Some suggested time-pass (including ideas for Python scripts involving vectors and tensors)

Actually, I am busy writing down some notes on scalars, vectors and tensors, which I will share once they are complete. No, nothing great or very systematic; these are just a few notings here and there taken down mainly for myself. More like a formulae cheat-sheet, but the topic is complicated enough that it was necessary that I have them in one place. Once ready, I will share them. (They may get distributed as extra material on my upcoming FDP (faculty development program) on CFD, too.)

While I remain busy in this activity, and thus stay away from blogging, you can do a few things:


1.

Think about it: You can always build a unique tensor field from any given vector field, say by taking its gradient. (Or, you can build yet another unique tensor field, by taking the Kronecker product of the vector field variable with itself. Or, yet another one by taking the Kronecker product with some other vector field, even just the position field!). And, of course, as you know, you can always build a unique vector field from any scalar field, say by taking its gradient.

So, you can write a Python script to load a B&W image file (or load a color .PNG/.BMP/even .JPEG, and convert it into a gray-scale image). You can then interpret the gray-scale intensities of the individual pixels as the local scalar field values existing at the centers of cells of a structured (squares) mesh, and numerically compute the corresponding gradient vector and tensor fields.

Alternatively, you can also interpret the RGB (or HSL/HSV) values of a color image as the x-, y-, and z-components of a vector field, and then proceed to calculate the corresponding gradient tensor field.

Write the output in XML format.


2.

Think about it: You can always build a unique vector field from a given tensor field, say by taking its divergence. Similarly, you can always build a unique scalar field from a vector field, say by taking its divergence.

So, you can write a Python script to load a color image, and interpret the RGB (or HSL/HSV) values now as the xx-, xy-, and yy-components of a symmetrical 2D tensor, and go on to write the code to produce the corresponding vector and scalar fields.


Yes, as my resume shows, I was going to write a paper on a simple, interactive, pedagogical, software tool called “ToyDNS” (from Toy + Displacements, Strains, Stresses). I had written an extended abstract, and it had even got accepted in a renowned international conference. However, at that time, I was in an industrial job, and didn’t get the time to write the software or the paper. Even later on, the matter kept slipping.

I now plan to surely take this up on priority, as soon as I am done with (i) the notes currently in progress, and immediately thereafter, (ii) my upcoming stress-definition paper (see my last couple of posts here and the related discussion at iMechanica).

Anyway, the ideas in the points 1. and 2. above were, originally, a part of my planned “ToyDNS” paper.


3.

You can induce a “zen-like” state in you, or if not that, then at least a “TV-watching” state (actually, something better than that), simply by pursuing this URL [^], and pouring in all your valuable hours into it. … Or who knows, you might also turn into a closet meteorologist, just like me. [And don’t tell anyone, but what they show here is actually a vector field.]


4.

You can listen to this song in the next section…. It’s one of those flowy things which have come to us from that great old Grand-Master, viz., SD Burman himself! … Other songs falling in this same sub-sub-genre include, “yeh kisine geet chheDaa,” and “ThanDi hawaaein,” both of which I have run before. So, now, you go enjoy yet another one of the same kind—and quality. …


A Song I Like:

[It’s impossible to figure out whose contribution is greater here: SD’s, Sahir’s, or Lata’s. So, this is one of those happy circumstances in which the order of the listing of the credits is purely incidental … Also recommended is the video of this song. Mona Singh (aka Kalpana Kartik (i.e. Dev Anand’s wife, for the new generation)) is sooooo magical here, simply because she is so… natural here…]

(Hindi) “phailee huyi hai sapanon ki baahen”
Music: S. D. Burman
Lyrics: Sahir
Singer: Lata Mangeshkar


But don’t forget to write those Python scripts….

Take care, and bye for now…

 

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How time flies…

I plan to conduct a smallish FDP (Faculty Development Program), for junior faculty, covering the basics of CFD sometime soon (may be starting in the second-half of February or early March or so).

During my course, I plan to give out some simple, pedagogical code that even non-programmers could easily run, and hopefully find easy to comprehend.


Don’t raise difficult questions right away!

Don’t ask me why I am doing it at all—especially given the fact that I myself never learnt my CFD in a class-room/university course settings. And especially given the fact that excellent course materials and codes already exist on the ‘net (e.g. Prof. Lorena Barba’s course, Prof. Atul Sharma’s book and Web site, to pick up just two of the so many resources already available).

But, yes, come to think of it, your question, by itself, is quite valid. It’s just that I am not going to entertain it.

Instead, I am going to ask you to recall that I am both a programmer and a professor.

As a programmer, you write code. You want to write code, and you do it. Whether better code already exists or not is not a consideration. You just write code.

As a professor, you teach. You want to teach, and you just do it. Whether better teachers or course-ware already exist or not is not a consideration. You just teach.

Admittedly, however, teaching is more difficult than coding. The difference here is that coding requires only a computer (plus software-writing software, of course!). But teaching requires other people! People who are willing to seat in front of you, at least faking listening to you with a rapt sort of an attention.

But just the way as a programmer you don’t worry whether you know the algorithm or not when you fire your favorite IDE, similarly, as a professor you don’t worry whether you will get students or not.

