Lee, Thanks. This is absolutely stunning. I am forwarding this to FRIAM as we speak and will reply in greater detail later.
Nick Nicholas S. Thompson Emeritus Professor of Psychology and Biology Clark University http://home.earthlink.net/~nickthompson/naturaldesigns/ -----Original Message----- From: [email protected] [mailto:[email protected]] Sent: Sunday, December 06, 2015 7:34 PM To: Nick Thompson <[email protected]> Subject: Re: [FRIAM] bubbles > Ok, so the grandkids are messing about with bubbles. When two bubbles > of equal size conjoin, the "membrane" between them appears to be a flat circle. > How general is this, we asked? So what if the conjoined bubbles are of > unequal size. Our experiments seemed to suggest that the answer was, > "No!", and that the smaller bubble bulged into the larger one. Why > would that be? Am I correct that a bubble will expand (if it can do > so without > breaking) until the pressure inside equals the pressure outside? No, I think you're not correct there (though the statement is very attractive). There are THREE forces acting on any small region of a bubble (be it simple or compound): the force exerted on the region by the pressure of the gas inside the bubble, the force exerted on the region by the pressure of the gas outside the bubble, AND the force exerted on the region by the elastic properties of the material forming the bubble itself. This latter is proportional to (a particular numerical measure of) the curvature of the bubble; the measure in question is (I think) what's called the "Gaussian curvature", and has an interesting property (classically phrased in terms of "curvatura integra"): as the bubble changes its shape (say, as a balloon is inflated, or a soap bubble detached from the bubble- blowing frame adjusts itself to the fact that the soap is accumulating at the bottom of the bubble because of gravity), without (of course) adding or removing any material, THE TOTAL CURVATURE REMAINS CONSTANT. If, therefore, the AREA OVER WHICH THIS CONSTANT CURVATURE IS DISTRIBUTED IS CHANGING (because the pressure either inside or outside the balloon is being increased by inflation/deflation or temperature change), then THE CURVATURE PER UNIT AREA changes inversely to that change. Now, when the balloon/bubble is in equilibrium, the three forces must sum (vectorially) to 0. Therefore the pressure inside is NEVER equal to the pressure outside, unless the contribution from the curvature is 0: there always has to be MORE PRESSURE INSIDE to keep the bubble inflated! Further analysis (with more idealization) shows that a surface of constant curvature has very constrained geometry. Essentially (for closed bubbles; NOT for fancy soap films on frames) it has to be spherical; the curvature of a sphere of radius R is proportional (by a universal constant) to 1/R. That means that a "sphere of curvature 0" has radius 1/0; with some magical thinking you can convince yourself that such a sphere is actually A FLAT PLANE (and without any magic you can calculate that the curvature of a plane is, indeed, 0). Now consider a compound bubble. It should made out of spherical pieces, right? (Wave your hands at this point). In case the bubble is compounded of two equal spheres, it will have three pieces, two congruent parts of the original spheres, with a planar face between them (which is a disk); the case you first observed. The pressure inside each part will be the same, and larger than the pressure outside; across the planar disk, the pressure differential is 0. If the bubble is compounded of UNequal spheres, then the three pressure differentials (between the part interior to the remains of Original Sphere 1 and the outside, between the part interior to the remains of Original Sphere 2 and the outside, and between the two remains) will all be different, so the interior wall between the two remains will be a piece of a THIRD (non-original) sphere. BEST OF ALL, the dihedral angles anywhere along the triple boundary circles are all 120 degrees (I think). I can never figure out how to send this to FRIAM without it bouncing, so it's just going to you. Forward ad lib. Look up "Plateau's problem" in Wikipedia, too. (The "double bubble theorem", done rigorously, is actually very hard, and was only quite recently proved, by a professor at Williams College and some of his genuinely talented undergraduates. But the account I've given is pretty much what everybody has believed for a long time, and it's truthy enough for you and your grandkids, I think.) ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College to unsubscribe http://redfish.com/mailman/listinfo/friam_redfish.com
