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Education Science

Physicists Finally Solve the Falling-Paper Problem 325

Posted by michael from the falling-down dept.
neutron_p writes "The so-called "falling paper" problem has long intrigued scientists. James C. Maxwell pondered the tumbling motions of playing cards in 1853. Why don't flat things fall straight down? Pieces of paper fall down, then rise into the air, then glide along, then again rise... It occurs in a seemingly chaotic manner. Now researchers at Cornell University have solved the falling paper problem by calculating the motions of a scientific journal page in flight and there were a few surprises." There's also a story in the Cornell Sun.
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Physicists Finally Solve the Falling-Paper Problem More

Physicists Finally Solve the Falling-Paper Problem

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  • Yup (Score:5, Funny)

    by Anonymous Coward on Sunday October 31, 2004 @01:19PM (#10679803)
    Now they just need to solve the 8+ folds problem...

  • The site is already getting quite slow... (Score:5, Informative)

    by IversenX ( 713302 ) on Sunday October 31, 2004 @01:20PM (#10679808) Homepage
    better save it here for posterity :-)

    Image: The seemingly chaotic motions of this page from a scientific journal became part of a computer modeling exercise to show why flat things don't fall straight down., J. Wang and U. Pensavento/Cornell University. Copyright Physical Review Letters 2004

    The same falling-paper principles apply, the physicists believe, to naturally flat things like leaves. If they are right, Wang and Pensavento may have finally solved the mystery of why autumn leaves depart from a neighbor's tree on a windless day . . .

    . . . rise into the air . . . . . . rise again . . .

    . . . glide along . . .

    . . . and have to be raked from yards that don't contain a single tree.

    As Wang explains, "Leaves and paper fall and rise in a seeming chaotic manner. As they fall, air swirls up around their edges, which makes them flutter and tumble. Because the flow changes dramatically around the sharp edges of leaves and paper, known as flow singularity, it makes the prediction of the falling trajectory a challenge."

    Among the first scientists to be intrigued by the behavior of falling paper was Scottish physicist James C. Maxwell, who pondered the tumbling motions of playing cards in 1853. But while Maxwell was a brilliant mathematician, he lacked the today's computer-modeling techniques, not to mention access to fast, powerful computers. Wang and Pensavento put those advanced tools to good use to show why the falling trajectory of thin flat things -- and the behavior of airflow and other forces -- is not predicted by the classical aerodynamic theory.

    "There were a few surprises," Wang notes. "We found the flat paper rises on its own as it falls, which would not happen if the force due to air is similar to that on an airfoil. Instead, the force depends strongly on the coupling between the rotating and translational motions of the object."

    Wang and Pesavento also showed that the falling-paper effect is almost twice as effective for slowing an object's descent, compared with the parachute effect (that is, if an object falls straight down). And that evidently benefits trees and other plants that need to disperse seeds some distance from the point of origin. Plants with flattened seedpods also take advantage of the falling-paper effect.

    The research was funded by National Science Foundation, the U.S. Air Force Office of Scientific Research and the Packard Foundation.

    Says the professor who does not use the falling-paper effect to grade student essays and forecast their future: "What is predictable is that as the autumn leaves tumble down, they drift in particular directions, depending on the way they turn. This may explain, Wang adds, "why you are getting the leaves from your neighbor."

    Source: Cornell University

    • Re:The site is already getting quite slow... (Score:5, Informative)

      by R.Mo_Robert ( 737913 ) on Sunday October 31, 2004 @01:55PM (#10680022)

      The site is already getting quite slow... better save it here for posterity :-)

      Or use the Coral cache version (remember, just appennd .nyud.net:8090 after the domain--I don't know why Slashdot doesn't do this more often): http://www.physorg.com.nyud.net:8090/news1630.html [nyud.net]

    • Wang and Pesavento also showed that the falling-paper effect is almost twice as effective for slowing an object's descent, compared with the parachute effect (that is, if an object falls straight down). And that evidently benefits trees and other plants that need to disperse seeds some distance from the point of origin. Plants with flattened seedpods also take advantage of the falling-paper effect.

      Interesting. It better explains why deciduous (hardwood) trees tend to take over the coniferous (pine fores

      • The only problem with that is that many of those broad-leaf trees don't have seeds that are leaf-shaped. The oak has the acorn, the walnut has the... walnut, and the same with chestnuts.

