"diaw's reply: Sure, the N-S are definitely in tensor (vector) form. What I'm exploring, I guess, are the inherent symmetries, properties of tensor operators & ?functional? groups. I've basically uncovered a 'Reynold's-type operator' & I'm working on. I'm doing this to look for the mechanisms (force-balances) at work. I'm treating the N-S terms as vector (tensor) operators in terms of force balances, but, go one step up in investigating the tensor operators from which the vector components arise. (Again, physicist's brain - think in pictures)."

There's been a huge amount of work (a lot of it in geophysical fluid dynamics; there's a book by R. Salmon on this) on symmetry/geometry of the (inviscid) equations of motion. This is all related to the observation that the inviscid equations possess a Lie-Poisson structure (i.e. they are Hamiltonian). This approach has made very little progress (the workers in this field will disagree with this comment) in useful results over the past 30 or so years. The viscous term in the full equations also screws up this approach since the equations cease to be Hamiltonian.

The problem I see with your approach, and it's made by a lot of physicists, is that looking the same and being the same mathematically can be very different things - in theorems there are usually subtle statements which preclude its applicability to a very similar looking problem.(*)

You probably what to look up some work on semigroups - there's a small book by M. I. Vishik "Asymptotic Behaviour of Solutions of Evolutionary Equations" which may help.

"In terms of incompressible fluids, there is an 'unstable' vibration ..."

You have to be careful that you're not talking about a comutational mode here. If you consider the simple equation u_t = u_x (so time and space are interchangable) and discretize it with, for example, an forward Euler step in time and central differences in space then there is a spatial computational mode analogous to that obtained in time stepping schemes such as leap-frog!

".. - it can provide local instability - instead of the usual concept of a global instability ..."

Have a look at the work of P. Heurre and his co-workers (there's a review article in the Annual Review of fluid mechanics).

"I'm planning to write that up soon - the only question is which Journal would find this of interest."

You could try the proceedings of the Royal Society of London (series A). It's got relatively high standards (like JFM it rejects over half of submissions out right) though and so you'd need to tighten up your use of terminology. There's also a numerical analysis IMA journal or even one of SIAM's (IMA is free to publish which is an advantage).

Tom.

(*) As an example there is a very famous paper where the author observes that the linear terms in the differential equations he is considering is of Sturm-Liouville type and so the solution to his problem can expanded as an infinite series of eigenfunctions of this operator. He then goes on to discuss the consequences of this in the full nonlinear problem using completely formal manipulations. The problem is that the particular "Sturm-Liouville problem" doesn't (in general) have a infinite set of eigenfunctions - and even if it did the series expansion would not be the complete solution!

This actually went unnoticed for 30ish years.

Hi guys....

Since you guys discussed lots about the convective & diffusion terms of the NS equation.... Would you mind if I ask a quesiton about the effect of this artificial viscosity on the convective lag effect of a flow following a sudden change in flow condition (e.g. sudden closure of a valve)? Will that cause any error when approximating the convective time lag?

It will have an effect - however if it is significant then your grid lengths and/or time step are too big. If coded correctly the effect of the artificial viscosity will diminish with increasing resolution (you also need a small enough time step to resolve the time-lag along with any transient features arising from the sudden change). If you are using a turbulence model then the uncertainty in the closure should be a much larger effect.

diaw's reply:

Tom, Thanks once again for your very comprehensive insights & links.

Tom wrote:

You could try the proceedings of the Royal Society of London (series A). It's got relatively high standards (like JFM it rejects over half of submissions out right) though and so you'd need to tighten up your use of terminology. There's also a numerical analysis IMA journal or even one of SIAM's (IMA is free to publish which is an advantage).

diaw's reply:

Those are exceptional Journals & I'd be more than honoured to try & submit to those.

A minor personal ramble:

In terms of the selection of appropriate terminology & suitably qualified, or interested collaborators. I currently am reasearching in a geographic region not overly rich in academic wisdom & experience in my current area of research. I happened to move towards this field during the initial part of a post-graduate study program & it has grown from there. I am desperately short of local folks I could send my work to for comment & insightful guidance. In many ways, the work seems to have grown past most of the local expertise, to the point where it becomes a little desperate at times - the moment one begins to move into a reasonable level of detail, folks eyes glaze & we cannot go much further. This is unfortunate.

I would love to be able to locate an academic forum where this level of discussion could take place on a routine basis - perhaps a cross-discipline group. I am reading as much as I can so that I can taylor my thought processes & language accordingly, but, there is no faster way to do this than peer-to-peer communication.

This is why I love it when we're able to debate freely on this forum. This medium has brought me tremendous guidance & applied wisdom for experts such as Tom and many others. For this I am very grateful.

desA

"diaw's reply: Those are exceptional Journals & I'd be more than honoured to try & submit to those."

"honoured" is a bit strong - I expect to be able to publish in these journals. Actually, in my opinion, if your going to publish papers you should aim/set yourself a high standard so that publishing in these types of journals is not a problem.

You could submit to one of these journals on the off chance (if it gets past the editorial vetting stage) that even if it gets rejected you may end up with useful comments from referees. In practise this probably won't work since, although I always try to give helpful pointers to the authors when rejecting papers, most people will not (I sometimes fall into this category as well - you don't get paid for refereeing papers and so getting a paper which you consider a waste of your time can be rather irksome). However it may be worth trying.

"I would love to be able to locate an academic forum ..."

you really need to find a supervisor that is interested in what you are trying to do and also works in the relevant field. You also have to factor in the possibility of him/her letting you work on your own project and not one of theirs.

Have you tried looking for positions in more relevant universities - in the UK for example Bath, Oxford, Imperial College, Reading and Leicester spring to mind. One possible problem may be whether your background in physics is enough to let you work in a relevant Mathematics department! PhDs in the UK tend to be shorter than other countries (3 years with possibly a 4th while writing up) with little or no taught courses to bring the student up to speed. This makes moving from a physics undergraduate degree to mathematics postgrad one rather difficult - but not impossible.

Artifical viscosity methods are so old, just forget it,