I won’t come back for some 2–3 weeks or more. The reason is this.
As you know, I had started writing some notes on FVM. I would then convert my earlier, simple, CFD code snippets, from FDM to FVM. Then, I would pursue modeling Schrodinger’s equation using FVM. That was the plan.
But before getting to the nitty-gritties of FVM itself, I thought of jotting down a note, once and for all, putting in writing my thoughts thus far on the concept of flux.
If you remember, it was several years ago that I had mentioned on this blog that I had sort of succeeded in deriving the Navier-Stokes equation in the Eulerian but differential form (d + E for short).
… Not an achievement by any stretch of imagination—there are tomes written on say, differentiable manifolds and whatnot. I feel sure that deriving the NS equations in the (d + E) form would be less than peanuts for them.
Yet, the fact of the matter is: They actually don’t do that!
Show me a single textbook or a paper that does that. If not at the UG level, then at least at the PG level, but one that is written using the language of only plain calculus, as used by engineers—not that of advanced analysis.
And as to the UG/PG books from engineering:
What people normally do is to derive these equations in its integral form, whether using the Lagrangian or the Eulerian approach. That is, they adopt either the (i + L) approach or the (i + D) approach.
At some rare times, if they at all begin fluid dynamics with a differential form of the NS equations, then they invariably follow the Lagrangian approach, never the Eulerian. That is, they invariably begin with only (d + L)—even in those cases when their objective is to obtain (d + E). Then, after having derived (d +L) , they simply invoke some arbitrary-looking vector calculus identities to “transform” those equations from (d + L) to (d +E).
They never discuss the context, meaning, or proofs of those identities. None from fluid dynamics or CFD side does that. And neither do the books on maths written for scientists and engineers.
The physical bases of the “transformation” process must remain a mystery.
When I started working through it a few years ago, I realized that the one probable reason why they don’t use the (d +E) form right from the beginning is because: forget the NS equations, no one understands even the much simpler idea of the flux—if it is to be couched entirely in the settings of (d+E). You see, the idea of the flux too always remains couched in the integral form, never the differential. For example, see Narasimhan [^]. Or, any other continuum mechanics books that impresses you.
It’s no accident that the Wiki article on Flux [^] says that it
needs attention from an expert in Physics.
And then, more important for us, the text of the article itself admits that the formula it notes, for a definition of flux in differential terms, is
an abuse of notation
See the section here [^].
Also, ask yourself, why is a formula that is free of the abuse of notation not being made available? In spite of all those tomes having been written on higher mathematics?
Further, there were also other related things I wanted to write about, like an easy pathway to the idea of tensors in general, and to that of the stress tensor in particular.
So, I thought of writing it down it for once and for all, in one note. I possibly could convert some parts of it into a paper later on, perhaps. For the time being though, the note would be more in the nature of a tutorial.
I started writing down the note, I guess, from 17 August 2018. However, it kept on growing, and with growth came reorganization of material for a better hierarchy or presentation. It has already gone through some 4–5 thorough re-orgs (meaning: discarding the earlier LaTeX file entirely and starting completely afresh), and it has already become more than 10 LaTeX pages. Even then, I am nowhere near finishing it. I may be just about half-way through—even though I have been working on it for some 7–8 hours every day for the past fortnight.
Yes, writing something in original is a lot of hard work. I mean “original” not in the sense of discovery, but in the sense of a lack of any directly citable material whatsoever, on the topic. Forget copy-pasting. You can’t even just gather a gist of the issue so that you could cite it.
And, the trouble here is, this topic is otherwise so very mature. (It is some 150+ years old.) So, you know that if you go even partly wrong, the whole world is going to pile on you.
And that way, in my experience, when you write originally, there is at least 5–10 pages of material you typically end up throwing away for every page that makes it to the final, published, version. Yes, the garbage thrown out is some 5–10 times the material retained in—no matter how “simple” and “straightforward” the published material might look.
Indeed, I could even make a case that the simpler and the more straight-forward the published material looks, if it also happens to be original, then the more strenuous it has been, on the part of the author.
Few come to grasp this simple an observation, ever, in their entire life.
As a case in point, I wish to recall here my conference paper on diffusion. [To be added here soon enough.]
I have many times silently watched people as they were going through this paper for the first time.
Typically, when engineers read it, they invariably come out with a mild expression which suggests that they probably were thinking of something like: “isn’t it all so simple and straight-forward?” Sometimes they even explicitly ask: “And, what do you say was the new contribution here?” [Even after having gone through both the abstract and the conclusion part of it, that is.]
On the other hand, on the four-five rare occasions when I have had the opportunity to watch professional mathematicians go through this paper of mine, in each case, the expression they invariably gave at the end of finishing it was as if they still were very intently absorbed in it. In particular, they never do ask me what was new about it—they just remain deeply engaged in what looks like an exercise in “fault-finding”, i.e., in checking if any proof, theorem or lemma they had ever had come across could be used in order to demolish the new idea that has been presented. Invariably, they give the same argument by way of an objection. Invariably, I explain why their argument does not address the issue I have raised in the paper. Invariably they chuckle and then go back to the paper and to their intent thinking mode, to see if there is any other weakness to my basic argument…
Till date (even after more than a decade), they haven’t come back.
But in all cases, they were very ready to admit that they were coming across this argument for the first time. I didn’t have to explain it to them that though the language and the tone of the paper looked simple enough, the argument itself was not easy to derive originally.
No, the notes which I am currently working on are nowhere near as original as that. [But yes, original, these are.]
Yet, let me confess, even as I keep prodding through it for the better part of the day the way I have done over the past fortnight or so, I find myself dealing with a certain doubt: wouldn’t they just dismiss it all as being too obvious? as if all the time and effort I spent on it was, more or less, ill spent? that it was all meaningless to begin with?
Anyway, I want to finish this task before resuming blogging—simply because I’ve got a groove about it by now… I am in a complete and pure state of anti-procrastination.
… Well, as they say: Make the hay while the Sun shines…
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(Marathi) “dnyaandev baaL maajhaa…”
Singer: Asha Bhosale
Lyrics: P. Savalaram
Music: Vasant Prabhu