And then, one big advantage of being a senior professor is that you can always “c” your more junior colleagues, where “c” stands for {convince, confuse, cajole, coax, compel, …} to attend. That’s why, I am not worried—not at least for the time being—about whether I will get students for my course or not. Students will come, if you just begin teaching. That’s my working mantra for now…


But of course, right now, we are busy with our accreditation-related work. However, by February/March, I will become free—or at least free enough—to be able to begin conducting this FDP.


As my material for the course progressively gets ready, I will post some parts of it here. Eventually, by the time the FDP gets over, I would have uploaded all the material together at some place or the other. (May be I will create another blog just for that course material.)

This blog post was meant to note something on the coding side. But then, as usual, I ended up having this huge preface at the beginning.


When I was doing my PhD in the mid-naughties, I wanted a good public domain (preferably open source) mesh generator. There were several of them, but mostly on the Unix/Linux platform.

I had nothing basically against Unix/Linux as such. My problem was that I found it tough to remember the line commands. My working memory is relatively poor, very poor. And that’s a fact; I don’t say it out of any (false or true) modesty. So, I found it difficult to remember all those shell and system commands and their options. Especially painful for me was to climb up and down a directory hierarchy, just to locate a damn file and open it already! Given my poor working memory, I had to have the entire structure laid out in front of me, instead of remembering commands or file names from memory. Only then could I work fast enough to be effective enough a programmer. And so, I found it difficult to use Unix/Linux. Ergo, it had to be Windows.

But, most of this Computational Science/Engineering code was not available (or even compilable) on Windows, back then. Often, they were buggy. In the end, I ended up using Bjorn Niceno’s code, simply because it was in C (which I converted into C++), and because it was compilable on Windows.

Then, a few years later, when I was doing my industrial job in an FEM-software company, once again there was this requirement of an integrable mesh generator. It had to be: on Windows; open source; small enough, with not too many external dependencies (such as the Boost library or others); compilable using “the not really real” C++ compiler (viz. VC++ 6); one that was not very buggy or still was under active maintenance; and one more important point: the choice had to be respectable enough to be acceptable to the team and the management. I ended up using Jonathan Schewchuk’s Triangle.

Of course, all this along, I already knew about Gmsh, CGAL, and others (purely through my ‘net searches; none told me about any of them). But for some or the other reason, they were not “usable” by me.

Then, during the mid-teens (2010s), I went into teaching, and software development naturally took a back-seat.

A lot of things changed in the meanwhile. We all moved to 64-bit. I moved to Ubuntu for several years, and as the Idea NetSetter stopped working on the latest Ubuntu, I had no choice but to migrate back to Windows.

I then found that a lot of platform wars had already disappeared. Windows (and Microsoft in general) had become not only better but also more accommodating of the open source movement; the Linux movement had become mature enough to not look down upon the GUI users as mere script-kiddies; etc. In general, inter-operability had improved by leaps and bounds. Open Source projects were being not only released but also now being developed on Windows, not just on Unix/Linux. One possible reason why both the camps suddenly might have begun showing so much love to each other perhaps was that the mobile platform had come to replace the PC platform as the avant garde choice of software development. I don’t know, because I was away from the s/w world, but I am simply guessing that that could also be an important reason. In any case, code could now easily flow back and forth both the platforms.

Another thing to happen during my absence was: the wonderful development of the Python eco-system. It was always available on Ubuntu, and had made my life easier over there. After all, Python had a less whimsical syntax than many other alternatives (esp. the shell scripts); it carried all the marks of a real language. There were areas of discomfort. The one thing about Python which I found whimsical (and still do) is the lack of the braces for defining scopes. But such areas were relatively easy to overlook.

At least in the area of Computational Science and Engineering, Python had made it enormously easier to write ambitious codes. Just check out a C++ code for MPI for cluster computing, vs. the same code, written in Python. Or, think of not having to write ridiculously fast vector classes (or having to compile disparate C++ libraries using their own make systems and compiler options, and then to make them all work together). Or, think of using libraries like LAPACK. No more clumsy wrappers and having to keep on repeating multiple number of scope-resolution operators and namespaces bundling in ridiculously complex template classes. Just import NumPy or SciPy, and proceed to your work.

So, yes, I had come to register in my mind the great success story being forged by Python, in the meanwhile. (BTW, in case you don’t know, the name of the language comes from a British comedy TV serial, not from the whole-animal swallowing creep.) But as I said, I was now into academia, into core engineering, and there simply wasn’t much occasion to use any language, C++, Python or any other.

One more hindrance went away when I “discovered” that the PyCharm IDE existed! It not only was free, but also had VC++ key-bindings already bundled in. W o n d e r f u l ! (I would have no working memory to relearn yet another set of key-bindings, you see!)

In the meanwhile, VC++ anyway had become very big, very slow and lethargic, taking forever for the intelli-sense ever to get to produce something, anything. The older, lightweight, lightening-fast, and overall so charming IDE i.e. the VC++ 6, had given way, because of the .NET platform, to this new IDE which behaved as if it was designed to kill the C++ language. My forays into using Eclipse CDT (with VC++ key-bindings) were only partially successful. Eclipse was no longer buggy; it had begun working really well. The major trouble here was: there was no integrated help at the press of the “F1” key. Remember my poor working memory? I had to have that F1 key opening up the .chm helpf file at just the right place. But that was not happening. And, debug-stepping through the code still was not as seamless as I had gotten used to, in the VC++ 6.

