Past exam of the fluids and waves course of the University of Oxford / 2023 / 1 / c

With zero internal pressure and negligible surface tension, pressure continuity sets the liquid-side boundary value to $p(R,t)=0$. Evaluating the pressure field of a pulsating sphere there givesThus the Rayleigh equation for an inviscid spherical bubble, together with its initial conditions, isFor $p_{\infty }>0$, initially $\ddot{R}(0)=-p_{\infty }/(\rho a)<0$, so the physically relevant branch describes Rayleigh collapse of a spherical cavity.

Write $P=p_{\infty }/\rho >0$. Multiplying the radius ordinary differential equation by $2R^2\dot{R}$ gives an exact derivative:The initial conditions therefore give the first integralChoose the negative square root because the cavity starts accelerating inward. It cannot turn around while $0<R<a$, since the squared speed is then strictly positive. Separating variables, with $s=R/a$, proves the collapse time of a spherical cavity isHenceThe improper integral is finite: its integrand behaves as $s^{3/2}$ near zero and as $[3(1-s){]}^{-1/2}$ near one. The divergent inward speed as $R\to 0$ signals a singular endpoint of this ideal incompressible flow model; the finite collapse time follows within that model.