From: Leigh Palmer <palmer@sfu.ca>
Subject: Re: bubble in container
X-Xxmessage-Id: <A764DB7F6C011C1F@rs15-annex3.sfu.ca>
X-Xxdate: Mon, 28 Dec 92 00:27:43 GMT
Organization: Simon Fraser University
X-Useragent: Nuntius v1.1.1d16
Date: Tue, 29 Dec 1992 00:26:32 GMT

In article <1992Dec28.214917.27561@CSD-NewsHost.Stanford.EDU> Vaughan R. 
Pratt, pratt@Sunburn.Stanford.EDU asks several questions about this 
fascinating problem, for which I thank him:

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

My hydrostatic solution applies to the initial and final states of the 
system, both of which are static. The original question asked if the 
pressure changed as the bubble rose. I take it that if the initial and 
final pressures differ, then the pressure must have changed. I will note 
that the solution applies to any intermediate height of the bubble as 
well.

>[interesting argument on another, slightly related, hydrodynamic problem 
>omitted here, since I'm not solving a hydrodynamic problem]
>
>...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,...

These are not limits to the validity of my solution in the ideal case. I 
invented the inverted test tube variant for a practical barostat (note
the 
"stat" suffix, as in "thermostat") because when containers are not 
absolutely rigid and fluids are slightly compressible, it helps to have a 
larger bubble.

>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.

Viscosity plays no role in a static problem, but hydrostatic equilibrium 
obtains during the bubble's rise in the limit of zero viscosity.

>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.

I guess that is correct for the hydrodynamic case.

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

The surface tension does not matter at all because, in the case of either
a 
bubble or a test tube, the geometry of the surface is the same anywhere
it 
is brought to rest. Any surface tension contribution to the pressure will 
be constant, which I will subsume into my "P".

>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.

I guess that's why I solved an easier problem. :-))

I infer that Sushil had already discovered the result I've derived for
you 
before he asked his question. He just wanted someone to confirm it for
him. 
I felt the same way the first time I figured it out while contemplating a 
Cartesian diver.

Am I correct, Sushil?

Leigh
Newsgroups: sci.physics