        Of course, there are those cool trees with the whirly-bird seeds. I love those!
    • The U.S. military have a longstanding interest in the dynamics of falling pieces of paper and did extensive research on the topic in arriving at the ideal dimensions for propaganda leaflets dropped from aircraft and leaflet bombs that would provide reasonable assurances that the leaflets hit the intended target.

      I served in an Army psychological warfare unit in Viet Nam that had produced and delivered, by 1970, enough leaflets to cover the entire country of South Viet Nam to a depth of more than 6 inches.

      • It's not surprising to see the Air Force funding further study on this subject.

        What is surprising is the reason the Air Force is so interested in this subject. Nothing related to aeronautical engineering or operational missions... They're looking to improve their staffing and paperwork flow. Because right now, throwing your Staff Summary package out the window at HQ appears to work better than walking the damn thing around directorate admin offices.

  • That's my prof! (Score:4, Interesting)

    by beefstu01 ( 520880 ) on Sunday October 31, 2004 @01:21PM (#10679815)
    Prof. Wang from TaM was my math teacher. Smart lady. She went crazy explaning the use of hyperbolic trig functions. At the time I had no idea what she was talking about, but now I see it actually has a use. Her other research is in the fields of insect flight. Looks like Calculus isn't useless after all.

    • Re:That's my prof! (Score:3, Insightful)

      by torako ( 532270 )
      It strikes me that you believe that calculus is useless in the first place... it's probably the single most useful field of mathematics: You need calculus to

      describe and find solutions to motion problems

      maximize or minimize functions (e.g. optimizing the cost of producing something)

      talk about any relationship between two variables that's close to zero

      straighten out complicated functions to handle them with computers (e.g. Taylor-series)

      I could go on and on about that.. Come on!

  • NewsFlash!! (Score:3, Funny)

    by thegoofeedude ( 771803 ) on Sunday October 31, 2004 @01:21PM (#10679817) Homepage
    Paper is affected by air as it falls! Astounding. ;-)

    • Re:NewsFlash!! (Score:4, Funny)

      by xsupergr0verx ( 758121 ) on Sunday October 31, 2004 @01:34PM (#10679904)
      The real news is that they actually found an interesting use for those "pages of a scientific journal."

      • Re:NewsFlash!! (Score:5, Funny)

        by Mark_in_Brazil ( 537925 ) on Sunday October 31, 2004 @06:11PM (#10681426)
        The real news is that they actually found an interesting use for those "pages of a scientific journal."
        There was a joke among physicists - I first heard it in the early 1990s- that said that a certain physics journal (I believe it was Physical Review Letters, or "PRL") was growing so quickly that its expansion was actually faster than the speed of light. There was, however, no violation of relativity, because no information was being conveyed.

        --Mark
    • More than that, air is affected by paper falling thru it. And around and around it goes...

  • Bah. (Score:5, Funny)

    by Black Parrot ( 19622 ) on Sunday October 31, 2004 @01:22PM (#10679827)


    This is just a rehash of an old study showing why open-faced peanut butter sandwiches always land face down.

  • Paper! (Score:5, Funny)

    by The-Bus ( 138060 ) on Sunday October 31, 2004 @01:23PM (#10679834)
    Has anyone combined this with other falling-object problems?

    For example, if one butters one side of the paper, will it still land face down, even if it's floating about?

    Since cats fall on their feet, what happens if you wrap playing cards on each of their legs? Will their happy flight downwards be interrupted by randomly flying limbs?

    What if you wrap the cat in a piece of paper that has been formed to make a Moebius strip, butter the other side of the animal, then tie it together to another cat? I suspect this may be the way to create time travel or a perpetual motion machine.

    I hereby ask everyone to funnel funds towards this dynamic Cat, Toast, and Paper Research. I approximate we have about 4 years to prepare to salute our new Paper Machie Strawberry Jelly Cat Overlords.

    • Re:Paper! (Score:5, Funny)

      by techno-vampire ( 666512 ) on Sunday October 31, 2004 @01:28PM (#10679860) Homepage
      What if you wrap the cat in a piece of paper that has been formed to make a Moebius strip, butter the other side of the animal, then tie it together to another cat? I suspect this may be the way to create time travel or a perpetual motion machine.

      It's probably been asked before, but this gave me an idea: take a long strip of bread, butter one side of it, twist it and connect the ends to make a mobius strip, then drop it. What happens?

    • Re:Paper! (Score:2, Funny)

      by xbytor ( 215790 )
      That should read "Strawberry Jelly Hypo-allergenic Cat Overlords". Funding committees are very picky about things like that.