But with PyCharm + Visual Studio key bindings, most my concerns got evaporated. Being an interpreted language, Python always would have an advantage as far as debug-stepping through the code is concerned. That’s the straight-forward part. But the real game-changer for me was: the maturation of the entire Python eco-system.

Every library you could possibly wish for was there, already available, like Aladdin’s genie standing with folded hands.

OK. Let me give you an example. You think of doing some good visualization. You have MatPlotLib. And a very helpful help file, complete with neat examples. No, you want more impressive graphics, like, say, volume rendering (voxel visualization). You have the entire VTK wrappped in; what more could you possibly want? (Windows vs. Linux didn’t matter.) But you instead want to write some custom-code, say for animation? You have not just one, not just two, but literally tens of libraries covering everything: from OpenGL, to scene-graphs, to computational geometry, to physics engines, to animation, to games-writing, and what not. Windowing? You had the MFC-style WxWidgets, already put into a Python avatar as WxPython. (OK, OpenGL still gives trouble with WxPython for anything ambitious. But such things are rather isolated instances when it comes to the overall Python eco-system.)

And, closer to my immediate concerns, I was delighted to find that, by now, both OpenFOAM and Gmsh had become neatly available on Windows. That is, not just “available,” i.e., not just as sources that can be read, but also working as if the libraries were some shrink-wrapped software!

Availability on Windows was important to me, because, at least in India, it’s the only platform of familiarity (and hence of choice) for almost all of the faculty members from any of the e-school departments other than CS/IT.

Hints: For OpenFOAM, check out blueCFD instead of running it through Dockers. It’s clean, and indeed works as advertised. As to Gmsh, ditto. And, it also comes with Python wrappers.

While the availability of OpenFOAM on Windows was only too welcome, the fact is, its code is guaranteed to be completely inaccessible to a typical junior faculty member from, say, a mechanical or a civil or a chemical engineering department. First, OpenFOAM is written in real (“templated”) C++. Second, it is very bulky (millions of lines of code, may be?). Clearly beyond the comprehension of a guy who has never seen more than 50 lines of C code at a time in his life before. Third, it requires the GNU compiler, special make environment, and a host of dependencies. You simply cannot open OpenFOAM and show how those FVM algorithms from Patankar’s/Versteeg & Malasekara’s book do the work, under its hood. Neither can you ask your students to change a line here or there, may be add a line to produce an additional file output, just for bringing out the actual working of an FVM algorithm.

In short, OpenFOAM is out.

So, I have decided to use OpenFOAM only as a “backup.” My primary teaching material will only be Python snippets. The students will also get to learn how to install OpenFOAM and run the simplest tutorials. But the actual illustrations of the CFD ideas will be done using Python. I plan to cover only FVM and only simpler aspects of that. For instance, I plan to use only structured rectangular grids, not non-orthogonal ones.

I will write code that (i) generates mesh, (ii) reads mesh generated by the blockMesh of OpenFOAM, (iii) implements one or two simple BCs, (iv) implements the SIMPLE algorithm, and (v) uses MatPlotLib or ParaView to visualize the output (including any intermediate outputs of the algorithms).

I may then compare the outputs of these Python snippets with a similar output produced by OpenFOAM, for one or two simplest cases like a simple laminar flow over step. (I don’t think I will be covering VOF or any other multi-phase technique. My course is meant to be covering only the basics.)

But not having checked Gmsh recently, and thus still carrying my old impressions, I was almost sure I would have to write something quick in Python to convert BMP files (showing geometry) into mesh files (with each pixel turning into a finite volume cell). The trouble with this approach was, the ability to impose boundary conditions would be seriously limited. So, I was a bit worried about it.

But then, last week, I just happened to check Gmsh, just to be sure, you know! And, WOW! I now “discovered” that the Gmsh is already all Python-ed in. Great! I just tried it, and found that it works, as bundled. Even on Windows. (Yes, even on Win7 (64-bit), SP1).

I was delighted, excited, even thrilled.

And then, I began “reflecting.” (Remember I am a professor?)

I remembered the times when I used to sit in a cyber-cafe, painfully downloading source code libraries over a single 64 kbps connection which would shared in that cyber-cafe over 6–8 PCs, without any UPS or backups in case the power went out. I would download the sources that way at the cyber-cafe, take them home to a Pentium machine running Win2K, try to open and read the source only to find that I had forgot to do the CLRF conversion first! And then, the sources wouldn’t compile because the make environment wouldn’t be available on Windows. Or something or the other of that sort. But still, I fought on. I remember having downloaded not only the OpenFOAM sources (with the hope of finding some way to compile them on Windows), but also MPICH2, PetSc 2.x, CGAL (some early version), and what not. Ultimately, after my valiant tries at the machine for a week or two, “nothing is going to work here” I would eventually admit to myself.

And here is the contrast. I have a 4G connection so I can comfortably seat at home, and use the Python pip (or the PyCharm’s Project Interpreter) to download or automatically update all the required libraries, even the heavy-weights like what they bundle inside SciPy and NumPy, or the VTK. I no longer have to manually ensure version incompatibilities, platform incompatibilities. I know I could develop on Ubuntu if I want to, and the student would be able to run the same thing on Windows.

