A virtual redirect

For those of you stumbling across this blog looking for my blog, I have to warn you, this isn't it. This was a required 'blog' for my Continuum Mechanics and Fluid Dynamics class, where I commented on lectures and readings. If you want to read my actual journal (not that I have anything deep in there, just comments about what's up in my life) check out my google+ .

So I started reading the paper for the project, and it started by doing nondimensionalization just like we did in class on Monday. So that was exciting.

The lecture notes for lecture 10 are, again, pretty reasonable. It would be nice if it were more explicit in them that we aren't taking any particular initial drop condition - I kept going back and trying to figure out what we were assuming about initial conditions, until I realized that the final product of the first section would be a DE for the drop, and that we didn't actually start with any particular L and h when doing the nondimensionalization. Only in the last section did we figure out what the shape of the drop can be.

That lecture seemed pretty short compared to previous ones... I didn't see how the first part, on nondimensionalization, tied in to the second part, about the effects of a thin layer of fluid. Both parts seemed pretty straightforward though, I don't have any particular questions about them.

Lecture 8

So in a fit of not working on a hum paper, I've worked ahead on the reading and the homework a bit. Lecture 8 is a pretty reasonable lecture, not too bad. I'd appreciate a review of how we know the thickness of the boundary layer in both cases - I especially didn't get where the last boundary layer number came from, I don't see how to get that. Overall, I'm happy with these last two lectures and the homework :)

Lecture 7

This lecture was a good bit easier - probably because last time I finally got all that surface tension stuff :) As seems to be the norm, the trickiest part is the setup of the linearized equations - there's a lot of information in a pretty small space, so I'll need to reread that again to get it fully, but that seems to be the way things generally work in this class, so I'm not too worried, it's getting easier to do that.

Lecture 6:

This lecture was trickier... I could definitely do with seeing the linearization from page 6.5 more explicitly. Or maybe I just need to go back and reread the previous lecture notes for similar things and then I'll get it. I'm not sure what else specifically to ask about - the material all seemed to make sense, but it feels like there's just a lot of it, a bunch of slightly different things to derive. I didn't really spend that much time on it either, I had a clinic presentation this week and the homework due tomorrow has been pretty hard. Ah well, I'll take another look at all the stuff and try to digest it all.

Lecture 5

That reading had an interesting collection of phenomena in it. It both started and ended on notes I didn't really get though - I wasn't understanding why the surface tension is given by "surface energy = T * surface area", it wasn't obvious to me that T had to be a constant that doesn't depend on any other parts of the geometry. And at the end, there was a note about sonoluminescence, and I didn't really get how an oscillating bubble would give rise to that. But that's okay, the middle is the part that's important for now, I guess. I mostly got that - though perhaps I should read it over again, I thought I'd gotten the homework too but going over it in class last time I realized that I'd missed some problems I thought I had right, so I should give the reading another read-over. I'm still not clear on the beginning of the bubble-oscillation section, what we do to get the order-epsilon terms.

Ok, so this reading was a bit more dense than the previous one... I definitely got lost in some of the parts of it. The part I'd most like to see again is the part where we apply the surface conditions - I kind of got the parts before then, and after we get the wave equation out of it the rest is simple - well, I guess not that simple, but the wave equation and analysis of it is pretty familiar. Well, except the part about the kdv equation - I don't think I've seen that equation before, haven't seen solutions to it.

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.

2.13, 3.1

I'm making posts really early, 'cause I'll be gone all weekend...

This section felt a lot like 2.11, there was a good bit of algebra to derive some wave behavior. I understood it better than 2.11 though. I was kind of curious about Love waves - is it possible to give a quick hand-wavy explanation of them like you did with the Raleigh waves (which, I think, did help in understanding the section)? On a more practical note, I'm a little bit curious about what happens around the depth where the retrograde motion turns into prograde motion. It seems like there might be some interesting phenomena there, once some of the terms cancel out and we can see the effects of things we'd neglected elsewhere, but I'm not really sure.

3.1

Phew, I think I'm getting better at understanding this tensor algebra stuff! It's getting quicker to read these types of sections, though it still takes some time to get through it all. One part that I wasn't able to figure out though was the note about 'effective pressure'. It seemed like a silly trick that didn't give me anything useful; did I miss something, is there some physical meaning to it that's good to know or some way in which it needs to be used?

2.13

This section was a good bit shorter! I'm very happy with it - we defined a conservation of energy law, which is cool. I didn't really have any issues with this section, I thought it was reasonable and made sense to me. I'll still have to go back over the previous section to make sense of all the derivations though.

2.11

Oof! This section dove into more detailed algebra than the previous ones... or is it just me? I'll probably have to read over some of those derivations again, and I'll probably grasp some of this better once I start doing the homework and get more of a handle on it. I was a bit curious about the potentials though. In electrostatics/electrodynamics, the potentials actually do have interesting physical interpretations. Are there physical interpretations for these potentials, phi and A? They're certainly interesting mathematical tricks to make the equations more solvable, but it would be even better if they had some sort of interpretation on their own as well.

