Reading 4/21/2008: Lecture 9

I'll admit when I first saw the words "Dimensional Analysis," I felt my skin flush; my heart started beating faster; my mind began racing; and I scanned the exits of the room I was in. Classic fight or flight, with an emphasis on the latter.

But as I read, I realized my first reaction was silly. That said, I couldn't stop thinking that a lot of this was hand-waving. Can we legitimately summarize this discussion (or least summarize the justification) by observing that "in physics, almost everything is continuous" so arguments like this just work?

More precisely, what exactly is "length scale" or "characteristic length" supposed to represent? Is this along the lines of the length of the box everything is contained in, or is this the length of the smallest phenomenon observable/significant? What about in problems with a large container and small phenomena of global significance?

Also, why do we put a bar on the velocity scale U?

Finally, how is Reynolds number in any way well-defined? Can't I just say the scales are approximately this or that and get entirely different values?

Onto the next section, when can we legitimately make the lubrication assumption and get realistic results? I want to say for "slippery" fluids, but what does that even mean?

When we get a time estimate for the length of time needed to remove an adhering object, what assumptions are we making about the way it's pulled off? I feel like this should be clear, but wasn't really for me.

Overall, really cool stuff. I'm amazed that despite the sophistication of the equations, we can get tangible and useful numerical results.

Reading 4/14/2008: Lecture 7

First derivation is very cool. Energy minimization leads to the fluid cylinder instability. Makes sense, and the derivation is simple enough.

Very cool to see Bessel functions popping up, though given the type of equations and the space on which we're solving them, this doesn't seem particularly surprising, comparing to experience with Math 180.

Overall, it all makes sense to me. It's very interesting to see that what is fundamentally a stability analysis can be performed by linearizing and solving the system and then looking at solutions for which the waves grow. I'm a little unclear as to where the "asymmetric modes" part of the final paragraph comes from, but the fact that wavelengths greater than a threshold value grow to infinity actually makes sense to me.

NB: Sorry I've missed so many blog entries. If I can find time, I'm going to go back and write them, but these past two weeks have been absolutely vicious.

Reading 4/2/2008: Lecture 3 Notes

I wish I'd had more time to blog recently, but life has been a little too crazy.

At any rate, this is cool stuff. It's nice to see how the free-surface boundary conditions play out in the mathematical PDEs framework, and it's even cooler to see a fairly rigorous proof of Bernoulli's theorem. Obviously the same concepts are there, but, well I'm a mathematician, so it's better now.

Definitely cool to see the Fourier transform appear in the end, too. I'd be curious to hear about the general applicability of the FT in fluids - it's certainly a big hammer and great for making this smooth. (No pun intended.)

The series expansion strangely reminded by of perturbation theory from big quantum; I'm guessing this is a fairly standard approach, I think it's more the notation. That said, I wonder how applicable the linearized equations are and/or what are their drawbacks?