Gone are those days. And how swiftly, it seems now.

How time flies…


I will be able to come back only next month because our accreditation-related documentation work has now gone into its final, culminating phase, which occupies the rest of this month. So, excuse me until sometime in February, say until 11th or so. I will sure try to post a snippet or two on using Gmsh in the meanwhile, but it doesn’t really look at all feasible. So, there.

Bye for now, and take care…


A Song I Like:

[Tomorrow is (Sanskrit, Marathi) “Ganesh Jayanti,” the birth-day of Lord Ganesha, which also happens to be the auspicious (Sanskrit, Marathi) “tithee” (i.e. lunar day) on which my mother passed away, five years ago. In her fond remembrance, I here run one of those songs which both of us liked. … Music is strange. I mean, a song as mature as this one, but I remember, I still had come to like it even as a school-boy. May be it was her absent-minded humming of this song which had helped? … may be. … Anyway, here’s the song.]

(Hindi) “chhup gayaa koi re, door se pukaarake”
Singer: Lata Mangeshkar
Music: Hemant Kumar
Lyrics: Rajinder Kishan

 

A prediction. Also, a couple of wishes…

The Prediction:

While the week of the Nobel prizes always has a way to generate a sense of suspense, of excitement, and even of wonderment, as far as I am concerned, the one prize that does that in the real sense to me is, of course, the Physics Nobel. … Nothing compares to it. Chemistry can come close, but not always. [And, Mr. Nobel was a good guy; he instituted no prize for maths! [LOL!]]. …

The Physics Nobel is the King of all awards in all fields, as far as I am concerned.

That’s why, this year, I have this feeling of missing something. … The reason is, this year’s Physics Nobel is already “known”; it will go to Kip Thorne and pals.

[I will not eat crow even if they don’t get it. [… Unless, of course, you know a delicious recipe or two for the same, and also demonstrate it to me, complete with you sampling it first.]]

But yes, Kip Thorne richly deserves it, and he will get it. That’s the prediction. I wanted to slip it in even if only few hours before the announcement arrives.

I will update this post later right today/tonight, after the Physics Nobel is actually announced.


Now let me come to the couple of wishes, as mentioned in the title. I will try to be brief. [Have been too busy these days… OK. Will let you know. We are going in for accreditation, and so, it’s been all heavy documentation-related work for the past few months. Despite all that hard-work, we still have managed to slip a bit on the progress, and so, currently, we are working on all week-ends and on most public holidays, too. [Yes, we came to work yesterday.] So, it’s only somehow that I manage to find some time to slip in this post—which is written absolutely on the fly, with no second thoughts or re-reading before posting. … So excuse me if there is a bit of lack of balance in the presentation, and of course, typos etc.]


Wish # 1:

The first wish is that a Physics Nobel should go, in a combined way, to what actually are two separate, but very intimately related, and two most significant advances in the physical understanding of man: (i) chaos theory (including fractals) and (ii)catastrophe theory.

If you don’t like the idea of two ideas being given a single Nobel, then, well, let me put it this way: the Nobel should be given for achieving the most significant advancements in the field of the differential nonlinearities, for a very substantial progress in the physical understanding of the behaviour of nonlinear physical systems, forging pathways for predictive capacity.

Let me emphasize, this has been one of the most significant advances in physics in the last century. No, saying so is emphatically not a hyperbole.

And, yes, it’s an advance in physics, primarily, and then, also in maths—but only secondarily.

… It’s unfortunate that an advancement which has been this remarkable never did register as such with most of the S&T “manpower”, esp., engineers and practical designers. It’s also unfortunate that the twin advancement arrived on the scene at the time of bad cultural (even epistemological) trends, and so, the advancements got embedded in a fabric of hyperbole, even nonsense.

But regardless of the cultural tones in which the popular presentations of these advancements (esp. of the chaos theory) got couched, taken as a science, the studies of nonlinearity in the physical systems has been a very, very, original, and a very, very creative, advancement. It needs to be recognized as such.

That way, I don’t much care for what it helped produce on the maths side of it. But yes, even a not very extraordinarily talented undergraduate in CS (one with a special interest in deterministic methods in cryptography) would be able to tell you how much light got shone on their discipline because of the catastrophe and chaos theories.

The catastrophe theory has been simply marvellous in one crucial aspect: it actually pushed the boundaries of what is understood by the term: mathematics. The theory has been daring enough to propose, literally for the first time in the entire history of mankind, a well-refined qualitative approach to an infinity of quantitative processes taken as a group.

The distinction between the qualitative and the quantitative had kept philosophers (and laymen) pre-occupied for millenia. But the nonlinear theory has been the first theoretical approach that tells you how to spot and isolate the objective bases for distinguishing what we consider as the qualitative changes.

Remove the understanding given by the nonlinear theory—by the catastrophe-theoretical approach—and, once in the domain of the linear theory, the differences in kind immediately begin to appear as more or less completely arbitrary. There is no place in theory for them—the qualitative distinctions are external to the theory because a linear system always behaves exactly the same with any quantitative changes made, at any scale, to any of the controlling parameters. Since in the linear theory the qualitative changes are not produced from within the theory itself, such distinctions must be imported into it out of some considerations that are in principle external to the theory.

