Bubbles (2D side view)

Gas bubbles in a liquid

Buoyancy against drag. A bubble of radius R is pushed up by rho_l g V and slowed by drag. With the added mass of the liquid (rho_l V / 2) the equation of motion is dv/dt = 2g - (3 C_D / 4R) v^2, so the terminal speed obeys v^2 C_D(Re) = 8 g R / 3, with Re = rho_l d v / mu. The page solves this by bisection for 180 diameters and interpolates.

• Clean bubble (mobile surface, pure water): C_D = (16/Re) [1 + 1 / (8/Re + 0.5 (1 + 3.315 / sqrt(Re)))] (Mei, Klausner and Lawrence 1994). At small Re it tends to 16/Re, which gives the Hadamard-Rybczynski speed, exactly 1.5 times the rigid-sphere value.
• Contaminated bubble (surfactant makes the surface rigid): Schiller-Naumann C_D = 24/Re (1 + 0.15 Re^0.687), with the Stokes law v = g d^2 / (18 nu) at small Re.
• Large bubbles are limited by interface waves, v = sqrt(2 sigma / (rho d) + g d / 2) (Mendelson 1967). The two regimes are joined with a smooth minimum, so the speed peaks near 2 mm (about 29 cm/s) and then falls to about 23 cm/s near 5 mm: the famous maximum of the curve in the first graph.
• Shape. The aspect ratio 1 / (1 + 0.163 Eo^0.757) with Eo = rho g d^2 / sigma squashes large bubbles into oblate ellipsoids. Clean bubbles above about 1.3 mm also zigzag (drawn as a spiral at a Strouhal number of about 0.12; the path is illustrative).

Birth, growth and pressure. A bubble leaving a pore of radius r_o has the Tate radius (3 r_o sigma / 2 rho g)^(1/3). Gas inside follows the isothermal ideal-gas law with the Laplace pressure, (p + 2 sigma / R) R^3 = K, where p = p_above + rho g (depth); K is solved for R with Newton iterations, so a bubble that rises 25 cm in water grows only a few per cent, but in a vacuum chamber it swells visibly. Dissolved gas adds to K by diffusion: dK/dt = 3 R_gas T R D Sh (c_inf - c_s) / (2 M), with the Ranz-Marshall Sherwood number Sh = 2 + 0.6 Re^0.5 Sc^(1/3). That is why champagne bubbles (CO2, supersaturation s) swell as they rise, while air bubbles hardly change.

Coalescence. Touching bubbles merge with a probability that depends on the liquid (high in pure water, very low in soapy or salty water, where drainage of the thin film between them is slowed). A follower in the wake of a leader is sped up by up to 45 % (wake drafting), which is what drives merging in clean water. At the top, bubbles form a raft whose lifetime depends on the liquid.

Soap bubbles in air

Laplace pressure. A soap film has two surfaces, so the excess pressure is Delta p = 4 gamma / R (a gas bubble in a liquid has one surface: 2 sigma / R). It follows from the energy E = 2 gamma 4 pi R^2: Delta p = dE/dV. Smaller bubbles have the higher pressure, which is why a small bubble connected to a big one empties into it.

Why a bubble falls (or floats). Net weight = film mass rho_w 4 pi R^2 h + inside air - displaced air. Warm breath is lighter than room air, so a fresh bubble can hover or rise until its air cools (time constant about 4 s per 2 cm of radius). Drag uses the sphere law above with the added mass of air; the wind is a uniform breeze plus a sum of sines for gusts.

Colour = thin-film interference. Light reflected from the front and back of the film interferes. For index n = 1.33 and thickness h, the reflectance (exact two-beam sum, averaged over polarisations) is R = F sin^2(beta) / (1 + F sin^2(beta)), with beta = 2 pi n h cos(theta_t) / lambda and F = 4 r^2 / (1 - r^2)^2. A film around 100 nm reflects best at lambda = 4 n h = 532 nm (green); below about 20 nm it is a "black film" and soon bursts. The spectrum is turned into colour with CIE colour-matching functions (Wyman et al. fit) and a 6500 K white light, stored in a thickness x angle table used by the shader. The bubble then reflects a procedural room (window and sky), brightened so the faint 7 % reflection is visible. The thickness varies over the surface: gravity drains the top (thin) toward the bottom (thick), and swirls grow with age.

Lifetime. Drainage dh/dt = -h / tau_d - evaporation (1 - RH) is a phenomenological model, not a derivation: real drainage depends on surfactant, viscosity and marginal regeneration. The bubble pops when the top of the film is thinner than a random threshold of 8 to 38 nm. When it pops, the hole opens at the Taylor-Culick speed sqrt(2 gamma / (rho h)) (about 10 m/s); the page shows it 60 times slower.

What is simplified

• No bubble-induced liquid flow, no hydrodynamic bubble-bubble forces beyond wake drafting, no mass loss from the tank liquid.
• The aquarium and flute are rigid containers; the foam raft is a single layer.
• Film thickness evolves by a lumped law; the colours are physically computed for that thickness but the pattern of swirls is procedural noise.
• Reflected radiance comes from a simple analytic room, not a full ray trace.

In-page numerical checks

Five independent tests run at start-up: (1) the scanned film reflection peak against 4 n h and the black-film limit; (2) the numerical bubble speed against Stokes' law and the 3/2 Hadamard-Rybczynski ratio; (3) the Laplace pressure from a finite-difference energy derivative; (4) the time integrator reaching the tabulated terminal speed and the isothermal growth law; (5) the total-internal-reflection angle asin(1/n) found by scanning, and the Tate radius.