2.8-2.9

I thought these sections were pretty interesting, but I wasn't sure of the use of the quasistatic approximation. We use it to be able to talk about path-independent quantities and then to justify more symmetries. However, it seems like motion in which there is no acceleration isn't too interesting, since it excludes wave phenomena and all sorts of other things. I'm having a little bit of trouble following through on where we need the quasistatic approximation and when we don't, and whether we'll need to keep in mind that some of the things we derive aren't valid when acceleration is present.

It's cool how we now have a potential energy function. That seems like it'll be very important, and adds a lot to my understanding of the system.

2.10

Wow, section 2.10 was really cool! I don't really have any particular questions about it, I thought it made sense, but I thought it was really cool how we can take nonlinear equations, identify when they're "almost linear", and rewrite them in a form that's probably way easier to solve.

Section 2.7

Section 2.7 is comparatively short, but I'm not completely clear on how the symmetries in the elastic constants play out. Sure, I can write them out and convince myself that there's 36 of them, but is there some better way of seeing how symmetries reduce the number of degrees of freedom of a tensor?

Sections 2.4-2.6

The derivation of the use of ellipsoids to visualize the tensor fields is interesting, but I'm having a lot of trouble figuring out how I would ever read one, or how to handle the hyperbola case. Are we going to use this more, and if so, are there some nice easy examples for me to look at so I can try to figure it out?

I got the explanation of why the stress tensor had to be symmetric, that made a lot of sense and was neat.

responses

So, I'm reading over other people's thoughts before I go to class, and I thought I could respond to some of them! W00ts!

Michael mentions that "A question about 1.6 and partial derivatives: in 1.6 a partial derivative was defined, and shown to transform like a tensor. But if I'm remembering correctly, in G.R. we saw a different 'partial derivative' (the "box operator"?) that was developed specifically because the old "standard" partial derivative did NOT transform in a covariant fashion. Is this simply because of some extra consideration arising in a non-Euclidean space, or am I getting these two cases mixed up or confused in some sad way?"

I think the reason that we needed a 'new' gradient operator is because in GR (or even in the special-relativity treatment of E&M) we want our vectors to be invariant not just under the standard rotations that we know and love in 3-space, but also under Lorentz transforms (essentially, 'rotations' in 4-space). In special relativity, as applied to E&M, needed the "box" operator because we needed the minus sign on the t component to make the physics and math work out.

And on a miscellaneous note, in response to " In composing this blog, I've been wishing that I could use cut-and-paste in the "Compose" window, but it doesn't seem to work. Is this a bug or a feature? Or is it just a quirk of my browser (Safari)? Thanks for any insight or advice!"

It's probably a bug. In Firefox under Linux, I can copy/paste fine into the text box, but it does behave strangely in other respects and isn't just a text box, it's got all sorts of fancy javascript behind it... yep, according to http://help.blogger.com/bin/answer.py?hl=en&answer=42247 , Safari isn't fully supported yet. Perhaps, if you want to copy/paste, temporarily switch over to the "Edit HTML" mode first (that looks like it just gives a standard text box) instead of the default "Compose" mode?

Sections 2.1-2.3

Ok, so I'm trying to get an early start on this semester, and hence I'm making a blog post about the reading due this wednesday :) I think I'm getting how we're defining the stress tensor; it seems to me to be pretty close qualitatively to the
Maxwell stress tensor from E&M; as then, it seems like the most concise way to put it is that the {ij} component of the stress tensor is the force in the i direction on a unit area normal to the j direction. I'm not sure I'm visualizing the "make a cut in the medium" explanation though. Am I getting this right, or do should I think about the problem setup some more?

Also, in response to Steven Ning wanting to get more fancy LaTeX than the online image converter offers - they cite the script they use as being available at http://www.nought.de/tex2im.html ; it's a bash script which uses "latex" and "convert" to make the images. If you have a unix system with bash available, it seems like it should be a pretty simple matter to stick '\include{amsmath}' into the script and use it, though I haven't tried it myself yet.

Reading for 1/28, 1.1-1.5

Whoa, expressing the standard dot products and cross products (and, later, div and curl) in terms of tensor rank-reductions is really cool!

I'm not sure what to make of the fact that the gradient of a scalar doesn't transform the same way as a 'normal' vector. What is the distinction, what does contravariance actually mean? I don't get it, and the text doesn't go into detail.

Frist psot.

Hey all!

I'm Max, and I'm currently an HMC senior majoring in both Physics and Computer Science. I'm taking the class because fluid dynamics is super-cool! And there's a pretty high chance I'll be using a lot of this stuff in grad school. I'd like to come out of this class with a much better understanding of tensors and how to use them - I never felt like I completely got them when taking GR, or when we used the Maxwell stress tensor in Big E&M, so I'm looking forward to a more in-depth treatment of them. My favorite two equations are the Lorentz transformations. These two go together; I couldn't pick just one, and I didn't want to convert them to a single equation that's less understandable.








A fun fact about me is that I've gotten my computer to beat expert minesweeper in 41 seconds without cheating :)