People often confuse such imports with “applications.” No, when it comes to the linear theory, it’s not the considerations of applications which can be said to be driving any divisions of qualitative changes. The qualitative distinctions are basically arbitrary in a linear theory. It is important to realize that that usual question: “Now where do we draw the line?” is basically absolutely superfluous once you are within the domain of the linear systems. There are no objective grounds on the basis of which such distinctions can be made.

Studies of the nonlinear phenomena sure do precede the catastrophe and the chaos theories. Even in the times before these two theories came on the scene, applied physicists would think of certain ideas such as differences of regimes, esp. in the areas like fluid dynamics.

But to understand the illuminating power of the nonlinear theory, just catch hold of an industrial CFD guy (or a good professor of fluid dynamics from a good university [not, you know, from SPPU or similar universities]), and ask him whether there can be any deeper theoretical significance to the procedure of the Buckingham Pi Theorem, to the necessity, in his art (or science) of having to use so many dimensionless numbers. (Every mechanical/allied engineering undergraduate has at least once in life cursed the sheer number of them.) The competent CFD guy (or the good professor) would easily be at a loss. Then, toss a good book on the Catastrophe Theory to him, leave him alone for a couple of weeks or may be a month, return, and raise the same question again. He now may or may not have a very good, “flowy” sort of a verbal answer ready for you. But one look at his face would tell you that it has now begun to reflect a qualitatively different depth of physical understanding even as he tries to tackle that question in his own way. That difference arises only because of the Catastrophe Theory.

As to the Chaos Theory (and I club the fractal theory right in it), more number of people are likely to know about it, and so, I don’t have to wax a lot (whether eloquently or incompetently). But let me tell you one thing.

Feigenbaum’s discovery of the universal constant remains, to my mind, one of the most ingenious advancements in the entire history of physics, even of science. Especially, given the experimental equipment with which he made that discovery—a handheld HP Calculator (not a computer) in the seventies (or may be in the sixties)! … And yes, getting to that universal constant was, if you ask me, an act of discovery, and not of invention. (Invention was very intimately involved in the process; but the overall act and the end-product was one of discovery.)

So, here is a wish that these fundamental studies of the nonlinear systems get their due—the recognition they so well deserve—in the form of a Physics Nobel.

…And, as always, the sooner the better!


Wish # 2:

The second wish I want to put up here is this: I wish there was some commercial/applied artist, well-conversant with the “art” of supplying illustrations for a physics book, who also was available for a long-term project I have in mind.

To share a bit: Years ago (actually, almost two decades ago, in 1998 to be precise), I had made a suggestion that novels by Ayn Rand be put in the form of comics. As far as I was concerned, the idea was novel (i.e. new). I didn’t know at that time that a comics-book version of The Fountainhead had already been conceived of by none other than Ayn Rand herself, and it, in fact, had also been executed. In short, there was a comics-book version of The Fountainhead. … These days, I gather, they are doing something similar for Atlas Shrugged.

If you think about it, my idea was not at all a leap of imagination. Newspapers (even those in India) have been carrying comic strips for decades (right since before my own childhood), and Amar Chitrakatha was coming of age just when I was. (It was founded in 1967 by Mr. Pai.)

Similarly, conceiving of a comics-like book for physics is not at all a very creative act of imagination. In fact, it is not even original. Everyone knows those books by that Japanese linguistics group, the books on topics like the Fourier theory.

So, no claim of originality here.

It’s just that for my new theory of QM, I find that the format of a comics-book would be most suitable. (And what the hell if physicists don’t take me seriously because I put it in this form first. Who cares what they think anyway!)

Indeed, I would even like to write/produce some comics books on maths topics, too. Topics like grads, divs, curls, tensors, etc., eventually. … Guess I will save that part for keeping me preoccupied during my retirement. BTW, my retirement is not all that far away; it’s going to be here pretty soon, right within just five years from now. (Do one thing: Check out what I was writing, say in 2012 on this blog.)

But the one thing I would like write/produce right in the more immediate future is: the comics book on QM, putting forth my new approach.

So, in the closing, here is a request. If you know some artist (or an engineer/physicist with fairly good sketching/computer-drawing skills), and has time at hand, and has the capacity to stay put in a sizeable project, and won’t ask money for it (a fair share in the royalty is a given—provided we manage to find a publisher first, that is), then please do bring this post to his notice.

 


A Song I Like:

And, finally, here is the Marathi song I had promised you the last time round. It’s a fusion of what to my mind is one of the best tunes Shrinivas Khale ever produced, and the best justice to the words and the tunes by the singer. Imagine any one else in her place, and you will immediately come to know what I mean. … Pushpa Pagdhare easily takes this song to the levels of the very best by the best, including Lata Mangeshkar. [Oh yes, BTW, congrats are due to the selection committe of this year’s Lata Mangeshkar award, for selecting Pushpa Pagdhare.]

(Marathi) “yeuni swapnaat maajhyaa…”
Singer: Pushpa Pagdhare
Music: Shrinivas Khale
Lyrics: Devakinandan Saraswat

[PS: Note: I am going to come back and add an update once this year’s Physics Nobel is announced. At that time (or tonight) I will also try to streamline this post.

Then, I will be gone off the blogging for yet another couple of weeks or so—unless it’s a small little “kutty” post of the “Blog-Filler” kind or two.]

 

Fluxes, scalars, vectors, tensors…. and, running in circles about them!

