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Blog - fluid flows and infinite dimensional manifolds (part 4)

Created stub for Blog - fluid flows and infinite dimensional manifolds (part 4). While I still need to complete Blog - fluid flows and infinite dimensional manifolds (part 2), I would nevertheless start to collect material for the follow up post. That should help in making up my mind about what should go where.

Right now I have three topics in mind that could be talked about here:

  • viscosity and the relation of Euler's equation and the Navier-Stokes equation for Newtownian incompressible fluids.

  • Jacobi fields and how to get an idea how fast geodesics diverge,

  • Hamiltonian formulation for fluid flows.

It's all very stubby right now...

Comments

  • 1.

    I think there should be one blog post dedicated to the comparison of nonviscous fluids described by Euler's equation and viscous fluids described by the Navier-Stokes equations, with some history of the subject, boundary layers and singular perturbation theory. And why no one seems to have succeeded in generalizing the framework of infinite dimensional manifolds to the Navier-Stokes equations.

    And another one about how pressure comes from the curvature of the Diffeomorphism groups, Jacobi's equation and how one can put a bound on the reliability of weather forecasts with this model.

    Comment Source:I think there should be one blog post dedicated to the comparison of nonviscous fluids described by Euler's equation and viscous fluids described by the Navier-Stokes equations, with some history of the subject, boundary layers and singular perturbation theory. And why no one seems to have succeeded in generalizing the framework of infinite dimensional manifolds to the Navier-Stokes equations. And another one about how pressure comes from the curvature of the Diffeomorphism groups, Jacobi's equation and how one can put a bound on the reliability of weather forecasts with this model.
  • 2.

    Be careful, or

    $$ \lim_{t \to +\infty} \frac{finished blog articles}{blog articles started} = 0 $$ But all this stuff sounds great.

    Comment Source:Be careful, or $$ \lim_{t \to +\infty} \frac{finished blog articles}{blog articles started} = 0 $$ But all this stuff sounds great.
  • 3.

    I have been strongly discouraged to finish the series on energy balance and atmospheric radiation by the discussions I had about the zero dimensional climate model. Maybe I should just ignore that and finish both "good vibrations" and "the color of the night". I will think about it :-)

    With regard to this blog post, I now have an explanation of the geodesic deviation equation. But I need a good exposition of how to calculate the sectional curvature in $SDiff(S^2)$.

    Comment Source:I have been strongly discouraged to finish the series on energy balance and atmospheric radiation by the discussions I had about the zero dimensional climate model. Maybe I should just ignore that and finish both "good vibrations" and "the color of the night". I will think about it :-) With regard to this blog post, I now have an explanation of the geodesic deviation equation. But I need a good exposition of how to calculate the sectional curvature in $SDiff(S^2)$.
  • 4.
    edited May 2012

    I've compiled Dominic Orchard's modified Numerical Recipes in C Navier-Stokes code which outputs an mpeg which is on my website. HTH

    Comment Source:I've compiled Dominic Orchard's modified Numerical Recipes in C Navier-Stokes code which outputs an mpeg which is on my [website](http://stuttard.org). HTH
  • 5.

    That's nice, but somehow it does not run in my Firefox browser.

    However, the quest to calculate sectional curvatures for diffeomorphism groups can hardly be handled by numerical approximations - it is the "analytical" part of the series. But of course I would also like to delve into numerical waters...later...much later.

    Comment Source:That's nice, but somehow it does not run in my Firefox browser. However, the quest to calculate sectional curvatures for diffeomorphism groups can hardly be handled by numerical approximations - it is the "analytical" part of the series. But of course I would also like to delve into numerical waters...later...much later.
  • 6.

    Runs fine on my and Glyn Adgie's FFs on archlinux and windows7. IE still isn't html5 compliant.

    Best wishes

    Comment Source:Runs fine on my and Glyn Adgie's FFs on archlinux and windows7. IE still isn't html5 compliant. Best wishes
  • 7.

    I came across this computational anatomy site where diffeomorphisms are being used in mathematical medicine.

    It has lots of simple interactive (jquery) sliders for operating group actions and investigating basic symmetries.

    It defines an anatomy as:

    "an orbit under groups of diffeomorphisms."

    Comment Source:I came across [this](http://cis.jhu.edu/education/introPatternTheory/) computational anatomy site where diffeomorphisms are being used in mathematical medicine. It has lots of simple interactive (jquery) sliders for operating group actions and investigating basic symmetries. It defines an anatomy as: "an orbit under groups of diffeomorphisms."
  • 8.

    Runs fine on my and Glyn Adgie's FFs on archlinux and windows7. IE still isn't html5 compliant.

    Hm, I run FF on windows7 and just see a grey area where the video is supposed to be...is it the html 5 video element ? Then I have no idea what could go wrong. When last I looked, IE was far behind with its htmll 5 support, but both Safari and FF looked good (but that is almost a year ago).

    Comment Source:<blockquote> <p> Runs fine on my and Glyn Adgie's FFs on archlinux and windows7. IE still isn't html5 compliant. </p> </blockquote> Hm, I run FF on windows7 and just see a grey area where the video is supposed to be...is it the html 5 video element ? Then I have no idea what could go wrong. When last I looked, IE was far behind with its htmll 5 support, but both Safari and FF looked good (but that is almost a year ago).
  • 9.
    edited May 2012

    Sorry Tim.

