Cartesian Diver (3D)

What happens in the bottle

The diver is a small glass tube with an air bubble trapped under its closed end and open at the bottom. Its density is only slightly below that of water, so it floats. Squeezing the bottle raises the pressure of the closed air space at the top, and water transmits that pressure unchanged (Pascal's law) to the diver. The trapped air is compressed and water enters the tube.

Boyle's law

At constant temperature the air obeys p V = n R T = const, so V_a = p0 V_a0 (T / T0) / p with p = p0 + dp + rho g h, where dp is the squeeze and h the depth of the diver.

Why the diver sinks

The diver (glass plus ballast) has mass m and the glass occupies V_g. Counting the water inside the tube as part of the load, the net downward force is F = g [ m - rho (V_g + V_a) ]. The diver is neutral when V_a* = m / rho - V_g. Compressing the air below V_a* makes F positive and the diver sinks.

An unstable balance

Deeper water has higher pressure, so a sinking diver is compressed more and sinks faster: the neutral depth h* = (n R T / V_a* - p0 - dp) / (rho g) is an unstable balance point. That is why a real diver either shoots to the bottom or floats up, and never hovers on its own. Letting go of the squeeze lowers dp, moves h* below the diver and it rises again.

Critical squeeze and temperature

From the surface the diver sinks once dp > dp_c = p0 (V_a0 (T/T0) / V_a* - 1). Warm water raises V_a at the same pressure, so a warm bottle needs a harder squeeze; a cold bottle needs a lighter one.

Motion

The model integrates m_eff dv/dt = F - (6 pi mu r + 0.5 rho Cd A |v|) v with a fourth-order Runge-Kutta step. The effective mass includes the water inside the tube and an added mass of 30 % of the displaced fluid. The squeeze reaches the diver with a short first-order delay (default 0.12 s).

Simplifications

The bottle is rigid except for the squeeze, the air in the tube is treated as an ideal isothermal gas, the tube shape is a vertical cylinder with a fixed 1.4 ml cavity, the sloshing of water and the tilt of the diver are not modelled, and the diver is held at the surface when it floats.

1. Pick Classic bottle, press Squeeze (30 kPa) and watch the air volume graph: the point slides left along the Boyle isotherm and the diver dives.
2. Release. The neutral depth in the readouts moves far below the bottom and the diver rises. Open the force-versus-depth graph to see the single unstable zero crossing.
3. Add ballast with the slider until the squeeze needed to sink is only a few kPa (the right-hand graph shows it directly), then hold the squeeze at that value: the diver barely dips.
4. Switch to Warm water and compare the squeeze needed to sink with the cold bottle.
5. Pick Thick glycerol and watch the viscous drag cap the diving speed.