0. This post is written for those who know something about Thermal Engineering (i.e., fluid dynamics, heat transfer, and transport phenomena) say up to the UG level at least. [A knowledge of Design Engineering, in particular, the tensors as they appear in solid mechanics, would be helpful to have but not necessary. After all, contrary to what many UGC and AICTE-approved (Full) Professors of Mechanical Engineering teaching ME (Mech – Design Engineering) courses in SPPU and other Indian universities believe, tensors not only appear also in fluid mechanics, but, in fact, the fluids phenomena make it (only so slightly) easier to understand this concept. [But all these cartoons characters, even if they don’t know even this plain and simple a fact, can always be fully relied (by anyone) about raising objections about my Metallurgy background, when it comes to my own approval, at any time! [Indians!!]]]

In this post, I write a bit about the following question:

Why is the flux \vec{J} of a scalar \phi a vector quantity, and not a mere number (which is aka a “scalar,” in certain contexts)? Why is it not a tensor—whatever the hell the term means, physically?

And, what is the best way to define a flux vector anyway?


1.

One easy answer is that if the flux is a vector, then we can establish a flux-gradient relationship. Such relationships happen to appear as statements of physical laws in all the disciplines wherever the idea of a continuum was found useful. So the scope of the applicability of the flux-gradient relationships is very vast.

The reason to define the flux as a vector, then, becomes: because the gradient of a scalar field is a vector field, that’s why.

But this answer only tells us about one of the end-purposes of the concept, viz., how it can be used. And then the answer provided is: for the formulation of a physical law. But this answer tells us nothing by way of the very meaning of the concept of flux itself.


2.

Another easy answer is that if it is a vector quantity, then it simplifies the maths involved. Instead of remembering having to take the right \theta and then multiplying the relevant scalar quantity by the \cos of this \theta, we can more succinctly write:

q = \vec{J} \cdot \vec{S} (Eq. 1)

where q is the quantity of \phi, an intensive scalar property of the fluid flowing across a given finite surface, \vec{S}, and \vec{J} is the flux of \Phi, the extensive quantity corresponding to the intensive quantity \phi.

However, apart from being a mere convenience of notation—a useful shorthand—this answer once again touches only on the end-purpose, viz., the fact that the idea of flux can be used to calculate the amount q of the transported property \Phi.

There also is another problem with this, second, answer.

Notice that in Eq. 1, \vec{J} has not been defined independently of the “dotting” operation.

If you have an equation in which the very quantity to be defined itself has an operator acting on it on one side of an equation, and then, if a suitable anti- or inverse-operator is available, then you can apply the inverse operator on both sides of the equation, and thereby “free-up” the quantity to be defined itself. This way, the quantity to be defined becomes available all by itself, and so, its definition in terms of certain hierarchically preceding other quantities also becomes straight-forward.

OK, the description looks more complex than it is, so let me illustrate it with a concrete example.

Suppose you want to define some vector \vec{T}, but the only basic equation available to you is:

\vec{R} = \int \text{d} x \vec{T}, (Eq. 2)

assuming that \vec{T} is a function of position x.

In Eq. 2, first, the integral operator must operate on \vec{T}(x) so as to produce some other quantity, here, \vec{R}. Thus, Eq. 2 can be taken as a definition for \vec{R}, but not for \vec{T}.

However, fortunately, a suitable inverse operator is available here; the inverse of integration is differentiation. So, what we do is to apply this inverse operator on both sides. On the right hand-side, it acts to let \vec{T} be free of any operator, to give you:

\dfrac{\text{d}\vec{R}}{\text{d}x} = \vec{T} (Eq. 3)

It is the Eq. 3 which can now be used as a definition of \vec{T}.

In principle, you don’t have to go to Eq. 3. In principle, you could perhaps venture to use a bit of notation abuse (the way the good folks in the calculus of variations and integral transforms always did), and say that the Eq. 2 itself is fully acceptable as a definition of \vec{T}. IMO, despite the appeal to “principles”, it still is an abuse of notation. However, I can see that the argument does have at least some point about it.

But the real trouble with using Eq. 1 (reproduced below)

q = \vec{J} \cdot \vec{S} (Eq. 1)

as a definition for \vec{J} is that no suitable inverse operator exists when it comes to the dot operator.


3.

Let’s try another way to attempt defining the flux vector, and see what it leads to. This approach goes via the following equation:

\vec{J} \equiv \dfrac{q}{|\vec{S}|} \hat{n} (Eq. 4)

where \hat{n} is the unit normal to the surface \vec{S}, defined thus:

\hat{n} \equiv \dfrac{\vec{S}}{|\vec{S}|} (Eq. 5)

Then, as the crucial next step, we introduce one more equation for q, one that is independent of \vec{J}. For phenomena involving fluid flows, this extra equation is quite simple to find:

q = \phi \rho \dfrac{\Omega_{\text{traced}}}{\Delta t} (Eq. 6)

where \phi is the mass-density of \Phi (the scalar field whose flux we want to define), \rho is the volume-density of mass itself, and \Omega_{\text{traced}} is the volume that is imaginarily traced by that specific portion of fluid which has imaginarily flowed across the surface \vec{S} in an arbitrary but small interval of time \Delta t. Notice that \Phi is the extensive scalar property being transported via the fluid flow across the given surface, whereas \phi is the corresponding intensive quantity.

