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Parameters
Model checks
The science
Forces on a bubble
A bubble of volume V feels buoyancy (ρ - ρg) V g upward and drag Fd = ½ Cd ρ A u2 downward. Accelerating it also drags fluid along, an added mass of about ½ ρ V, so (ρg + ½ρ) V du/dt = (ρ - ρg) V g - Fd. For small Reynolds numbers a contaminated bubble behaves like a rigid sphere and reaches Stokes' terminal speed u = 2 r2 ρ g / (9 μ); at intermediate Re the Schiller-Naumann drag Cd = 24/Re (1 + 0.15 Re0.687) applies; large wobbling bubbles rise at about √(2σ/(ρ d) + g d / 2) (Mendelson). The page uses the larger of the first two and the Mendelson value.
Expansion with depth
The pressure is p = patm + ρ g z. At constant temperature the gas obeys Boyle's law, p V = constant, so the radius grows as r = r0 (p0/p)1/3 and the bubble speeds up as it rises. The water density, viscosity (Vogel) and surface tension change with temperature.
Honest simplifications: spherical shape at all sizes, a clean-bubble drag modelled only as two thirds of the rigid-sphere drag, no bubble-bubble interaction or wakes, no gas dissolution, and a constant added-mass coefficient. Real clean bubbles in the 1 mm range rise considerably faster than shown.
What to try
- Choose Tiny 50 micron and see the Stokes regime: it takes minutes to surface. Then Aquarium 1 mm and Wobbling 3 mm.
- Raise the water temperature: the viscosity drops, small bubbles speed up while large ones barely change.
- Release from 10 m and watch the bubbles swell as the pressure falls from about 2 bar to 1 bar.
- Switch to Clean water: the model drag drops and the speed rises.