    • ROTFL!

      This is the winner of the award "most interesting insightful funny joke on slashdot" ever.
    • For those who got lost, there, let me recap the parent's sources:

      "Has anyone combined this with other falling-object problems?"

      A reference to this story.

      "if one butters one side of the paper [...] Since cats fall on their feet [...]"

      A reference to the ever-so-often-handed around buttered cat story. It's been cited dozens of times, but I've never found a good attribution. The earliest reference I can find is here:

      http://w2s.co.uk/timo/jokes/joke1a.html [w2s.co.uk]

      The person claims to have been the author, and an

    • Re:Paper! (Score:2, Informative)

      by imaginate ( 305769 )
      I don't know the answers to your questions, but check this out:

      cat falling in zero g [216.40.242.213]

      It's a video of a cat on the "vomit comit". Most amusing. Get it before it's /.ed!

    • Re:Paper! (Score:5, Funny)

      by Jonas the Bold ( 701271 ) on Sunday October 31, 2004 @03:28PM (#10680494)
      What if you wrap the cat in a piece of paper that has been formed to make a Moebius strip, butter the other side of the animal, then tie it together to another cat? I suspect this may be the way to create time travel or a perpetual motion machine.

      You're theory is good, except that it fails to take into account the sheer impossiblity of attaching anything to a cat.
  • I'd thought that if the page weren't straight up and down, you'd get lift on the leading edge, causing that edge to rise up, slowing the fall until it gets high enough to stall. When I RTFA, it said that the page's rotation was enough to change the direction from clean drop to a sideways motion. Go know.

  • Usefulness (Score:5, Interesting)

    by ornil ( 33732 ) on Sunday October 31, 2004 @01:24PM (#10679837)
    The article says that the slowing-down effect for paper-like objects is much larger than normal "parachuting" effect. I wonder if this could be used in some way for parachutes.

    • Re:Usefulness (Score:3, Funny)

      by 26199 ( 577806 )

      Probably not, since it's unpredictable... which could translate to, say, random 30ft drops. Which would be rather unpleasant if you happened to be 29ft from the ground.

      You'd need a parachute to deploy when you got close to the ground ;-)

      • Re:Usefulness (Score:3, Interesting)

        by 3770 ( 560838 )
        You are thinking about the first possible application which would be to somehow attach yourself to one huge piece of flat material and somehow use that to slow your fall.

        But if you on the other hand had a parachute which somehow was made up of thousands or maybe millions of small pieces of flat objects which could rotate independently you might achieve the same effect, and the random falls would average out.

        It is also reasonable to believe that the smaller the object the smaller the random drops.

        My imagi

        • Re:Usefulness (Score:5, Interesting)

          by ajna ( 151852 ) on Sunday October 31, 2004 @02:30PM (#10680219) Homepage Journal
          if you on the other hand had a parachute which somehow was made up of thousands or maybe millions of small pieces of flat objects which could rotate independently

          Again your caveat about not fully understanding the issues involved after reading a single non-technical article applies, but I got the impression that the phenomenon requires rotational and translational motion to be decoupled. Thus rotating independently may well be insufficient to allow for the effect of falling slower than via "parachuting".

    • Re:Usefulness (Score:4, Insightful)

      by Aglassis ( 10161 ) on Sunday October 31, 2004 @01:42PM (#10679944)
      This effect isn't completely new (at least I don't think so). The space shuttles would roll and yaw back and forth a few degrees on reentry to slow down faster. If you ignore the horizontal speed of the spacecraft, this is somewhat similar to a piece of paper falling (but obviously more controlled--sometimes). Seems to me that the two items might be conceptually related. That being the case, I wouldn't be suprised if we saw a new style of atmospheric slowdown in future space probes.

    • Re:Usefulness (Score:3, Insightful)

      by PerpetualMotion ( 550623 )
      It's called a hand glider. Unfortunatly, it cannot be packed into a backpack and deployed after falling 1,000 feet out of a plane.

      Kites do not work well as parachutes.
    • Parachutes made of paper?

    • Re:Usefulness (Score:5, Funny)

      by JeanPaulBob ( 585149 ) on Sunday October 31, 2004 @01:46PM (#10679975)
      I wonder if this could be used in some way for parachutes.

      Step 1: Flatten self into a 1mm-thick sheet.