Now express \Omega_{\text{traced}} in terms of the imagined maximum normal distance from the plane \vec{S} up to which the forward moving front is found extended after \Delta t. Thus,

\Omega_{\text{traced}} = \xi |\vec{S}| (Eq. 7)

where \xi is the traced distance (measured in a direction normal to \vec{S}). Now, using the geometric property for the area of parallelograms, we have that:

\xi = \delta \cos\theta (Eq. 8)

where \delta is the traced distance in the direction of the flow, and \theta is the angle between the unit normal to the plane \hat{n} and the flow velocity vector \vec{U}. Using vector notation, Eq. 8 can be expressed as:

\xi = \vec{\delta} \cdot \hat{n} (Eq. 9)

Now, by definition of \vec{U}:

\vec{\delta} = \vec{U} \Delta t, (Eq. 10)

Substituting Eq. 10 into Eq. 9, we get:

\xi = \vec{U} \Delta t \cdot \hat{n} (Eq. 11)

Substituting Eq. 11 into Eq. 7, we get:

\Omega_{\text{traced}} = \vec{U} \Delta t \cdot \hat{n} |\vec{S}| (Eq. 12)

Substituting Eq. 12 into Eq. 6, we get:

q = \phi \rho \dfrac{\vec{U} \Delta t \cdot \hat{n} |\vec{S}|}{\Delta t} (Eq. 13)

Cancelling out the \Delta t, Eq. 13 becomes:

q = \phi \rho \vec{U} \cdot \hat{n} |\vec{S}| (Eq. 14)

Having got an expression for q that is independent of \vec{J}, we can now use it in order to define \vec{J}. Thus, substituting Eq. 14 into Eq. 4:

\vec{J} \equiv \dfrac{q}{|\vec{S}|} \hat{n} = \dfrac{\phi \rho \vec{U} \cdot \hat{n} |\vec{S}|}{|\vec{S}|} \hat{n} (Eq. 16)

Cancelling out the two |\vec{S}|s (because it’s a scalar—you can always divide any term by a scalar (or even¬† by a complex number) but not by a vector), we finally get:

\vec{J} \equiv \phi \rho \vec{U} \cdot \hat{n} \hat{n} (Eq. 17)


4. Comments on Eq. 17

In Eq. 17, there is this curious sequence: \hat{n} \hat{n}.

It’s a sequence of two vectors, but the vectors apparently are not connected by any of the operators that are taught in the Engineering Maths courses on vector algebra and calculus—there is neither the dot (\cdot) operator nor the cross \times operator appearing in between the two \hat{n}s.

But, for the time being, let’s not get too much perturbed by the weird-looking sequence. For the time being, you can mentally insert parentheses like these:

\vec{J} \equiv \left[ \left( \phi \rho \vec{U} \right) \cdot \left( \hat{n} \right) \right] \hat{n} (Eq. 18)

and see that each of the two terms within the parentheses is a vector, and that these two vectors are connected by a dot operator so that the terms within the square brackets all evaluate to a scalar. According to Eq. 18, the scalar magnitude of the flux vector is:

|\vec{J}| = \left( \phi \rho \vec{U}\right) \cdot \left( \hat{n} \right) (Eq. 19)

and its direction is given by: \hat{n} (the second one, i.e., the one which appears in Eq. 18 but not in Eq. 19).


5.

We explained away our difficulty about Eq. 17 by inserting parentheses at suitable places. But this procedure of inserting mere parentheses looks, by itself, conceptually very attractive, doesn’t it?

If by not changing any of the quantities or the order in which they appear, and if by just inserting parentheses, an equation somehow begins to make perfect sense (i.e., if it seems to acquire a good physical meaning), then we have to wonder:

Since it is possible to insert parentheses in Eq. 17 in some other way, in some other places—to group the quantities in some other way—what physical meaning would such an alternative grouping have?

That’s a delectable possibility, potentially opening new vistas of physico-mathematical reasonings for us. So, let’s pursue it a bit.

What if the parentheses were to be inserted the following way?:

\vec{J} \equiv \left( \hat{n} \hat{n} \right) \cdot \left( \phi \rho \vec{U} \right) (Eq. 20)

On the right hand-side, the terms in the second set of parentheses evaluate to a vector, as usual. However, the terms in the first set of parentheses are special.

The fact of the matter is, there is an implicit operator connecting the two vectors, and if it is made explicit, Eq. 20 would rather be written as:

\vec{J} \equiv \left( \hat{n} \otimes \hat{n} \right) \cdot \left( \phi \rho \vec{U} \right) (Eq. 21)

The \otimes operator, as it so happens, is a binary operator that operates on two vectors (which in general need not necessarily be one and the same vector as is the case here, and whose order with respect to the operator does matter). It produces a new mathematical object called the tensor.

The general form of Eq. 21 is like the following:

\vec{V} = \vec{\vec{T}} \cdot \vec{U} (Eq. 22)

where we have put two arrows on the top of the tensor, to bring out the idea that it has something to do with two vectors (in a certain order). Eq. 22 may be read as the following: Begin with an input vector \vec{U}. When it is multiplied by the tensor \vec{\vec{T}}, we get another vector, the output vector: \vec{V}. The tensor quantity \vec{\vec{T}} is thus a mapping between an arbitrary input vector and its uniquely corresponding output vector. It also may be thought of as a unary operator which accepts a vector on its right hand-side as an input, and transforms it into the corresponding output vector.


