forces attributable to viscosity, surface tension, and
turbulence, and imagine that a small tube of air penetrates the water
from bottom to top (whether by external intervention, spontaneous
symmetry breaking, or whatever, is immaterial).  The water is now free
to fall, with the air blasting freely up the tube.  While the water is
in free fall the pressure *throughout the container* equals that of the
air, which in the absence of any countervailing forces is uniformly
distributed.  When the water reaches and settles down at the bottom,
the pressure remains unchanged throughout the air-filled region, but
increases linearly with depth in the water-filled region.  The pressure
at the bottom is now that of the air plus  rho g h  where h is the
height of the water.  This is your delta P, valid at the bottom but
nowhere else.

This argument is valid for any ratio of water to air volume, whence it
remains valid for small air bubbles.

Now take viscosity etc. into account.  The water is indeed no longer in
free fall.  However the same factor slowing down the progress of the air
bubble also slows down the propagation of the pressure differential
between the bubble and the side of the tube.

The above argument shows that your formula is not valid (except at the
bottom) for sufficiently low surface tension, viscosity etc.  Can you
give limiting conditions under which your formula *is* valid?  E.g. an
arbitrarily small air bubble, arbitrarily narrow container, arbitrarily
high viscosity,...

Viscosity:  The problem with increasing the viscosity would seem to be
that it retards transmission of the pressure defect to the side of the
container.  It is not at all clear to me whether more or less viscosity
is better for your formula.

Width:  If the water is much wider than it is deep then the time of
transit of the bubble would be short compared to the time for its
pressure differential to move out to the side, and conversely for a
narrow container.  Thus a narrow container would seem to help your
formula.

Surface Tension:  Enough surface tension will hold the bubble together
and prevent the "tube of air," presumably helping your formula.

The combination of high surface tension, low to medium viscosity, and a
narrow container would seem like the optimal combination for which your
formula would be a good approximation.  There should be a single
formula combining these factors to give a measure of goodness of your
approximation.  Finding a reasonable such formula seems like an
extremely hard hydrodynamics problem.
-- 
Vaughan Pratt				There's safety in certain numbers.
Newsgroups: sci.physics