Young–Laplace equation

The Young–Laplace equation relates the pressure difference across a curved interface to surface tension and the sum of the interface’s principal curvatures.

Mean curvature: A measure of a surface’s local bending, defined as the average of its principal curvatures. The equation’s pressure jump is proportional to the sum of principal curvatures, twice the mean curvature.
Capillary action: The rise or fall of a liquid in a narrow space due to interfacial forces and wetting. Curvature in a capillary meniscus sets the pressure difference driving liquid motion.
Young's equation: The relation between contact angle and interfacial tensions at a solid–liquid–gas contact line. It complements the pressure-curvature relation by describing equilibrium at a three-phase contact line.
Young–Laplace equation for a spherical interface: The spherical-interface form of the pressure-curvature relation, with equal principal curvatures. It reduces the general equation to a pressure jump proportional to twice the inverse radius.
Principal curvature: The maximum or minimum normal curvature of a surface at a point, measured along perpendicular directions. These two directional curvatures supply the equation’s local geometric terms.
Soap bubble: A thin liquid film enclosing gas, with two curved interfaces separating gas from air. Its two interfaces produce twice the pressure jump of a single spherical interface.
Laplace–Beltrami operator: A differential operator that generalizes the Laplacian to curved spaces and surfaces. For a surface, its action on position encodes mean curvature used in capillary shape equations.
Tolman length: A length describing curvature-dependent changes in surface tension for small interfaces. It marks a limitation of treating surface tension as constant at molecular-scale curvature.
Surface tension: The force per unit length, or equivalently energy per unit area, associated with an interface. It sets the proportionality between interface curvature and pressure difference.
Alveolus: A small air sac in the lung where oxygen and carbon dioxide are exchanged. Alveolar curvature and surfactant affect the pressure needed to keep air sacs open.

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