BALLOONS — credits, sources and provenance ========================================== An independent browser reimplementation. Blowing up a balloon belongs to nobody; the TWO-BALLOON EXPERIMENT is a piece of published physics, credited below. No code, no asset and no data file in this app comes from any existing product. THE ORIGINAL WORK ----------------- * The problem was posed by J. S. Miller, "Pressure within a Bubble", American Journal of Physics 20(2), 115 (1952). DOI 10.1119/1.1933135. * It was first explained theoretically by DAVID R. MERRITT and FRED WEINHAUS, "The pressure curve for a rubber balloon", American Journal of Physics 46(10), 976-977, October 1978. DOI 10.1119/1.11486. (Several secondary sources, including Wikipedia, give the page range as 976-978; the scanned article's own page footers read 976 and 977.) * Companion paper: F. Weinhaus and W. Barker, "On the equilibrium states of interconnected bubbles or balloons", American Journal of Physics 46(10), 978-982 (1978). DOI 10.1119/1.11487. * Y. Levin and F. L. da Silveira, "Two rubber balloons: Phase diagram of air transfer", Physical Review E 69, 051108 (2004). DOI 10.1103/PhysRevE.69.051108. * W. Dreyer, I. Mueller and P. Strehlow, "A study of equilibria of interconnected balloons", Q. J. Mech. Appl. Math. 35(3), 419-440 (1982). DOI 10.1093/qjmam/35.3.419. Cited but NOT read: paywalled with no abstract available. WHAT THIS APP DOES DIFFERENTLY FROM MERRITT & WEINHAUS ------------------------------------------------------ Merritt & Weinhaus use the James-Guth network relation, P = (C / (r0^2 r))[1 - (r0/r)^6], which is the NEO-HOOKEAN curve: one fitted constant, a peak locked at r = 7^(1/6) r0, and no second ascending branch. They report no C1, no C2, no shear modulus and no wall thickness. This app uses the two-term MOONEY-RIVLIN membrane instead, so that the second ascending branch - which they measured and explicitly could not explain - exists in the model at all. The neo-Hookean curve is still in the app, as the C2 = 0 control. THE MEMBRANE LAW ---------------- p(lam) = (4 t0 / r0) (C1 + C2 lam^2) (lam^-1 - lam^-7), lam = r / r0 DOCUMENTED. This is Eq. (28) of E. Verron and G. Marckmann, "Numerical analysis of rubber balloons", Thin-Walled Structures 41(8), 731-746 (2003), rewritten with C = C1 and aC = C2. The same expression appears in Mangan & Destrade, arXiv:2009.08752, Eqs. (2.14)/(3.3). MATERIAL CONSTANTS ------------------ C1 = 26 760 Pa, C2 = 1 731 Pa (C2/C1 = 0.06469) DOCUMENTED, qualified. R. Mangan and M. Destrade, "Gent models for the inflation of spherical balloons", arXiv:2009.08752, Table 1 - their Mooney fit to Merritt & Weinhaus's own measured curve. Their fitted exponent was n = 1.968, not exactly 2, so this app's classical n = 2 Mooney model is a near neighbour of their model rather than their model. r0 = 1.9173 cm DOCUMENTED (Mangan & Destrade, section 4). t0 = 0.50 mm QUALIFIED - an explicit ASSUMPTION in the source, not a measurement: "Because the exact thickness of the balloon used in the experiment is not known, we take it to be that of a typical rubber balloon, 0.5 mm." Only the product (t0/r0)*C1 is pinned by a pressure curve, so C1 and t0 trade off one for one and must be quoted together. The one study that MEASURED a party-balloon wall found 0.114-0.183 mm, average 0.140 mm (Lat. Am. J. Solids Struct., scielo.br/j/lajss/a/3mx4LPcdntsyFpFcvPgzxTM). That is 3.5x thinner. The disagreement is recorded here rather than resolved. THE MEASURED CURVE PLOTTED ON THE PAGE -------------------------------------- 14 points, MEASURED and published: A. Anssari-Benam, A. Bucchi and G. Saccomandi, J. Elasticity 151, 15-45 (2022), Table 6 - Merritt & Weinhaus's Fig. 1 digitised. DOI 10.1007/s10659-021-09823-x. Peak 1.955 kPa at lam 1.44; trough 1.34 kPa at lam 2.96; still rising again at lam 3.61. Independent confirmation of the peak location: J. Vandermarliere, "On