    My FF had DvX installed but I didn't notice. Glyn thought he'd tested FF with pure html5 but was in the wrong browser's window.

    It wouldn't have worked anyway as FF doesn't support mp4. So I got more than the test was wrong (I don't remebmer Popper having an answer to that?).

    I made html5 standards compliant webm and ogg versions of the video but they didn't work! FF has a bug. Wasted 3 hours, urrgh, won't go into it.

    Safari and Opera are supposed to be compliant so I will just reopen yet another what I think is a falsely closed a bugzilla report.

    Comment Source:Sorry Tim. My FF had DvX installed but I didn't notice. Glyn thought he'd tested FF with pure html5 but was in the wrong browser's window. It wouldn't have worked anyway as FF doesn't support mp4. So I got more than the test was wrong (I don't remebmer Popper having an answer to that?). I made html5 standards compliant webm and ogg versions of the video but they didn't work! FF has a bug. Wasted 3 hours, urrgh, won't go into it. Safari and Opera are supposed to be compliant so I will just reopen yet another what I think is a falsely closed a bugzilla report.
  • 10.

    Safari and Opera are supposed to be compliant so I will just reopen yet another what I think is a falsely closed a bugzilla report

    So does it work with Safari? Should I install and try it?

    I don't remebmer Popper having an answer to that?

    There are a lot of misconceptionis about Popper in the physics community, one is based upon Popper's usual trick to deliberatly oversimplify his points. Taken out of context, many of his statements seem to be quite naive. Like the one about black swans: "All swans are white" is falsifiable, because you only need to observe a black swan. Well, your first reaction would be of course "who painted the poor swan black"? And after you are convinced that no one painted the swan black, you'll run into the problem that no one believes you, and rightfully so, because your observation is not reproducable. Etc. etc. Popper knew all this, of course :-)

    Comment Source:<blockquote> <p> Safari and Opera are supposed to be compliant so I will just reopen yet another what I think is a falsely closed a bugzilla report </p> </blockquote> So does it work with Safari? Should I install and try it? <blockquote> <p> I don't remebmer Popper having an answer to that? </p> </blockquote> There are a lot of misconceptionis about Popper in the physics community, one is based upon Popper's usual trick to deliberatly oversimplify his points. Taken out of context, many of his statements seem to be quite naive. Like the one about black swans: "All swans are white" is falsifiable, because you only need to observe a black swan. Well, your first reaction would be of course "who painted the poor swan black"? And after you are convinced that no one painted the swan black, you'll run into the problem that no one believes you, and rightfully so, because your observation is not reproducable. Etc. etc. Popper knew all this, of course :-)
  • 11.
    edited May 2012

    I changed the name of this wiki entry from

    Blog - fluid flows and infinite dimensional manifolds III

    to

    Blog - fluid flows and infinite dimensional manifolds (part 4)

    Saying 'part $n$' is our blog-standard notation for multi-part posts, as you can see on Blog articles in progress.

    Also, I like to keep things simple and systematic, so instead of 'part 1', 'part 2', 'interlude' and 'part 3', I'm going to number them sequentially.

    Comment Source:I changed the name of this wiki entry from Blog - fluid flows and infinite dimensional manifolds III to [[Blog - fluid flows and infinite dimensional manifolds (part 4)]] Saying 'part $n$' is our blog-standard notation for multi-part posts, as you can see on [[Blog articles in progress]]. Also, I like to keep things simple and systematic, so instead of 'part 1', 'part 2', 'interlude' and 'part 3', I'm going to number them sequentially.
  • 12.
    edited July 2012

    You asked for comments, Tim. It looks great! The first place I ran into a little expository trouble was where you said

    Euler's equation is a geodesic equation:

    $$ \frac{d^2 \phi}{d t^2} = \partial_t u + \nabla_u u = \text{something coming from curvature} $$

    and I think:

    1) is that supposed to be the geodesic equation? It doesn't look like the versions of the geodesic equation I know!

    2) What's $\phi$?

    3) What's $u$?

    Of course I can guess answers to these questions, but I think you're going a bit too fast here.

    Comment Source:You asked for comments, Tim. It looks great! The first place I ran into a little expository trouble was where you said > Euler's equation is a geodesic equation: > $$ \frac{d^2 \phi}{d t^2} = \partial_t u + \nabla_u u = \text{something coming from curvature} $$ and I think: 1) is that supposed to be the geodesic equation? It doesn't look like the versions of the geodesic equation I know! 2) What's $\phi$? 3) What's $u$? Of course I can guess answers to these questions, but I think you're going a bit too fast here.
  • 13.

    Thanks, yes, re-reading it a couple of months later I realize that I should explain more. I don't know when I will have the time to come back to this topic, though.

    Comment Source:Thanks, yes, re-reading it a couple of months later I realize that I should explain more. I don't know when I will have the time to come back to this topic, though.
  • 14.

    Well, maybe I should finish off this article. Is that okay?

    Comment Source:Well, maybe I should finish off this article. Is that okay?
  • 15.

    Sure :-)

    Comment Source:Sure :-)
  • 16.

    Okay, I'll try to do that soon. I'm in the mood for taking things people have done here and putting them on the blog!

    Comment Source:Okay, I'll try to do that soon. I'm in the mood for taking things people have done here and putting them on the blog!
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