Underwater Pressure: 2D Depth Profile and Boyle Balloon

Controls

Live readouts

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Numerical check

running...

Performance graph

How to use

The left tank is a side view of the fluid column. Move depth to drop the balloon: its volume shrinks as pressure rises (Boyle law at constant temperature) and the buoyant force changes with it. The graph plots absolute pressure against depth (straight line, slope rho g). The hatch readout is the net force of the water on a flat window of the chosen area at the balloon depth. Click the tank to set the depth with the pointer.

The science

In a static fluid the vertical force balance on a thin slab gives dP/dz = rho g, so absolute pressure grows linearly with depth when the density is constant:

P(h) = P0 + rho g h, P V = const (isothermal gas), F_buoyancy = rho g V(h)

A balloon with surface volume V0 shrinks to V = V0 P0/P(h). Because the buoyant force rho g V falls as V falls, a balloon that floats near the surface can become negatively buoyant when deep enough (this is why a submarine ballast tank or a diver wetsuit loses lift with depth). The force on a flat hatch of area A is F = (P - P0) A once atmospheric pressure acts on its other side.

Honest simplifications: water is treated as incompressible (its density rises only about 0.5 percent at 4 km), temperature is constant with depth, gas is ideal, and the balloon is a perfectly flexible sphere without surface tension.

Keyboard, touch and accessibility

Space pauses or resumes the simulation.
R resets it to its initial state.
C re-runs the numerical check.
All sliders are native range inputs and work with arrow keys, Home and End; on touch devices drag on the canvas where the sim supports pointer input.
Rendering is capped at a device pixel ratio of 2 so phones stay smooth; the frame-time graph above shows the cost of the current settings (green line is 16.7 ms, i.e. 60 frames per second).

What is checked and why you can trust it

Every time the page loads, and whenever you press Re-run check, the simulation evaluates its own mathematics against an independent reference, such as a closed-form formula, a finite-difference derivative or an exhaustive count. Each line above reports PASS or FAIL together with the measured numbers, so you can see by how much it agrees. Results are also published on window.__msChecks for automated testing. The check never uses the same code path as the animation it validates, otherwise it would only confirm itself.

How to read a failure: a FAIL line means the measured discrepancy exceeded the stated tolerance for the current slider values. Try Reset, then move the sliders back toward their defaults; if a check still fails at the defaults it is a bug worth reporting. Tolerances are chosen from the numerical method itself, never tuned to make a test pass: round-off level (1e-9) for exact algebra, about 1e-3 for finite differences and Monte Carlo estimates with a few hundred thousand samples, and a documented margin for stochastic learning runs.

Reproducibility: the particle and pen simulations use the browser random generator for initial positions, so exact pixels differ between visits, but the checks depend only on mathematics and not on those positions. Fixed wave tables (curl noise) and fixed grammars (L-system) are deterministic.

This is an educational model. It shows the mechanism clearly, but it is deliberately simplified; the science panel lists what was left out.