      Step 2... Uh, actually, we seem to be running into a problem at step 1.
    • Yeah, it'd be great. You'd just sort of slowly and gently touch down on the ground...

      Or about 20 feet from the ground you'd flip up and over, and slam into the ground at about 80 mph.

      It'd be the MS Windows of parachutes.

    • Re:Usefulness (Score:3, Informative)

      by renoX ( 11677 )
      > normal "parachuting" effect.

      But I suspect that what they call the normal "parachuting" effect is what occurs with round parachute, now modern parachute are wing-like so they are more efficient..
      Now I'm not sure because of the imprecise wording of the articles :-(
    • Small model rockets use a 'streamer' instead of a parachute. It is a piece of paper an inch or two wide and a couple feet long.

      • Re:Usefulness (Score:2, Informative)

        by aixou ( 756713 )
        The 'streamer' is there so you can see where the rocket landed even if you're far away, not to slow down the descent. Usually the streamers are made with a metallic color so they reflect light. This has actually helped me find my model rocket a couple times after I blasted it off and it landed a quarter mile away.

        To slow down the descent you still want a good ol fashioned parachute to come out.
        • Not true, many model rockets are brightly colored themselves and have a non-reflective streamer but no parachute. The streamer flaps around and reduces the terminal velocity of the rocket significantly (probably due to effects discussed in this article), so it doesn't break into a million pieces when it hits the ground. It doesn't slow the rocket as much as a parachute, which can be an advantage on windy days where a rocket hanging from a parachute could drift for miles. But it does slow the descent; if
    • The effect he's describing is actually commonly used with model rockets, that often have streamers that deploy in mid air rather than parachutes so that they don't get carried away on the wind.

    • Re:Usefulness (Score:3, Informative)

      by Wizzy Wig ( 618399 )
      The article says that the slowing-down effect for paper-like objects is much larger than normal "parachuting" effect. "

      We already have "flat parachutes," they're called air foils - and, yes, they do provide a slower descent than a penumbral parachute.

  • But .... (Score:4, Funny)

    by Anonymous Coward on Sunday October 31, 2004 @01:24PM (#10679840)
    by calculating the motions of a scientific journal page in flight

    ... they still need to repeat the experiment with different types of journals; psychology, home decorating, sports and paranormal to be absolutely sure.

  • They used a scientific journal page... (Score:4, Funny)

    by Anonymous Coward on Sunday October 31, 2004 @01:25PM (#10679847)
    ..and it was well behaved and obeyed the laws of physics. I want to see what happens when they repeat it with a bible page.
  • Paperweight - stop it going floating in the first place.

  • Comment removed (Score:5, Funny)

    by account_deleted ( 4530225 ) on Sunday October 31, 2004 @01:28PM (#10679862)
    Comment removed based on user account deletion

  • Navier Stokes Equation (Score:5, Informative)

    by Sipos ( 731917 ) on Sunday October 31, 2004 @01:31PM (#10679878)
    This seemingly simple problem like many other (more important problems like understanding air turbalance) is an exercise in solving the navier-stokes equation for a fixed set of boundary or initial conditions. The Navier-Stokes equation [wikipedia.org] is the equation that describes the flow of fluids on the large scale. It is a non-linear partial differential equation and is in some cases extremely difficuilt to solve (There is a $1,000,000 prize for the answer to the question: Do smooth initial conditions always lead to smooth solutions?). This may not seem very significant but it is probably very difficuilt to solve.
    • Do smooth initial conditions always lead to smooth solutions?

      Well from personal experience I know that if she has not waxed then there will be a major reluctance to be a smoothie on my part. So the answer is yes.

      Where's my million bucks?

    • Re:Navier Stokes Equation (Score:5, Funny)

      by Anonymous Coward on Sunday October 31, 2004 @02:48PM (#10680300)
      "There is a $1,000,000 prize for the answer to the question: Do smooth initial conditions always lead to smooth solutions?"

      Hmm. Tell you what... I'll submit a "yes" and you submit a "no" and whichever of us wins will split the money with the loser. Sound like a good deal? :)

  • rolloverrover (Score:5, Insightful)

    by Madcapjack ( 635982 ) on Sunday October 31, 2004 @01:31PM (#10679888)
    "Wang and Pesavento also showed that the falling-paper effect is almost twice as effective for slowing an object's descent, compared with the parachute effect (that is, if an object falls straight down)."

    This might be useful for future Rover missions (or, um Beagle missions). You'll lose accuracy, but at least you wouldn't hit the ground like a falling rock.

