Yo—8: This tomboy is lovable. Also, a bit about the NS equations

“This tomboy is lovable.”

That was the note on which I wanted to begin my new year’s writing.

But I liked the way this tomboy handle maths so much, I had to begin messing around way before the new year’s eve would even arrive. …

OK. My puny ability to make puns is more or less fully exhausted by now. So, let’s go on to the thing straight. … More details, here [^]. And, for the maths part, here[^].

BTW, I don’t know about the other platforms, but at least on Ubuntu 14.04 LTS, after installation of the tomboy-latex plugin, you have to “enable” it from the “Preferences” dialog, and then close the software and restart it, before you can use it.

If you have a scatter-brain sort of a mind the way I do [heck, did I say something materialistic here?], you will find this tiny little piece of software a great help. … No need to compile the LaTeX document; the equations become directly visible as soon as you finish typing them in. Very handy.

… I am not sure if I am going to use the linking feature a lot. … But let’s see what the feel of the software becomes like, once there are, say, over 100 or so notes.

Anyway, that’s what my New Year’s Resolution for 2015 has been. To use Tomboy. … Low maintenance. Easy to keep. [If any pun has slipped in once again, then that is totally unintentional.]

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Still twiddling thumbs on the job front… Hope something works out soon. … If you (once again) go  jobless, you think of some terrific ideas like trying your hand at some funny writing, and out comes some pathetic, or (Hindi) “bakwaas” or even (Marathi) “bhakaas” sort of puns . … Only to be expected… OK, more, some time later. … But, yes, Tomboy is handy for jotting down research ideas.

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[As usual, a short and sweet Yo post has become lengthy… Might as well continue writing a bit….]

In the meanwhile, am continuing reading up here and there on CFD and all.

… Got confused about one seemingly small part of the derivation of the Navier-Stokes equations, in the conservative differential form, i.e., with a differential element fixed in space, i.e., an Eulerian infinitesimal element. Which means, none of the following three: (i) Eulerian integral, (ii) Lagrangian integral, or (iii) Lagrangian differential.

Instead, as I said, we are talking about the fourth possibility: Eulerian infinitesimal.

When it comes to equations put forth in the so-called conservative form, no text-book ever gives you a direct derivation in a differential form—I mean, with sufficient generality.

The modern texts almost always begin with the integral form of the equations, i.e., the Reynolds Transport Theorem (or RTT) for a finite (or integral) element, and only then, on the basis of the RTT, derive the conservative differential form. The derivation, thus, is indirect. As far as a derivation directly in the differential form (with an infinitesimal element) goes, text-books invariably choose a moving (i.e. Lagrangian) element, not a fixed (Eulerian) one.

But I wanted the odd combination: Eulerian, differential, and direct derivation.

True to my style, I thought that it would be very easy to do a derivation directly in this last form. So, instead of first working it out with paper and pencil, I directly started writing a small note in LaTeX, until I began stumbling around, whenever I tried to supply a “brief” conceptual explanation of what was going on in the derivation. Something wrong would sneak in into my derivation, and I could spot it only when I tried to explain the kind of assumptions I was implicitly making in conceptual terms. I tried may be some 2–3 times afresh, every time, directly in LaTeX, each time only to stumble on to some or the other error. (Of course, I was trying to build the derivation in general terms.)

Then, to get the matter straightened out, I consulted may be some 8–10 books/online notes, even one pedagogical paper on how the NS equations should be presented to students. This paper was written by a professor who had taught introductory FM course some 33 times! [(pdf) ^]. But still, a derivation of a sufficient generality but directly in the conservative differential form is not there even in this paper.

Finally, I found something mentioned in White’s introductory text on fluid mechanics. It was the only FM text to mention this point, even though, it does so only passingly. White, too, doesn’t give you a derivation directly in the required form, but it does only passingly mention a remark—which led to the lighting of the bulb for me!

Well, in a way, this is quite minor a matter—nothing even remotely like a new or a research idea. But still, I found myself getting confused about “such minor” an issue for an unexpectedly long time—for a few days or so (with some 2–3 hours per day, writing typing directly in LaTeX).

… May be, I will post something about it, some time later. (And, yes, I have made a Tomboy note about this confusion, too.) Bye for now.

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[This post was originally posted on January 7, 2015, but then, I decided to hold it back for a while. Reposting the same today, with a bit of a revision (2015.01.15)]

[E&OE]

 

 

 

 

 

 

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