Some Ideas Implicit in Modeling Something as a Continuum
This post concerns continua. It touches on certain basic conceptual ideas concerning continua but not with details of their applications. Of course, the ideas here remain relevant for applications too. Many engineers/physicists do not seem to be very clear on these ideas. … Most of these are the observations I have derived myself through self-study and thought. As such, their expression is likely to be a bit immature. However, a blog is an informal medium, and so, there is no harm sharing these…
What is a continuum? For the purposes of our discussion, the real number line, studied right in high-school, provides a suitable example. Take any line segment; no matter how small it may be, there always are an infinity of points lying on that segment. There are no gaps in any part of a line segment. As such, it is a continuously existing object—a continuum. Extend this same idea to a 2D or 3D embedding space, and you can say that there are an infinity of points on any surface or in any 3D volume. … We all learnt it in high-school. However, some of its implications for our more advanced courses in physics/engineering sciences are not always fully grasped.
Consider a simple physical situation concerning fluids.
Light up a mosquito coil, an incense stick (or, if you are a smoker, a cigarette) in a closed lighted room, and watch the smoke spread. (No fans—let the process be as dominated by natural convection as possible.) This is a routine example to illustrate the transition from the laminar to the turbulent flow. We will not go into those details. But what concerns us here is a certain basic conceptual point.
On their upward path, initially, the smoke lines are rather sharply defined. As the smoke goes further up and the turbulence sets in, the streak-lines mix up and become less sharply defined. As the smoke travels still further, the lines also become increasingly thin, and therefore, visually, they are less sharply defined. If you let the smoke fill the room, eventually, you will find that there no longer are smoke lines visible at a remote corner; instead, it’s all almost uniformly foggy out there.
You know that such mechanics of fluids is ruled by the Navier-Stokes equation—an equation that is very much continuum-mechanical. … Now, here is a question.
Consider two fluid (air) particles initially in close vicinity with each other. (You may consider smoke particles too, so long as remember that our basic concern is with the air motion; we consider smoke only because it makes the fluid movement visible.) The question is: as the fluid moves, do the neighboring particles remain neighbors always? Or do they drift apart?
… Think about it for a while before proceeding further…
If you have caught the drift of this post (and not just the drift of the smoke), you would notice that the question does not so much concern only the NS equations as it does the fundamental assumptions we bring to bear in our continuum—i.e. conceptual and therefore mathematical—modeling.
Let’s make the question more precise: consider two particles that are only an infinitesimal distance apart initially. How about these?
The answer, as far as I know it, based purely on my own reasoning (and I would be happy to correct my reasoning if you point out errors in it) is: in the NS (or any continuum theory), the two particles would always remain neighbors no matter how far away they travel.
Now, here, you are likely to disagree with me, basing your reasoning on the chaos theory—the extraordinary sensitivity to the initial conditions brought about by the differential non-linearity and all…
(Incidentally, from a general philosophic viewpoint, the naming of the non-linear phenomena is a curiosity—the terminology wrongly suggests that things are/can be defined purely by negation of something else—not in reference to some actually existing facts of reality, but by negation alone.)
But coming back to the chaos theory, though I have only surface knowledge (if at all that) concerning it, I, however, also believe (wrongly or rightly—but I think rightly) that the said sensitivity to the initial condition applies only to those particles which are a finite distance apart initially—not infinitesimal. … There is a huge mathematical difference between the two: the finite and the infinitesimal.
If you find it strange that despite chaos (or turbulence) the infinitesimally close particles don’t move a finite distance apart, consider that college physics experiment done with some peculiar jelly-like thick fluid. (I forgot where I read up on this experiment but it could easily be the Feynman Lectures.) Roughly speaking, the experiment goes like this.
You take that jelly-like fluid in a beaker and put a drop of ink or so in it. Then, with a stirrer, you gently stir the fluid, say in the CW sense. What now happens is that the portions near the stirrer keep sticking to it, and so, due to the stirring action, they get stretched and form roughly circular lines. As these fluid layers stretch, the ink particles move along too. (After all, the stickiness applies as much to the ink particles as to the fluid-stirrer interface). After a few rotations, you see a highly mixed up jelly. All the colored ribbons seem entangled with each other and it seems impossible to disentangle them.
