Reading 3/10/2008: Section 3.6

Late, I know, but better than never.

This stuff honestly is straightforward. Having seen complex variable before, the approach is a little weird (partial derivatives of a complex function and chain rule usage are a little suspect, but it works).

Laplace's equation is solved pretty thoroughly in a bunch of classes, so that's pretty much par for the course. It is really cool to see the stuff on p. 197 about using the real vs. complex part as the potential. Didn't know you could look at it that way.

What exactly happens physically at the interface where, mathematically, the pressure becomes negative? That was really my only major question from the chapter.

Reading 3/5/2008: Sections 3.4-3.5

Section 3.4: Very cool; we have a number that can tell is whether we make the assumption of diffusion-dominance or vorticity-dominance. Makes sense to me, though where do we get the characteristic length scale from? And why does it shrink with turbulence? Otherwise, everything is clear.

Section 3.5: The derivation of small Bernoulli is very straightforward, though the result is quite cool. For big Bernoulli, we demand that a flow is irrotational - where are some examples where this really breaks down in a big way? Also, how might one measure &phi, the velocity potential function, much less its time rate of change?

The connection with the Laplacian here makes sense given our assumptions, but is a nice touch. The dipole/etc. thing is worth discussing in class. I've never understood physicists' fascinating with dipoles, but maybe I can with a little more detail.

Reading 3/3/2008: Sections 3.2-3.3

Section 3.2: Okay, cool. Our equations reduce with the acoustic approximation to something much more tractable. Very nice. I'm a little curious why we can assume (3.58) can only be satisfied in the two ways mentioned in the book. If I had a little more time, I would sit down and just prove this, but I wonder if there's a quick answer?

What are the real-world implications of S-waves decaying so rapidly? If the waves are only significant very, very close to the source, where do they arise/where are they important in practice?

How is the scattering effect of particles in p. 163 accounted for in fluid models?

Overall, this section seemed pleasantly simple. We get some nasty dispersion relations, but they're easy enough to use, and reduce to forms that are fairly easy to work with. Cool stuff.

Section 3.3: A section with "Theorem" in the title. Yay, math. Honestly, everything here made good sense. I wish I could see a more rigorous proof of the theorem, but for our purposes, this seems pretty good to me.

The bit at the end about vortex tubes is awesome. So THAT'S what a tornado is...

Reading 2/27/2008: Section 3.1

Woo, Fluids.

So we can immediately dispense with &mu, simplifying things quite nicely. Very cool derivation, and seemingly quite rigorous. I'm not entirely clear why we can assume &xi_n,n is zero, but I'm assuming it's because S_mm is &Phi.

The derivation of the new equation of state is very cool. It's remarkable to see that dp can be characterized completely and uniquely in terms of &rho. I'm not quite sure where the book is going with the "exact differential" comment, but I'm assuming that means something to physicists that it lacks in meaning to me.

Why do we assume viscous stresses are linearly proportional to velocities? What is the origin of this postulate? That was one of the major aspects unclear in the section. Otherwise, the derivation of Newtonian viscosities was clear enough.

And holy cow, we have Navier-Stokes! If only we could solve them generally...

I'm a little unclear what is meant by a volume force in (3.27) - is this just to emphasize that this is a force separate from the external (e.g. gravitational) force?

On p. 151 I just want to point out that the word "magma" is bloody awesome. Everyone should incorporate it into their daily speech immediately. No, seriously. I mean it.

Overall a fantastic section.