🚀 go-pugleaf

In article <1993Jan8.021210.27077@murdoch.acc.Virginia.EDU>
crb7q@kelvin.seas.Virginia.EDU (Cameron Randale Bass) writes:

> ...
> Words are much more dangerous than equations.  Equations are well-defined
> and can be examined quantititively for correlations with experience.
> Words are fluid and mutable and ill-defined.
> ...

Alright!  Concur completely.  Precisely why I _stuck my neck out and sent
out the UC draft_ -- so others would whack me up the side of the head and
(maybe even occassionally) help me develop specific quantifications for it.

I rushed it because it has very specific experimental implications, some
of which seemed to be pretty relevant (and maybe even accurate).  This was
a terrible thing for me to do?  Maybe next time I should sit on it for, oh,
six months to a year and patent the daylights out of every other sentence?
Alas, I just don't have that high of an opinion of my own ideas!


> [terry writes:] ... _Do_ you call this a "shock wave," or not? ...
>
> Not exactly, there would be a rarefaction wave within the fluid.  Some
> people call your surface a 'contact wave'.  However, you are still limited
> in driving force to the ambient pressure...

Ah.  Thanks -- that will help in the lookup process.


> Shock tubes are run somewhat this way, but you don't obtain fusion.

The shock tube is nice as a verbal analogy, but it needs quantification.
E.g., how precisely does the spherical curvature and very small size of
the "bubble" vacuum scenario impact the behavior of contact waves?

I will try to do a literature search on this to find the relevant articles,
if they exist.  (Boy do I miss D.C.  UTA, maybe?  Do you guys let off-campus
computer types into your physics library without a leash?)


> One can do an analysis of this collapse (an empty bubble as per Rayleigh),
> but it breaks down at the time compressibility effects start to become
> important (and starts to impede the acceleration process).

Reasonable enough.  In UC I even mentioned compressibility slowdown in the
context of large, hard-to-rearrange molecules, and only proposed the wedge-
out as likely to apply to small, mobile molecules.  Unquantified, of course.


> An even better way, though, to get the fluid to higher thermal energy is
> to assume a spherical piston of some sort in a gas.  Give the 'piston' a
> massive whang, and a strong shock is created.  As the center is approched,
> the shock itself gets stronger.

Yes, I assume that this is the same Landau et al model you've mentioned
before as your approach to quantifying and formalizing cavitation problems.
($50K?  Wowzers!)  But do Landau et al explicitly include an analysis of the
contact wave case, or not?  (Sigh.  Sorry that I don't just look it up, but
I'm still trying to get a copy of the book.)


> One can solve this using similarity methods (c.f. Landau and Lifshitz,
> Fluid Mechanics, section 107). As the center of the the region approches,
> the energy within the shock goes as 
>
>     E ~ R^{5-2/a}
>
> Where $a$ is a similarity parameter (about 0.7 for polytropic gases with
> with gamma = 5/3 (monatomic) or gamma = 7/5 (diatomic).
>
> So, you can see that though the shock itself gets stronger, the energy it
> contains the decreases drastically as one approaches the center.

I hope I can interpret that correctly.  Please let me know if I messed up:

            E - energy
            R - Radius of shock front
          $a$ - Same as "a"?
            a ~ 0.7  (for a first approximation?)
   polytropic - Sorry, my physics dictionary is at home
        gamma - (??)  'aven't the foggiest
    R^{5-2/a} - R to the exponent ( 5 - 2/a ) => ( 5 - 2/0.7 ) => ~ 2.1 

I'll assume that to be the correct interpretation for the rest of the
discussion, and will apologize later if it is not... :)  I assume, by
the way, that this Landau et al equation is for shock waves only?  It
would seem plausible that modifications would be needed for contant waves.)

So I guess you are saying that total energy decreases something close to
R^2 as R drops towards zero.

Of course, since _volume_ decreases as R^3, this would mean that the energy
density (I'll call it D for the moment) in this shrinking region of space
is _increasing_ as:

    D  ~  R^2 / R^3  =  1/R

Now given that energy _density_ is what a lot of folks might call "heat,"
could you _please_ tell my why you seem to think everything that I say
is a violation of the second law, yet the equation (as I've been able to
interpret it) that you just gave is _not_?

Also, I note that the energy density increases to infinity as the shock
wave reaches the center, hmm?  And here I thought you disagreed with me!

