Ok, so I'm starting to read the next lecture (3, water waves) and I'm a bit confused at the beginning. I'm not sure I understand what the kinematic condition means - it seems to be getting something from nothing, a condition that seems to be just based on defining a surface "eta(x,t) = z". I could follow most of the rest of the reading okay. I was surprised at getting hyperbolic trig functions out of there into the dispersion relation, I'll definitely have to have another think at some point about what that means.

Ok, so I'm trying to get the reading done early...

To answer the question at the end of the first lecture notes, I'm pretty sure it is possible - you just need u_t to be equal to -(u \cdot \nabla) u.

The second lecture notes were interesting, it was nice to see the Euler and Nanvier-Stokes equations derived again. I guess these aren't quite the same Euler equations that we had before, since we're assuming incompressibility whereas before there'd been one equation with derivatives of the density. I guess it's a reasonable and useful assumption for a lot of cases.

Ok, so I really liked that section! It's a cute way of solving 2D problems, though it doesn't seem like it could generalize to 3D. It seems like a lot of it is kind of ad-hoc, in that I still have no idea how to properly pick a function which would give me the solution for any particular geometry. It's kind of like method of images in that respect, it takes a bit of intuition, which I guess I don't have yet. Is there some way of starting with a surface, and then going to the function? I guess setting the appropriate derivatives of the function to zero at the surface (representing zero velocity at the boundary). I don't think methods for actually coming up with what function to use were discussed in the text, and some more talk about that would be nice.

Oops! Forgot to post these earlier! Here's some questions for the next two days' worth of reading...

3.2-3.3

I was quite surprised when at the end of 3.2, we decided to drop the treatment of waves! I did have a question about S-waves before we go on though.

It seemed to me that, with the restoring force coming from the velocity-dependent viscosity, there wouldn't be an S-wave at all - a bit of fluid could never be made to return to its original position at all, even at very small distances. Actually, I guess that means you couldn't have any wave propagating from a one-time disturbance, but a wave that's been going fo
r all time with a continuous forcing term could exist.

3.3 was tricky as well, there were parts of the derivation of Helmholtz's theorem that I didn't quite get, I'll have to go back and stare at them a bit more... I was curious about the "Scalar function P" on page 168 though. Is there any physical meaning to it? It's nicer when the mathematical tricks we use also have a physical meaning, though that might be too much to ask considering how many of them we use :)

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3.4-3.5

Ooh, we know what the Reynold's number is now. It ended up seeming a bit out there - just an order-of-magnitude estimate, not to actually calculate something but to give us an idea of which regime the fluid is in. I'd appreciate a physical example of what "diffusion of vorticity" would look like. Maybe I have an idea, but I'm not sure I'm envisioning it right.