the inflation of a rubber balloon", The Physics Teacher 54(9), 566-567 (2016), DOI 10.1119/1.4967901 - a smartphone barometer sealed inside a balloon reads a peak at R/R0 about 1.44. HOW WELL THE SHIPPED MODEL FITS THAT DATA - stated because it is not perfect --------------------------------------------------------------------------- RMS 0.0982 kPa over the 14 points, R^2 0.954, worst 0.1476 kPa at lam 2.27. The peak is reproduced to 0.16% in pressure and 0.12% in stretch. The TROUGH IS NOT: the model puts it at lam 3.925 where the data puts it at 2.96, 32.6% too far out. Refitting C1 and C2 to the same data by exact normal equations (the pressure is linear in both) gives C1 = 26 179 Pa, C2 = 1 512 Pa, RMS 0.0625 kPa, R^2 0.981 - and still a trough at 4.156. The misplaced trough is a limitation of the two-term Mooney model, not of the published fit. A Gent-Gent model fits the data properly; this app does not ship one. THE CRITICAL MOONEY RATIO ------------------------- DOCUMENTED. Verron & Marckmann section 4.1: "a_C is a critical value of the material parameter approximately equal to 0.214 ... for a = 0.25, the pressure always increases as the balloon expands and no limit point exists." Mangan & Destrade Eq. (3.8), recovering Goriely et al., print lam_cr = (19 + 6 sqrt 11)^(1/6) and (C2/C1)_cr = (2 sqrt 11 - 3) / [5 (19 + 6 sqrt 11)^(1/3)]. This app derives alpha_cr = 3 w^(2/3) / (4w + 5) with w = 19 + 6 sqrt 11 from the double root of its own extremum quartic; the two expressions agree to 5.6e-17. RECONSTRUCTED - not measurements of anything -------------------------------------------- * RECONSTRUCTED - the BURST STRETCH, lam = 5.3. No peer-reviewed measurement of burst stretch or burst pressure for a party balloon was found. 5.3 is the limiting stretch implied by Mangan & Destrade's Gent fit (J_m = 53.33) to the same balloon, which is a model asymptote, not a burst. In the game each balloon's burst stretch is drawn at random from 4.70 to 5.80 around it (RECONSTRUCTED as well). Figures circulating on the open web (10-12 psi, 4.5-7x stretch) are unsourced and are roughly 50x above the measured curve at lam 3.6; they are not used here. * RECONSTRUCTED - THE PUMP: its ceiling (2.6x the peak pressure), its 0.60 s rise and 0.90 s bleed-down time constants, the one-way valve and its conductance, the five rounds, the target band, the hold time and the scoring. All invented to make a playable game. NOT MODELLED ------------ Hysteresis - real rubber follows a lower pressure curve on deflation than on inflation, so a real pair of connected balloons equalises with a smaller change in diameter than shown here. Also ignored: crystallisation, rate and temperature dependence, the neck and knot, the balloon's departure from a sphere, and any flow resistance other than a linear conductance. A PUBLISHED SENTENCE THIS APP CONTRADICTS ----------------------------------------- Wikipedia's "Two-balloon experiment" states: "Equilibria in which both balloons are on the right of the pressure peak also exist but are unstable." Unqualified, that is false for a real balloon. It follows from the James-Guth model used in that article, which has no second ascending branch; the same article records that real rubber has one, and the measured data show it. Past the TROUGH, dp/dV is positive again, so two equal balloons are a stable equilibrium - at lam 4.5 the measured growth rate of a perturbation is negative. The sentence is correct only between the peak and the trough. Reported as a disagreement, not as an error in the underlying physics. SOFTWARE -------- Everything is written for this app: the engine, the canvas renderer, the seeded generator (mulberry32), the closed-form cubic and quartic solvers, the charts. No third-party library, no CDN, no external font, no network request after load, no model call, no credits, no account. Licence: LICENSE.txt (MIT).