I'm just kidding, of course. The billiard-ball like behavior of gas molecules
under high pressures -- fugacity, as Dale mentioned -- will dominate in the
end stages and keep such silly predictions from having any meaning.  The
question instead is how _far_ you can go with such increases before other
effects begin bleeding away too much of the energy to give intensification.

More specifically, the exponent for the total inbound energy wave equation
should itself be a monotonically decreasing function f(R), reflecting the
increasing importance of new energy draining mechanisms as the density and
temperature of the matter inside a sphere of radius R increases.  Thus the
maximum energy density will be reached when:

    E = R^(f(R))   and   f(R) = 3

... where the first equation gives the total inbound energy of the wave
front.  My wedge-out postulate is that for contact waves and small (e.g.,
monatomic gases), the equation f(R) will remain relatively flat and below
3 for several orders of magnitude of R closer as it approaches R=0 than
for classic shock waves.

I do not know (yet) how to quantify that postulate.  Nor I am convinced
(yet) that such a quantification has _every_ been fully studied for the
case of contact waves in very small spherical implosions.


> Look at this another way.  The inbound surface increases its velocity.
> The velocity is limited by a) real gas effects b) energy.

I'm unsure what you mean by "real gas effects."

If by that you mean some kind of contrast to "ideal gases," you might note
that I _never_ assumed ideal gases for the wedge-out idea.  I don't see how
you could and still expect to get plausible results on the size scales we
are talking about, especially during the last few nanoseconds.  Indeed, that
is precisely why I keep insisting on _specific_ quantification of what goes
on at a near-molecular level, not just extrapolations from larger scales.

As for energy, see my earlier disucussion about energy _densities_ being
the key issue, not the total wave energy.


> ... even if you figure out a way to simply extract some tail of the
> standard energy distribution (without violating the laws of thermodynamics),
> you have to go way way up the tail to get to fusion energies.  One has to
> go so far up the tail that there are probably no such molecules actually
> present in the fluid, and you cannot get the fluid to give them to by
> itself you without violating the second law.

Phffft.  I never proposed this.  I mentioned a diffusion selection mechanism
simply to point out that such things happen (e.g., in Hilsch tubes).  I was
very specific in saying that to be really interesting some sort of further
_acceleration_ of the gas molecules would be needed -- i.e., wedge-out.

What is the point of arguing against assertions I never made?

Also, you seem to be returning to the theme that _any_ instance of one
particle gaining energy from two or more other ones is a violation of the
second law.  It is not, and I really don't care to rehash the issue again.

You are very welcome to argue that the _specific_ case of very small, very
round contact waves cannot do it, but the issue is whether there is enough
structure and "order" of some sort in such systems to allow amplification
effects to occur in them.  _That_ is the real issue, not the second law.

My entire argument has been based on the presence of just that kind of
order (via the combination of high symmetry and no void gases), so I'm
not about to throw without a much more detailed analysis of this case.


> There is a way to quantify this gain in velocity, but it is not in the
> thermodynamically organized fashion presented.  However, my specific
> second law objection was to a specific scheme.  I do have general second
> law objections, but it seems silly to bring them up in the absence of
> some quantification.
> ...
> It doesn't matter.  There will certainly be acceleration, but pressure-
> limited, and nothing truly exciting for ordinary fluids under ordinary
> conditions.

A strong assertion, but I've heard somewhere or the other that "words are
much more dangerous than equations."  :)  Would you care to quantify _your_
claim that while there will be acceleration, it will be "nothing truly
exciting for ordinary fluids under ordinary conditions?"  (BTW, since when
is severe cavitation an "ordinary" condition?)

As for "pressure limited," I would like to see (or develop) a very specific
quantification of small, highly spherical contact waves before responding.

I might note that my own analysis of that scenario is based heavily on
information and entropy arguments, and from that analysis I say that such
cases are very interesting, even for ordinary fluids.  Your analysis appears
to be based primarily on an extrapolation of large-scale shock wave models
to very small scales and the contact wave situation.

I think there is room for further analysis and (I suspect) quantification
that goes beyond the standard texts you've mentioned.

				Cheers,
				Terry Bollinger


P.S. --	Dale, I know you from the other group.  You've made some great
	observations and suggestions, but I am worried that this could
	be turning into one of your infamous ping-pong matches.  Could
	we maybe tone it down a bit after this exchange?  Feel free to
	get your response to this one in, but my next one is liable to
	be a lot shorter (up to and including zero bytes!)

	I look forward to being blasted by you again in the near future.