Gravitycapillary

Gravitycapillary refers to the physical regime in fluid dynamics where gravitational and capillary (surface-tension) forces are both important in shaping the behavior of liquid interfaces, such as the surface of a body of water or a liquid film. In this regime, wave motion and interfacial phenomena are governed by a balance between gravity, which tends to flatten large-scale disturbances, and surface tension, which resists curvature and dominates at small scales.

For a linear, inviscid, incompressible fluid with density ρ, surface tension σ, and gravity g, the dispersion relation for small-amplitude waves on deep water is well described by ω^2 = gk + (σ/ρ)k^3, where ω is angular frequency and k is wavenumber. Finite depth introduces a tanh(kh) factor. The gravity-capillary crossover occurs around a wavenumber kc = sqrt(ρg/σ) and a corresponding wavelength λc = 2π/kc = 2π sqrt(σ/(ρg)). For ordinary water (σ ≈ 0.072 N/m, ρ ≈ 1000 kg/m3, g ≈ 9.81 m/s2), λc is about 1.7 cm. Wavelengths longer than λc are gravity-dominated, while shorter wavelengths are dominated by capillarity. The phase and group velocities follow v_p = ω/k = sqrt((g/k) + (σ/ρ)k) and their behavior reflects the gradual transition between the two forces.

Applications and implications of gravity-capillary interactions appear in ocean waves, ripples and spraying, microfluidics, inkjet printing, and surface pattern formation. The concept has historical roots in early studies of capillary waves and remains a central framework for analyzing interfacial dynamics where both gravity and surface tension are present.