Key examples — surface tension and capillarity

Where the chapter’s machinery shows up across the bookshelf.

Example 1: Young-Laplace term in Rayleigh-Plesset

For an oscillating bubble of radius $R$ in liquid, the boundary condition at the wall is

The $2\sigma/R$ term is exactly the chapter’s Young-Laplace pressure jump. For small bubbles it dominates: a 1 μm bubble in water has a static $2\sigma/R$ of $\approx 1.4\,\text{atm}$ — comparable to atmospheric pressure itself. See Cavitation Ch 3.1.

Example 2: alveolar surfactant and the breath

A typical alveolus has radius $\sim 100\,\mu\text{m}$. Without surfactant, the Young-Laplace pressure across its air-liquid interface at $\sigma_\text{water} = 72\,\text{mN/m}$ would be $\Delta p = 2\sigma/R \approx 1.4\,\text{kPa}$. Pulmonary surfactant drops $\sigma$ to $\sim 5\,\text{mN/m}$ at small alveolar radii, reducing the pressure jump to $\approx 100\,\text{Pa}$ — manageable by ordinary breathing muscles. The molecular biology of surfactant production is one of the more elegant cases of physics-driven biological design.

Example 3: crevice nucleation in cavitation

A V-shaped crevice on a solid surface with half-angle $\beta$ stably traps a gas pocket when $\theta_c + \beta > 90°$. Hydrophobic crevices ($\theta_c \to 180°$) trivially trap gas at any opening angle; mildly-hydrophilic surfaces require sharper geometry. The size distribution of real crevices on engineering surfaces typically peaks in the range of 1–100 μm — exactly the right scale for cavitation nucleation at MPa tensions. See Cavitation Ch 2.2.

Example 4: cohesion-driven tensile strength of capillary water

Water inside a sufficiently fine capillary can sustain large negative pressures — this is how trees lift water tens of metres without pumping. The driving force is the capillary depression at the curved meniscus at the leaf-cell wall, where the radius of curvature is $\sim 10\,\text{nm}$ and the Laplace tension is $\sim 14\,\text{MPa}$. Trees exploit Jurin’s law on a heroic scale; the laboratory analogue is the Cavitation book’s tensile-strength-in-principle chapter.

Example 5: dimensionless numbers and droplet break-up

The Weber number $\mathrm{We} = \rho U^2 L/\sigma$ controls when a fluid jet or droplet breaks up. A water droplet falling through air starts to deform at $\mathrm{We} \sim 1$ and breaks up at $\mathrm{We} \sim 12$. For a 5 mm raindrop at terminal velocity $\sim 9\,\text{m/s}$, $\mathrm{We} \approx 6$ — close to the deformation regime but below break-up. Hailstones experience much higher $\mathrm{We}$ during formation, which is why large hailstones tend to be ragged rather than spherical. See scaling chapter.

Cross-book backlinks

• Cavitation Ch 3.1 — Rayleigh-Plesset derivation: Young-Laplace term in the boundary condition.
• Cavitation Ch 2.1 — homogeneous nucleation: surface energy ⅔ contribution to barrier.
• Cavitation Ch 2.2 — heterogeneous & crevices: contact-angle-driven trapping.
• Cavitation Ch 1.2 — tensile strength in principle: cohesion limit.
• Physics free-energy — chapter: the surface-area free-energy term.