= Pressure-work energy balance for a spherical bubble
{title2=$K=2\pi\rho R^3\dot R^2,\quad\dot K=(p_b-p_\infty)\dot V$}
For spherical expansion in an unbounded <incompressible flow>, continuity gives $u_r=R^2\dot R/r^2$. Integrating the <kinetic energy> outside the bubble gives $K=2\pi\rho R^3\dot R^2$. The <Rayleigh equation for an inviscid spherical bubble> then yields $\dot K=(p_b-p_\infty)\dot V$, where $V=4\pi R^3/3$. With $p_b=p_\infty V_0/V$ and constant $p_\infty$, integration gives $K-K_0=p_\infty[V_0\log(V/V_0)-V+V_0]$. This neglects <viscosity>, <surface tension> and external body-force differences, which contribute additional terms in other bubble models.
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