However—and here’s the dramatic part of the experiment—if you now slowly rotate the stirrer backwards (in the CCW sense), then the jelly-ribbons actually begin to “shrink” backwards. Once you complete the same number of rotations backwards, you once again get just a localized spot of the ink. The entanglement of the jelly-streaks disappears completely!!
The reason the experiment works even under actual physical (laboratory) conditions is because the fluid in question is thick. However, the principle does get established with this experiment. So long as you have viscosity, in principle, the same behavior can be expected.
During fluid mixing, local sub-regions of a viscous fluid never really tear away from each other (neither do they begin sticking with some other sub-regions). Continuity of the adjacent fluid particles is maintained…. Is “continuity” the right word here? … Actually, what gets preserved is not so much continuity as it is: connectivity.
Two fluid particles connected to each other always remain so. Regardless of the degree of internal mixing of the fluid. Nay, not just that. Regardless of any turbulence within the fluid. That’s the conclusion we seem to be reaching here, don’t we?
If you have seen those CFD simulations of vortex-shedding, you must be wondering: “how come?”After all, a vortex is defined by a finite quantity of fluid rotating within its local region. As vortices get shed, they drift away from each other. If one considers a fluid particle in between two adjacent vortices but closer to one of them, it is an easy guess that it would get sucked into the nearest vortex. Thus, there should be a separation between this particle and its neighbor that went to the other neighbouring vortex.
But rather than answer this question, I would let you figure it how—just in case, of course, you don’t know it already. …
Actually, connectivity or otherwise of adjacent points in vortex-shedding is a fairly well known result. … Typically, students do know the right answer about it. [Something similar used to be a routine orals question at COEP for a second-year course on fluid mechanics + heat transfer for us—the students of metallurgy.] What students don’t realize is that the right answer also applies as a generalization to all continuum phenomena—wherever the assumption of the continuum applies.
[Of course, I am only asserting a generalization here, without really having proved it, and so, I could be wrong. But I can’t think of any good argument why or how this could be a hasty generalization. Please let me know if you know of any.]
Something similar is what I had indicated during one of my past comments at iMechanica. The context, there, was mechanics of solids rather than of fluids, in particular, fracture mechanics. I had pointed out how I had had a discussion about these observations of mine with a graduate student of mathematics (among others) and how I was always told that (in my words) that “one body separates internally and becomes two bodies” is something that simply cannot be dealt with using the mathematics of continua. The issue is one of connectivity, really speaking. A continuum (of the type we conventionally use in physics and engineering to describe real things) locally does conserve connectivity.
I have something to say about implications of this all too, but some other time, may be, my next post here… Also, about singularities in continua—some basic conceptual comments, concerning the way we do our mathematics.
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Update on February 1, 2010
No one seems to have caught an obvious error in the above description. (Actually, the statements above are not so much wrong as they are in need of suitable qualifications throughout.). … I had thought that someone would catch me right within a day or so—at least that part of the writing above where I get to making that statement about two infinitesimally distant particles going into two different vortex regions. … In any case, right from the very first draft of this post, I had been dropping hints, too, concerning singularity and all… Now, more than two days later (and many hits later), still, none seems to have caught the error. What’s going on?
Apparently, going by the comments received at this blog (but moderated out) many people (notably including certain students (probably from an engineering department) at IIT Bombay) seem to have been more concerned with swearing at me rather than catching the technical/conceptual/mathematical/reasoning errors in the write-ups…. Hmmm…
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A Few Songs I Like (More or less at random)
1. (Marathi) “tuj saguN mhaNu ki nirguN re…”
Lyrics: Sant Dnyaneshwar
Singer: Pt. Bhimsen Joshi
2. (Marathi) “tyaa phulanchyaa ganDhakoshi…”
Lyrics: Sooryakant Khandekar
Singer: Hridaynath Mangeshkar
3. (Hindi) “chhupaa lo yun dil me pyaar meraa…”
Singers: Hemant Kumar, Lata Mangeshkar
Lyrics: Majrooh Sultaanpuri