General update: Will be away from blogging for a while

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).

And, worse:

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…


A Song I Like:
(Marathi) “dnyaandev baaL maajhaa…”
Singer: Asha Bhosale
Lyrics: P. Savalaram
Music: Vasant Prabhu

 

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Where are those other equations?

Multiple header images, and the problem with them:

As noted in my last post, I have made quite a few changes to the layout of this blog, including adding a “Less transient” page [^].

Another important change was that now, there were header images too, at the top.

Actually, initially, there was only one image. For the record, it was this: [^] However, there weren’t enough equations in it. So, I made another image. It was this [^]. But as I had already noted in the last post, this image was already crowded, and even then, it left out some other equations that I wanted to include.

Then, knowing that WordPress allows multiple images that can be shown at random, I created three images, and uploaded them. These are what is being displayed currently.

However, randomizing means that even after re-loading a page a couple of times, there still is a good chance that you will miss some or the other image, out of those three.

Ummm… OK.


A quick question:

Here is the problem statement:

There are three different header images for this blog. The server shows you only one of them during a single visit. Refreshing the page in the browser also counts as a separate visit. In each visit, the server will once again select an image completely at random.

Assume also that the PDF for the random sequence is uniform. That is to say, there is no greater probability for any of the three images during any visit. Cookies, e.g., play no role.

Now, suppose you make only three visits to this blog. For instance, suppose you visit some page on this blog, and then refresh the same page twice in the browser. The problem is to estimate the chances that you will get to see:

  • all of the three different images, but in only three visits
  • one and the same image, each time, during exactly three visits
  • exactly two different images, during exactly three visits

Don’t read further until you solve this problem, right now: right on-the-fly and right in your head (i.e. without using paper and pencil).

(Hint [LOL!]: There are three balls of different colors (say Red, Green, and Blue) in a box, and \dots.)


 

…No, really!

 


Ummm… Still with me?

OK. That tells me that you are now qualified to read further.

Just in case you were wondering what was there in the “other” header images, here is a little document I am uploading for you. Go, see it (.PDF [^]), but also note the caveat below.

Caveats: It is a work in progress. If you spot a mistake or even just a typo, then please do let me know. Also, don’t rely on this work.

For example, the definition of stress given in the document is what I have not so far read in any book. So, take it with a pinch of the salt—even if I feel confident that it is correct. Similarly, there might be some other changes, especially those related to the definition of the flux and its usage in the generic equation. Also, I am not sure if the product ansatz for the separation of variables technique began with d’Alembert or not. I vaguely remember its invention being attributed to him, but it was a long time ago, and I am no longer sure. May be it was before him. May be it was much later, at the hands of Fourier, or, even still later, by Lame. … Anyway let it be…

…BTW, the equations in the images currently being shown are slightly different—the PDF document is the latest thing there is.

Also, let me have your suggestions for any further inclusions, too, if any. (As to me: Yes, I would like to add a bit on the finite volume method, too.)

As usual, I may change the PDF document at any time in future. However, the document will always carry the date of compilation as the “version number”.


General update:

These days, I am also busy converting my already posted CFD snippets [^] into an FVM-based code.

The earlier posted code was done using FDM, not FVM, but it was not my choice—SPPU (Pune University) had thrust it upon me.

Writing an illustrative code for teaching purposes is fairly simple and straight-forward, esp. in Python—and especially if you treat the numpy arrays exactly as if they were Python arrays!! (That is, very inefficiently.) But I also thought of writing some notes on at least some initial parts of FVM (in a PDF document) to go with the code. That’s why, it is going to take a bit of time.

Once all this work is over, I will also try to model the Schrodinger equation using FVM. … Let’s see how it all goes…

…Alright, time to sign off, already! So, OK, take care and bye for now. …

 


A Song I Like:
(Hindi) “baharon, mera jeevan bhee savaron…”
Music: Khayyam
Singer: Lata Mangeshkar
Lyrics: Kaifi Aazmi


[The obligatory PS: In all probability, I won’t make any changes to the text of this post. However, the linked PDF document is bound to undergo changes, including addition of new material, reorganization, etc. When I do revise that document, I will note the updates in the post, too.]