I'm not sure what the point of finding an example of a situation in where a "slow manifold" doesn't exist does. I have yet to see an article where the slow manifold fails to exist in the context of removing the acoustic mode from the compressible equations. In the event you do then there are a huge number of researchers out there who cannot justify their constraint manifold and have computed flows with a nontrivial acoustic component.

An undergraduate can be pointed to the continuity equation and asked what happens when rho is constant. They will likely tell you the divergence of the velocity is zero. They can also be told this is "incompressible flow." It may be right but it is not a particularly informed or enlightened discussion. One can certainly assume a solenoidal flow at the outset but this is entirely unmotivated. The reason one does it, knowingly or not, is because the acoustics gets removed (isothermal, small Mach number).

The firm physical basis for the solenoidality of the velocity comes when one finds the slow manifold. The origin of the word is rather extraneous to the topic at hand.

The bottom line here is that people need to remove the acoustic component of the governing equations or they are wasting their time with their equation set and ought to just solve the compressible equations. It is useful to understand this process, particularly when one goes beyond isothermal flows. To quote esoteric scripture from Lions that "existence and smoothness for all times still hasn't been proven" may be formally correct in certain situations but makes no useful contribution to anyone contemplating solving acoustic-free flows. If Zank and Mattheus or Bayly et al. are out to lunch with their "slow manifold" efforts, you might write a journal article correcting them.

In order for the slow manifold concept to supplant what I have said you need to demonstrate that it exists under all possible choices of initial and boundary conditions under which my argument works! To my knowledge such a proof has not been giving in the literature.

In the case of rotating stratified flow, which has similarities to the acoustic problem (i.e. shallow water theory), you should look at

"Balance and the Slow Quasimanifold: Some Explicit Results" by Ford, McIntyre and Norton in Journal of the Atmospheric Sciences: Vol. 57, No. 9, pp. 1236–1254.

Here, by analogy with acoustic radiation from a vortex, it is shown that a "strict" slow manifold does not exist.

It finally dawned on me where this particular tangent of yours came from. In the context of the relationship between compressible/ incompressible/low Mach number equations I doubted whether there was a situation existed where the slow mainfold did not exist. You seem to have interpreted this as saying that given ANY two different time scales within a flow, does a "slow manifold exist?" That is an astonishly broad question to which you now seem devoted. It also has virtually nothing to do with the essential theme of the last 10+ threads. My comment was intended to mean that I was unaware of a context where the acoustic-free manifold does not exist. You have gone to great pains to change the subject to anything but the original theme. Do you have anything useful to contribute to the slow manifold topic in the contest of the title to this line of threads - compressible flow? You stated that the existence of the "slow manifold" is debateable in this context.

http://www.cfd-online.com/Forum/main.cgi?read=35513

Those are pretty strong words because it could call into question the constraint equations that people use. Feel free to discuss where Kreiss, Zank, Bayley, etc have made erroneous assumptions and conclusions. And what are the consequences of their flawed analysis? I don't think another etymology digression will be useful here.

Seriously! Enough of all this other extraneous garbage and vague suggestion.

If the shallow water problem has DIRECT relevance to a failure in the slow manifold's existence towards nonacoustic flows then please enlighten us. Such an observation is likely to undermine the works of Kreiss, Zank, Bayley and others. What is wrong with the works of Kreiss, Zank, Bayley - the basis of most of what I have been repeating over and over?

By the way, this comment of yours is beyond ridiculous:

"In order for the slow manifold concept to supplant what I have said you need to demonstrate that it exists under all possible choices of initial and boundary conditions under which my argument works! To my knowledge such a proof has not been giving in the literature."

That is far beyond what anyone needs to do. In the event anyone choses to solve a form of the incompressible or low-Mach number equations, they should make sure that their equations have a sound basis before they try to solve them. Kreiss, Zank, and Bayley have done exactly that. People move forward in science by solving tough problems and not posturing behind petty comments like "you need to demonstrate that it exists under all possible choices."

This discussion is becoming rather tiresome. As my last comments on this matter:

(1) At no point have I argued that Kreiss etc are wrong for the problems they have solved. All I have said is that your interpretation of their work in correcting Andy above is not correct (and is a rather over zealous interpretation of the cited works).

(2) There are times when the slow manifold can fail to exist, even for problems which have two distinct timesales, due to problems with the "slaving" process. The SWE example (which is of direct relevance to acoustics - write down the two sets of equations and compare them!) is an example of this.

(3) The definition of incompressible IS that volumes must be preserved (look up incompressible in the dictionary). This implies trivially that div(u)=0 (=> density is constant on particle paths). The anelastic approximation also filters acoustic modes but the resulting flow is not incompressible and would correspond to a different slow manifold.

1) The original topic was issues surrounding the incompressible equations and ultimately about their derivation and meaning. I have contended that to understand this derivation, one needs to pay attention to the slow manifold or else there may be a nontrivial acoustic component remaining. Since most people solving the equations are expressedly concerned with avoiding the step size restriction that acoustics brings, this is an important topic. This is when you chimed in and said, within this particular context, that the slow manifold may fail to exist. In the present context, in my opinion, you are coming extremely close to conflicting with the derivations by Kreiss, Zank at al., and Bayley. I'm not sure how all of this is an overzealous interpretation of these authors.

My comments with Andy were more about two people talking about different sides of an elephant. I was trying to get people to look beyond (not tell them they are wrong) the traditional discussion of what incompressiblity is because when the temperature changes or the composition changes, the original constraint loses it's clear meaning. The only clear place where my comments amount to a "correction" to anything Andy said (my interpretation) was with regard to the low-Mach number equations. Otherwise, most I what I said to him is within the theme of the previous paragraph.

2) I am bewildered how my mentioning of the slow manifold within this context has led you in to a far and wide search for cases when such a manifold fails to exist. If there are situations where one cannot make the non-acoustic governing equations, equations like the incompressible or low-Mach number equations, then that is germane to the thread and is an important thing for people to know. I see no evidence that anyone seeking to compute flows that have variations in temperature and/or composition will find it impossible to derive non-acoustic governing equations. Your SWE example doesn't seem to me to be a cause for concern for these people despite your insistence that it, effectively, should.

3) I don't think anyone here has suggested that div(u) isn't zero for incompressible flows. My recurrent comment is on where does this comes from and why. If people don't know this then when temperature or composition vary, they may well use the wrong constraint. Solving index-2 DAEs or index-reducing them is a serious pain and people need to be very clear that their chosen constraint is the correct one for their purposes. I also understand that div(u)=0 has the physical/mathematical interpretation of constant volumes however this is an interpretation of the result rather than an examination of how the result was obtained. It seems to me that Gough's anelastic approximation, as well as those of Rehm and Baum and others, have been refined and clarified more recently by Zank et al. and Bayley et al. I also contend that the quintessence of the thread topic is contained in these last two papers.