RayleighPlesset

The Rayleigh-Plesset equation describes the radial dynamics of a spherical gas bubble in a liquid. It is a foundational model in cavitation, sonolysis, and related fluid-structure interactions, and has been extended to account for viscosity and surface tension. The equation is named for Lord Rayleigh, who studied cavity collapse, and Milton Plesset, who incorporated viscous effects.

In its standard form, for a bubble of radius R(t) in an incompressible, non-flowing liquid with density ρ, dynamic viscosity μ, surface tension σ, and far-field pressure P∞(t), the Rayleigh-Plesset equation is

ρ [R d^2R/dt^2 + (3/2)(dR/dt)^2] = P_B(t) − P∞(t) − 4μ(dR/dt)/R − 2σ/R.

Here P_B(t) is the pressure inside the bubble, which evolves with the gas content and thermodynamic process inside the bubble. A common closure is a polytropic relation for the gas: P_B(t) = P_B0 (R0/R)^{3γ}, where R0 is the ambient bubble radius, P_B0 is the pressure at equilibrium, and γ is the polytropic exponent.

The model relies on several simplifying assumptions, including spherical symmetry, negligible gravity, and an incompressible liquid. At higher driving pressures or rapid motions, compressibility effects become important, and extended forms such as the Keller-Miksis or Gilmore equations are used to capture them.

Applications span cavitation damage assessment, sonoluminescence studies, medical ultrasound and drug delivery modeling, and various industrial processes where bubble dynamics influence fluid behavior. Numerical solutions require initial conditions for R and its time derivative, as well as appropriate gas and liquid properties.