    • Re:rolloverrover (Score:2, Interesting)

      by slaad ( 589282 )
      This might be useful for future Rover missions (or, um Beagle missions). You'll lose accuracy, but at least you wouldn't hit the ground like a falling rock.

      Or maybe for falling capsules [nytimes.com]...(just in case someone plugs something in upside down)

  • Related Stories (Score:4, Funny)

    by SuperJason ( 726019 ) on Sunday October 31, 2004 @01:34PM (#10679899) Homepage
    Notice the "Related Stories" section. It is blank. This doesn't relate to anything. Does that tell you something?
  • Finally, a possible answer to why Toast ALWAYS falls butter side down in uncontrolled experiments !

    Of course, this still doesn't mean we can get a perpetual motion engine by strapping said toast to a cats back, but we can hope !

    I see a new form of energy just round the corner, CatToastOnics !
  • Sounds like they need to figure out the motions of falling big old heavy solid objects before they worry about this one.

  • Original pages... (Score:5, Informative)

    by argent ( 18001 ) <peter@slashdot.2 ... m ['ong' in gap]> on Sunday October 31, 2004 @01:43PM (#10679954) Homepage Journal
    And it's another physorg dead-end. Rather than mirror it or anything, a little googling will find the original material. Here's The original spam-free press release [cornell.edu] and Professor Wang's home page [cornell.edu] with a full citation for the paper.

  • A bit of clarification (Score:5, Informative)

    by Underholdning ( 758194 ) on Sunday October 31, 2004 @01:45PM (#10679960) Homepage Journal
    Plants with flattened seedpods also take advantage of the falling-paper effect.
    A specific example of this is the sycamore seed. As a matter of fact, landing a helicopter without motor assistance is called "the sycamore landing". It utilizes the exact same theory these phycisists has explained. So - It's not the theory that's new - it's the level of detail.

    • Re:A bit of clarification (Score:5, Informative)

      by polecat_redux ( 779887 ) <spamwichNO@SPAMgmail.com> on Sunday October 31, 2004 @01:54PM (#10680018)
      landing a helicopter without motor assistance is called "the sycamore landing".

      For those interested, I believe the maneuver is more commonly referred to as an autorotation [fact-index.com].

    • Re:A bit of clarification (Score:5, Insightful)

      by Kiryat Malachi ( 177258 ) on Sunday October 31, 2004 @02:39PM (#10680274) Journal
      Not quite... sycamore seeds fall with a twirling motion, much like a helicopter rotor. But autorotation (the proper name for that maneuver) doesn't rely on the physics they're talking about at all - heli blades are shaped such that the movement of air around the blades forces them to rotate, and the rotation generates lift. Not enough to keep the copter flying unpowered, but enough to prevent it from falling straight to the ground. The motion in the article is definitely not related to rigid airfoils with a fixed axis of rotation - the motion described in this article is that of a thin unconstrained flexible flat sheet.

      Some people have made comments about using tumbling motion to build better parachutes - it probably wouldn't work for a parachute because a parachute requires some attachment of the load to the sheet, and that attachment will prevent the tumbling motion from happening, both by preventing the tumbling and also by loading specific points on the sheet instead of having the load effectively equally distributed.

  • Now I Can... (Score:3, Funny)

    by brandonp ( 126 ) <brandon.petersen@nOspAM.gmail.com> on Sunday October 31, 2004 @01:52PM (#10680009) Homepage
    Sure a cure for cancer would be nice, but atleast I can use this to calculate how many of those leaves from my neighbor's damned tree are going to end up on my lawn.

    Maybe now I can bill him for raking...

    Brandon Petersen
    Get Firefox! [spreadfirefox.com]
  • Now they can make even better-looking animation movies!
  • It's amazing how many basic problems were left behind in the multibillion race to find the latest useless particle.

    It's not that understanding particle physics is not useful. What bugs me it that it was done at the cost of neglecting other equally important areas of physics.

    • Re:Classic problems (Score:4, Insightful)

      by NichG ( 62224 ) on Sunday October 31, 2004 @04:25PM (#10680817)
      It doesn't necessarily follow that this problem was solved now because of lack of funding. Rather, I'd say it's more likely that it means that the difficulty of the two problems is about equal.

      We still can't solve the three-body problem analytically (except for some special cases), and thats been around for 400 years. And its not for lack of trying.

