I have taken off the slides for the Lecture # 3 of my introductory course on CFD, because it contained some serious errors. (….Yes, those were the serious errors—not just typos.)
I had caught the errors right the next day after posting it (about 10 days ago), but it so happened that in the meanwhile, I ended up (literally) wasting my time (and money) attending to certain highly promising promises of interviews for a professor’s job, as well as also actually attending a couple of highly promising interviews for promising positions of a professor, and subsequently, also trying to interact with the “management”s of these colleges.
I therefore couldn’t find the frame of the mind (and in fact also the necessary time at one stretch) to write down those lengthy equations in LaTeX, by way of rectifying the mistakes. I in fact was also traveling for these job-related matters. During these last 10 days, I visited some 4 places out of town, with two of them being more than a hundred km away. … No, I haven’t landed a professor’s job yet.
Anyway, back to the mistakes in the uploaded notes.
The mistakes in particular were concerned with the tensor calculus part of the slides. I caught the mistakes when I went a bit further in my notes, to the point of preparing the slides for the next couple of lectures—which would be: on the Navier-Stokes equations.
I had never worked out the full derivation of the Navier-Stokes equations in this way, i.e., for an infinitesimal CV, but using the Eulerian approach right from the beginning. … In the past, I had always relied on books, and never worked out my own derivations without referring to the proofs given in them. Many of these books are excellent. However, for the infinitesmal CV, they all always derive it only in the Lagrangian frame, and only later on do they use the vector calculus manipulations (or identities) to map the end-result to the Eulerian frame. Every one proceeds only that way. None does what I had unwittingly ended up attempting. …
… In fact, most of them use only the Reynolds’ Transport Theorem or RTT for short. (BTW, Reynolds had never himself stated this theorem in his entire life-time; the entire RTT movement was started only later, by an MIT professor.) Now, RTT is an integral approach, not differential. Usually, the books do the derivation using the RTT, and then proceed to get the differential form from this initial integral form. In the rare cases that they at all try to use an infinitesimal CV in an ab initio manner, they invariably use only the Lagrangian i.e. the non-conservative form.
Indeed, see the unanswered query on the Physics StackExchange here: [^]. … The first part of the question has gone unanswered for 3.5 years by now, after 10,000+ views. So, you know what I was getting at, here. And how, my errors, caught by me before engaging a single class based on these notes, therefore, might perhaps be excusable.
Anyway, what is more important is to note down the references which I found useful in working out this entire issue. These are the following two. (No, they of course don’t give you the derivation; they just deal with the basics of tensors and their calculus):
- “A brief introduction to tensors and their properties,” by Prof. Allan Bower of Brown (a fellow iMechanician!) [^]
- “Tensor derivative (continuum mechanics,” Wiki, section on divergence of a tensor [^] (and Prof. Piaras Kelly’s notes that it refers to, here [ (.PDF) ^] )
I then worked out the tensors appearing in the Navier-Stokes equations, in fully expanded components form. In this way, my path-way to the final Navier-Stokes equations now seems OK.
In other words, yes, I am now getting ready to answer that Physics StackExchange question, in my upcoming notes. … Give me a few days’ time, and both the components-wise worked out results, as well as the relevant portion excerpted for the slides of the Lecture # 3 of my CFD course, should be online.
But, also, please note, I haven’t run my work by anyone so far. So, it’s still an easy possibility that there are some elementary mistakes in it, too. At least, it would be easy enough for some unwarranted assumptions to creep in. (For instance, it was only during this recent phase of working out these things that I gathered for the first time in my life that there are some subtle pre-suppositions going into the Helmholtz decomposition theorem for the vector fields, too—assumptions like the field having to approach zero as distance tends to infinity—assumptions that I wasn’t at all aware of….) Therefore, I do plan to privately run my notes through a few mechanician friends/blogging acquaintances—even as I simultaneously post them here, within a few days’ time.
BTW, no, coming to those earlier errors in the Lecture #3 slides, even if someone had caught my errors (IMO, a low probability), none had pointed it out to me. None. I found it on my own. But only after publishing something else, in the first place!!
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A Song I Like:
You are (or at least should be) well-familiar with the well-worn out story by now.