# Balloons

> A browser app that inflates a rubber balloon and runs the two-balloon experiment, with the
> Mooney-Rivlin thin-shell pressure curve solved in closed form. Entirely client-side: no
> account, no model call, no credits, no network request after load.

URL: https://balloons.skillsafe.ai/

## What it is

Two things in one page. A game: hold a pump, inflate a balloon to a target stretch without
bursting it, five balloons a game. And a laboratory: connect two balloons through a tap and
watch which way the air actually goes.

## The physics

A thin incompressible Mooney-Rivlin spherical membrane of unstretched radius r0 and wall t0
holds a gauge pressure

    p(lam) = (4 t0 / r0) (C1 + C2 lam^2) (lam^-1 - lam^-7),   lam = r / r0

which is NOT monotonic: it rises, peaks, falls to a trough, and rises again. Everything else
follows from the shape of that one curve.

Shipped balloon: C1 = 26.76 kPa, C2 = 1731 Pa, r0 = 1.9173 cm, t0 = 0.50 mm (an assumption in
the source, not a measurement). Peak 1.952 kPa at lam 1.4383; trough 1.420 kPa at lam 3.9254.

## The headline

"The bigger balloon pushes harder, so air flows from big to small" is wrong about the middle
clause. Air always flows from higher pressure to lower; SIZE DECIDES NOTHING. Over 4 830
sampled pairs air went big-to-small 1 714 times and small-to-big 3 116 times.

Two exact statements bound when the naive rule IS right:

1. For a small balloon at lam_s, the rule fails on exactly one open interval of lam_b - the
   interval between the other two stretches carrying the same pressure. Outside it, including
   for balloons far bigger, the bigger balloon really does push harder.
2. Below lam-dagger = 1.1353 (a radius of 2.18 cm for this balloon) the rule NEVER fails. If
   the small balloon's pressure is under the trough pressure, the curve never comes back down
   to meet it. Two limp balloons obey the schoolroom rule exactly; the demonstration needs one
   of them properly inflated.

And one about the rubber rather than the balloons: if C2/C1 exceeds 0.214458 the curve has no
peak at all and the effect cannot happen. That critical ratio is (19 + 6 sqrt 11)^(1/6) in
stretch, derived here from the double root of the extremum quartic and agreeing with the
published closed form to 5.6e-17.

## Other results on the page

- Stability of a connected pair is decided by the sign of dp1/dV1 + dp2/dV2. Bisecting the
  MEASURED growth rate of a perturbation finds the pressure peak and trough to 6.3e-8 and
  2.0e-6, without ever evaluating dp/dlam.
- Wikipedia's unqualified "equilibria in which both balloons are on the right of the pressure
  peak also exist but are unstable" is false past the trough, where two equal balloons are
  stable. The app measures the growth rate and reports the sign.
- A single balloon inflated by ADDING AIR never snaps, at any ambient pressure, because p*V is
  a sum of strictly increasing terms. Compressibility never flips a stability verdict in
  5 594 equal-pressure pairs; it only slows the response, by at most 5.08%.
- The peak located from pressure VALUES stalls at 3.3e-7 because the curve is quadratic there;
  from the numerically differentiated pressure it reaches 5.7e-12.

## Honesty

The two-term Mooney model reproduces the measured peak to 0.16% but puts the trough at lam
3.93 where the measured data put it at 2.96. Hysteresis is not modelled. The burst stretch is
reconstructed, not measured. Full provenance: /CREDITS.txt.

## Credit

Problem posed by J. S. Miller, Am. J. Phys. 20(2), 115 (1952); explained by D. R. Merritt and
F. Weinhaus, "The pressure curve for a rubber balloon", Am. J. Phys. 46(10), 976-977 (1978).
An independent reimplementation; no code or asset from any existing product.

## Files

- /CREDITS.txt  provenance for every constant, and what differs from the original
- /LICENSE.txt  MIT