      However, only within the last 50 years or so could we make approximations to the solution that work for long enough to be interesting and give insight into the problem. It's the availability of computers that makes it possible.

      Fluid dynamics is a hot topic in astrophysics right now (simulating stars, gravitational collapse of nebulae, accretion discs and jets around blackholes, ...), and there's a lot of consideration being given to tricks to solve Navier-Stokes (and other more complicated models that include the fluid being conducting or charged, or in some GR framework). So it's reasonable to expect that with new algorithms popping up, and refinements on the old ones, suddenly some intractable problems become accessible.

      So I don't think that this was a 'problem left behind', as much as a problem which is just now becoming solvable. (Part of) the reason we spend billions on particle physics and not on this sort of problem is that the minimal 'thing' to advance the science in particle physics costs billions, whereas nowadays one can run fairly large-scale simulations (of classical systems) on a $2000 laptop: the biggest cost for those problems is hiring students/postdocs/professors to work on them. So really there what funding enables is diversity in the problems being tackled (how many laptops can you afford? how many grad students?), rather than the speed at which any one particular problem is solved.

      Of course, this isn't true of some problems (quantum systems) which you really do need 1000 cutting edge systems all networked together to solve even a simple problem. In that case, you're going to have to be willing to throw a fair amount of money at the problem before you can see any progress.

  • They do fall straight down... (Score:4, Interesting)

    by Anonymous Coward on Sunday October 31, 2004 @02:48PM (#10680306)
    If you're on the moon, where there is little or no atmosphere, they will fall straight down. Has anyone seen the video of the feather falling straight down without fluttering around at all?
    • Has anyone seen the video of the feather falling straight down without fluttering around at all?

      I have actually seen the real thing. In connection with our faculty there is a small museum. Among other things they have two vacuum tubes that can be turned upside down. In one there is a feather in the other there is a stone. Interesting to see them fall at exactly the same speed.
  • Was it just me who got amused that Pesavento in Portuguese means Wind-weight?

    I wonder if Wang means somethign related. :)
  • I know bad joke but I'm sorry I just couldn't resist :-)

    Actually as silly as this paper seems it reminds me of Einstein's explanatation of Brownian Motion. Published the same year as his Special Relativity paper it was titled "On the Motion--Required by the Molecular Kinetic Theory of Heat--of Small Particles Suspended in a Stationary Liquid". I doubt this paper is as significant, but it may not be as trivial as it sounds when reported in laymans terms either.

  • I'm surprised at what surprised these guys... (Score:5, Interesting)

    by Anonymous Coward on Sunday October 31, 2004 @03:21PM (#10680459)
    "There were a few surprises," Wang notes. "We found the flat paper rises on its own as it falls, which would not happen if the force due to air is similar to that on an airfoil. Instead, the force depends strongly on the coupling between the rotating and translational motions of the object."

    Anyone who has ever thrown playing cards, frisbee, venetian blind bomerang (you have to be old enough to have had wooden venetian blinds as a kid) would not be surprised at the quoted 'surprise'.

  • Ignoble Awards 2004/5 (Score:3, Funny)

    by Roger W Moore ( 538166 ) on Sunday October 31, 2004 @05:07PM (#10681035) Journal
    I think we may have a winner....

  • Experiment with a Ruler (Score:4, Funny)

    by florescent_beige ( 608235 ) on Sunday October 31, 2004 @09:57PM (#10682455) Journal
    Take a regular 12" ruler preferably one of those wooden ones or stiff plastic. Hold it on the long edges between your thumb and middle finger (I mean, your thumb on the 6" mark and your index finger on the 15cm mark). Heave it into the air at about 45 degs (up not down), trying to give it some backspin.

    With any luck it will fly around a bit, swoopishly. The circulation caused by the back-spin generates lift, same as airfoil-shape induced circulation (faster airflow on top, slower on the bottom) as per that well known Kutta-Joukowski formula s * b * mu * gamma.

    Which is apropos of nothing. Also, the Navier-Stokes equations can't be solved around a singularity like the edge without a simplification which usually takes the form of an assumed boundary layer of some sort (probably laminar at these Reynolds numbers which makes it a lot easier). Also, N-S is initial-condition sensitive because the solutions have bad scale missmatch, so you'll want to use your duodecaduple precision math library.

    I didn't really understand from the blurb if they were talking about bendy things like paper pages. That would make it a fluid-structural coupled problem. Very tricky. The hardest part of that is getting the fluids guys to return the structures guys' phone calls.

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