Re: Texts for eigenvalues, eigenvectors & pde's
 

I've given Tom's though-process a bit more thought & worked through a few 'scaling' examples. I have a few comments & thoughts, listed below:

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Tom wrote: (*repeat*)

In the case of classification of the equations "size does not matter" it is simply the fact that the terms are there in the first place (another way to think of this is that the Navier-Stokes equations need more boundary conditions than the Euler equations - does the effect of these extra bcs go away as the viscosity is reduced? where have the extra bcs gone?).

... You really need to put this into the context of a Reynolds number: from the observation that the viscosity is small it does not follow that the inertia terms are bigger (think of very slow flow or a very thin pipe).

As I've already stated its not the size of these terms that is important - it is there implications upon what is a well defined problem.

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diaw: (*new*)

Following on this 'equation connectivity' logic path, I re-tackled a few assymptotic pipe-flow analyses I had been working on, using a new scaling-rule I've been developing. Where I had typically *initially* discarded the 'v' velocity component as of small size, with v~0, I now retained it & continued the development. The result is that this 'v' component ends up multiplied by an 'amplification factor' which means that even a tiny 'v' velocity in the flow field can now have a significant influence - under certain circumstances.

This makes me think back to many of the under-graduate & graduate fluids/heat-transfer courses where we begin with the full N-S & then begin deleting various terms which we assume to be 'small', or of little influence. We then proceed to develop solutions.

I wonder if perhaps, with this assymptotic analysis, we may be violating the 'N-S equation connectivity' / 'shape' relationships? Note that this 'equation shape' will also have a geometric dependency built in... we have been taught differently!

Re: Texts for eigenvalues, eigenvectors & pde's
 

diaw:wonder if perhaps, with this assymptotic analysis, we may be violating the 'N-S equation connectivity' / 'shape' relationships? Note that this 'equation shape' will also have a geometric dependency built in... we have been taught differently!

If what you mean is that the equations change type under the approximation then the answer is yes. The standard example of this is the steady flow of an incompressible fluid past a semi-infinite flat plate. The full steady NS equations are then elliptic. If you now perform the boundary-layer approximation (to get the classical Blasius problem) then you end up with a parabolic system (the distance along the plate, x, is time-like). This is well-known and is the cause of the Goldstein/separation singularity within the boundary-layer equations which has no counterpart in the full system. Kevorkian and Cole introduce the concept of bi-characteristics to distinguish between the characteristics of the full-problem to those of the approximate one. Note this is only an issue in singular perturbation theory and not in regular problems - the NS equations usually give rise to singular perturbations.

Triple-deck (or interactive boundary-layer) theory partially corrects this by, using a careful analysis of the asymptotic expansion, allowing the boundary-layer to deflect the invisicd flow in an interactive way (basically by reintroducing some of the missing ellipticity).

One of the main tests of a singular perturbation theory is understanding and fixing problems that occur due to the "missing terms". The theory isn't invalid provided due care is taken in interpreting the results and ensuring the asymptotic approximations are uniformly valid within all parts of the flow domain.

Re: Texts for eigenvalues, eigenvectors & pde's
 

Thanks Tom for an excellent & very complete answer.

So, the trick seems to be in identifying & trying to maintain the 'shape' of the original flow equation in the approximation, through various means & patches. Makes a lot of sense.

Can I ask you to elaborate on the following sentence, if you will:

"Kevorkian and Cole introduce the concept of bi-characteristics to distinguish between the characteristics of the full-problem to those of the approximate one."

The words 'characteristics of the full-problem' raised interest. Do you have any references to work describing the the characterisitics of the full N-S equations? This would be of much interest to me at present.

Thanks again for your input.

diaw...

Re: Texts for eigenvalues, eigenvectors & pde's
 

diaw: So, the trick seems to be in identifying & trying to maintain the 'shape' of the original flow equation in the approximation, through various means & patches. Makes a lot of sense.

In its basic form this patching is usually called the van Dyke matching principle. A more complete version is called the matching through an intermediate variable. This has to be done correctly to ensure physically/mathematically consistent results.

diaw:"Kevorkian and Cole introduce the concept of bi-characteristics to distinguish between the characteristics of the full-problem to those of the approximate one."

The basic idea is look at projections of the characteristics onto surfaces. A simple example of this is the observation that the bi-characteristics of the Navier-Stokes equations at high Reynolds number are actually the characteristics of the Euler equations.

The book by Kevorkian and Cole "Mutiple scales and singular perturbations" (Springer) discusses this as does,if I have the reference right, "unsteady viscous flow" by Telionis - this one is mainly about the boundary-layer equations and relevant numerics if I recall.

The book by Kevorkian and Cole also discuss the method of matching via an intermediate variable.

diaw:The words 'characteristics of the full-problem' raised interest. Do you have any references to work describing the the characterisitics of the full N-S equations? This would be of much interest to me at present.

Off the top of my head the only paper I can think of is that of Wang (1971) "On the determination of the zones of influence and dependence for three-dimensional boundary-layer equations" in JFM vol 48 #2.

As I recall he starts with the full (steady) NS equations and identifies what changes occur (and there implications) when certain terms are removed (i.e. going to the boundary layer approximation). The interesting observation is that, although it is the v-momentum equation that gets most of its terms deleted with the solenoidal/continuity equation remaining unchnaged, it is the characteristics related to mass balance which are removed in the approximation! (You could guess this from my earlier post where I mention the NS-BL limit). This paper is well worth reading (in my opinion - for sub-characteristic read, give or take a slight redefinition, bi-characteristic). I think the above reternce is correct - I don't have my copy at hand to check.

The only (real) characteristic direction of the full unsteady NS equations is time (parabolic) and in the steady state there are no real characteristics (elliptic). This is usually just accepted in the literature without reference (i.e. it's an old result - it's a bit like working with the boundary layer equations in that hardly anybody references Prandtl's original paper).

Re: Texts for eigenvalues, eigenvectors & pde's
 

Thanks so much Tom for your very detailed posting, it is very, very helpful. I'll get hold of the papers & work through the logic.

As aluded to in a previous thread (the 'marathon thread'), I'm currently working in the time-zone pre steady-state, for the N-S, within constrained flow paths. It has become very interesting & profitable in terms of research. The influence of the various terms of the N-S is intriguing, to say the least. This is where an understanding of characteristics will be useful.

Again, many thanks for your kind contribution.

diaw...