From: Leigh Palmer <palmer@sfu.ca>
Subject: Re: bubble in container
X-Xxmessage-Id: <A761D905EC011C23@rs17-annex3.sfu.ca>
X-Xxdate: Sat, 26 Dec 92 17:40:21 GMT
Organization: Simon Fraser University
X-Useragent: Nuntius v1.1.1d16
Date: Sat, 26 Dec 1992 17:39:15 GMT

In article <1992Dec26.033119.5443@lynx.dac.northeastern.edu> michael 
kagalenko, mkagalen@lynx.dac.northeastern.edu writes:

>I can't bear a fealing of pity to the students of SFU, since all of the 
>above is incorrect. If bubble of air is not accelerating, the pressure
>does not depend on its position. (Doubts ? Recall conservation of the 
>momentum from your "Physics 101" :) I'd like to point out that Arhimed
>force on the bubble does not depend on its position as well, provided
>liquid is incompressible & container abs. rigid

Hmm... I'll try to relieve that "unbearable fealing" you're having. The 
phenomenon, as I said, is counterintuitive. That is why physics is of 
value; it replaces intuition when intuition fails us.

First, note that the volume, V, of the bubble is constant throughout 
this process. Thus, if the system is in thermal equilibrium at 
temperature T*, the pressure, P, in the bubble will remain constant 
throughout the process. This does not depend upon a particular model for 
the gas in the bubble, but only upon the *existence* of an equation of 
state P(V,T). You may think of it as an ideal gas, however, if you wish 
to do so. The practical efficacy of this phenomenon depends upon the 
bubble being very compressible, and a gas is a good choice.

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.

Third, consider the pressure at other points in the fluid in hydrostatic 
equilibrium in a uniform gravitational field which are located a 
distance z above the bubble. The pressure 

                        P(z) = P - rho g z,

where the symbols used are the same as before.

By raising the altitude of the bubble, and consequently of the 
(constant) P isobar, we decrease z everywhere in the fluid, thus 
increasing the pressure correspondingly.

Cute, eh? This mechanism, applied to bubbles in magma chambers closed at 
their tops, has been proposed as the source of the tremendous pressures 
associated with explosive volcanic eruptions. The energy which is seen 
in such eruptions is not stored in the relatively incompressible magma, 
but in the bubbles. The extreme pressures are necessary to rupture the 
capsules in the first place, and the bubbles (mostly steam, I believe) 
transport this energy to the surface by rising relatively isochorically. 
in the magma chamber.

This sort of counterintuitive phenomenon, when introduced to a class of 
students at SFU, usually initiates a lively discussion rather than any 
feeling of distress. I feel that students receive valuable exercise in 
thinking when discussing such things. You need not pity them; they do 
quite well. Last week one of our physics majors successfully defended 
his physics PhD thesis at Princeton, and another did so at Cambridge. We 
don't have results like that every week, but we do have a large number 
of our undergraduates out in the world doing quite well for themselves.

Leigh

*This is an Ansatz I intentionally neglected to state in order to 
heighten the sense of mystery by misdirection. You bit because you 
thought it was a mechanics problem.
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