YoungLaplace

The Young-Laplace equation, often written as ΔP = γ(1/R1 + 1/R2), relates the pressure difference ΔP across a curved fluid interface to the surface tension γ and the interface curvature characterized by the principal radii R1 and R2. It is named for Thomas Young, who introduced the concept of surface tension in liquids, and Pierre-Sigismond Laplace, who extended the analysis to curved interfaces in capillarity problems.

In the spherical case, where the two principal radii are equal (R1 = R2 = R), the equation simplifies to ΔP = 2γ/R. The equation can be derived from a balance of forces on a small element of the interface or from an energy perspective that views surface tension as the energy cost of increasing surface area for a fixed volume.

Applications of the Young-Laplace equation span a range of phenomena, including capillary rise of liquids in tubes, the stability and pressure of droplets and gas bubbles, and the internal pressure difference across bubbles. It also plays a role in biological contexts, such as the mechanics of alveoli in the lungs, where surfactants alter the effective surface tension.

Limitations include the assumption of static equilibrium and constant surface tension, with neglect of gravitational effects for small interfaces. Dynamic situations require fluid dynamics with interfacial forces. Generalizations address anisotropic surface tensions, membranes, or varying γ, extending the basic relation to more complex interfacial systems.