6. “Where am I?…”

Now is the time to take a pause and ponder about a few things. Let me begin doing that, by raising a few questions for you:

Q. 6.1:

What kind of a bargain have we ended up with? We wanted to show how the flux of a scalar field \Phi must be a vector. However, in the process, we seem to have adopted an approach which says that the only way the flux—a vector—can at all be defined is in reference to a tensor—a more advanced concept.

Instead of simplifying things, we seem to have ended up complicating the matters. … Have we? really? …Can we keep the physical essentials of the approach all the same and yet, in our definition of the flux vector, don’t have to make a reference to the tensor concept? exactly how?

(Hint: Look at the above development very carefully once again!)

Q. 6.2:

In Eq. 20, we put the parentheses in this way:

\vec{J} \equiv \left( \hat{n} \hat{n} \right) \cdot \left( \phi \rho \vec{U} \right) (Eq. 20, reproduced)

What would happen if we were to group the same quantities, but alter the order of the operands for the dot operator?  After all, the dot product is commutative, right? So, we could have easily written Eq. 20 rather as:

\vec{J} \equiv \left( \phi \rho \vec{U} \right) \cdot \left( \hat{n} \hat{n} \right) (Eq. 21)

What could be the reason why in writing Eq. 20, we might have made the choice we did?

Q. 6.3:

We wanted to define the flux vector for all fluid-mechanical flow phenomena. But in Eq. 21, reproduced below, what we ended up having was the following:

\vec{J} \equiv \left( \phi \rho \vec{U} \right) \cdot \left( \hat{n} \otimes \hat{n} \right) (Eq. 21, reproduced)

Now, from our knowledge of fluid dynamics, we know that Eq. 21 seemingly stands only for one kind of a flux, namely, the convective flux. But what about the diffusive flux? (To know the difference between the two, consult any good book/course-notes on CFD using FVM, e.g. Jayathi Murthy’s notes at Purdue, or Versteeg and Malasekara’s text.)

Q. 6.4:

Try to pursue this line of thought a bit:

Start with Eq. 1 again:

q = \vec{J} \cdot \vec{S} (Eq. 1, reproduced)

Express \vec{S} as a product of its magnitude and direction:

q = \vec{J} \cdot |\vec{S}| \hat{n} (Eq. 23)

Divide both sides of Eq. 23 by |\vec{S}|:

\dfrac{q}{|\vec{S}|} = \vec{J} \cdot \hat{n} (Eq. 24)

“Multiply” both sides of Eq. 24 by \hat{n}:

\dfrac{q} {|\vec{S}|} \hat{n} = \vec{J} \cdot \hat{n} \hat{n} (Eq. 25)

We seem to have ended up with a tensor once again! (and more rapidly than in the development in section 4. above).

Now, looking at what kind of a change the left hand-side of Eq. 24 undergoes when we “multiply” it by a vector (which is: \hat{n}), can you guess something about what the “multiplication” on the right hand-side by \hat{n} might mean? Here is a hint:

To multiply a scalar by a vector is meaningless, really speaking. First, you need to have a vector space, and then, you are allowed to take any arbitrary vector from that space, and scale it up (without changing its direction) by multiplying it with a number that acts as a scalar. The result at least looks the same as “multiplying” a scalar by a vector.

What then might be happening on the right hand side?

Q.6.5:

Recall your knowledge (i) that vectors can be expressed as single-column or single-row matrices, and (ii) how matrices can be algebraically manipulated, esp. the rules for their multiplications.

Try to put the above developments using an explicit matrix notation.

In particular, pay particular attention to the matrix-algebraic notation for the dot product between a row- or column-vector and a square matrix, and the effect it has on your answer to question Q.6.2. above. [Hint: Try to use the transpose operator if you reach what looks like a dead-end.]

Q.6.6.

Suppose I introduce the following definitions: All single-column matrices are “primary” vectors (whatever the hell it may mean), and all single-row matrices are “dual” vectors (once again, whatever the hell it may mean).

Given these definitions, you can see that any primary vector can be turned into its corresponding dual vector simply by applying the transpose operator to it. Taking the logic to full generality, the entirety of a given primary vector-space can then be transformed into a certain corresponding vector space, called the dual space.

Now, using these definitions, and in reference to the definition of the flux vector via a tensor (Eq. 21), but with the equation now re-cast into the language of matrices, try to identify the physical meaning the concept of “dual” space. [If you fail to, I will sure provide a hint.]

As a part of this exercise, you will also be able to figure out which of the two \hat{n}s forms the “primary” vector space and which \hat{n} forms the dual space, if the tensor product \hat{n}\otimes\hat{n} itself appears (i) before the dot operator or (ii) after the dot operator, in the definition of the flux vector. Knowing the physical meaning for the concept of the dual space of a given vector space, you can then see what the physical meaning of the tensor product of the unit normal vectors (\hat{n}s) is, here.

Over to you. [And also to the UGC/AICTE-Approved Full Professors of Mechanical Engineering in SPPU and in other similar Indian universities. [Indians!!]]

A Song I Like:

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