State Bernoulli's expression for the pressure in an unsteady potential flow with conservative force $-\nabla \chi$.

A spherical bubble in an incompressible liquid of density $\rho$ has radius $R(t)$. If the pressure far from the bubble is $p_{\infty}$ and inside the bubble is $p_{b}$, show that

$$p_{b}-p_{\infty}=\rho\left(\frac{3}{2} \dot{R}^{2}+R \ddot{R}\right)$$

Calculate the kinetic energy $K(t)$ in the flow outside the bubble, and hence show that

$$\dot{K}=\left(p_{b}-p_{\infty}\right) \dot{V}$$

where $V(t)$ is the volume of the bubble.

If $p_{b}(t)=p_{\infty} V_{0} / V$, show that

$$K=K_{0}+p_{\infty}\left(V_{0} \ln \frac{V}{V_{0}}-V+V_{0}\right)$$

where $K=K_{0}$ when $V=V_{0}$.