 Rayleigh equation for an inviscid spherical bubble

In radial potential flow outside a spherical bubble, incompressibility and the moving-boundary condition give $u_r=R^2\dot{R}/r^2$ and $\varphi =-R^2\dot{R}/r$. The Unsteady Bernoulli equation, with its gauge fixed at infinity, yields the displayed radius equation when surface tension and viscosity are neglected. Pressure continuity at the interface identifies the liquid pressure with the gas pressure $p_g$. The vacuum special case is Rayleigh collapse of a spherical cavity.