Rayleigh-Plesset equation ($R\ddot{R}+\frac{3}{2}{\dot{R}}^2=(p_B-p_{\infty }-2\sigma /R-4\mu \dot{R}/R)/\rho$)

The Rayleigh-Plesset equation balances the radial inertia of an incompressible liquid around a spherical bubble against internal and ambient pressure, surface tension and viscous stress. Removing the last two effects and setting the bubble pressure to zero gives the following vacuum-collapse case.

A vacuum cavity of radius $a(t)$ in an infinite inviscid incompressible fluid of density $\rho$ and far-field pressure $p_0$ obeysIf $a(0)=a_0$ and $\dot{a}(0)=0$, its collapsing branch satisfies

For a spherical vacuum cavity initially at rest with radius $a$, in an inviscid liquid of constant mass density $\rho$ and positive constant far-field pressure $p_{\infty }$, the Rayleigh equation for an inviscid spherical bubble has the first integral $R^3{\dot{R}}^2=2p_{\infty }(a^3-R^3)/(3\rho )$. To obtain it, multiply the radius equation by $2R^2\dot{R}$ and integrate the resulting exact derivative. Choosing the negative square root and integrating $dt=dR/\dot{R}$ givesThis improper integral converges at both endpoints, proving finite-time Rayleigh collapse of a spherical cavity in the ideal model. It has numerical value approximately $0.914681$. The prediction neglects viscosity, surface tension and liquid compressibility.