Consider the spherically symmetric motion induced by the collapse of a spherical cavity of radius $a(t)$, centred on the origin. For $r<a$, there is a vacuum, while for $r>a$, there is an inviscid incompressible fluid with constant density $\rho$. At time $t=0, a=a_{0}$, and the fluid is at rest and at constant pressure $p_{0}$.

(a) Consider the radial volume transport in the fluid $Q(R, t)$, defined as

$$Q(R, t)=\int_{r=R} u d S$$

where $u$ is the radial velocity, and $d S$ is an infinitesimal element of the surface of a sphere of radius $R \geqslant a$. Use the incompressibility condition to establish that $Q$ is a function of time alone.

(b) Using the expression for pressure in potential flow or otherwise, establish that

$$\frac{1}{4 \pi a} \frac{d Q}{d t}-\frac{(\dot{a})^{2}}{2}=-\frac{p_{0}}{\rho}$$

where $\dot{a}(t)$ is the radial velocity of the cavity boundary.

(c) By expressing $Q(t)$ in terms of $a$ and $\dot{a}$, show that

$$\dot{a}=-\sqrt{\frac{2 p_{0}}{3 \rho}\left(\frac{a_{0}^{3}}{a^{3}}-1\right)}$$

[Hint: You may find it useful to assume $\dot{a}(t)$ is an explicit function of a from the outset.]

(d) Hence write down an integral expression for the implosion time $\tau$, i.e. the time for the radius of the cavity $a \rightarrow 0$. [Do not attempt to evaluate the integral.]