RayleighPlessetligningen

RayleighPlessetligningen, commonly known in English as the Rayleigh–Plesset equation, is a nonlinear ordinary differential equation that describes the radial dynamics of a spherical gas bubble in an incompressible, viscous liquid subjected to time-dependent external pressure. It is derived from the Navier–Stokes equations under the assumption of spherical symmetry and appropriate boundary conditions at the bubble interface.

The standard form for the bubble radius R(t) is

ρ (R d^2R/dt^2 + (3/2) (dR/dt)^2) = p_g(R) - p_∞(t) - 4 μ (dR/dt)/R - 2 σ / R.

Here, ρ is the liquid density, μ the dynamic viscosity, σ the surface tension, and p_∞(t) the far-field pressure. The gas pressure inside the bubble, p_g(R), is often modeled with a polytropic relation, p_g(R) = p_g0 (R0/R)^{3γ}, where R0 is the ambient radius and γ the polytropic exponent.

The equation is widely used to study cavitation, bubble oscillations under acoustic driving, and related phenomena in sonolumescence and medical ultrasound. It assumes an incompressible liquid, a single spherical bubble, and neglects effects such as thermal conduction and gravity. For fast or highly nonlinear dynamics, compressibility-corrected forms (e.g., Keller–Miksis) may be employed.