Past exam of the mathematics course of the University of Cambridge / 2021 / ib / Paper 4 / 16A (Fluid Dynamics)

In spherical symmetry, the velocity is radial and constant over each sphere. HenceApply the divergence theorem to the fluid shell between radii $R_1$ and $R_2$. Since the flow is incompressible,Thus $Q$ is independent of $R$ and is a function of time alone. This is the flux law for spherically symmetric incompressible radial flow.

The radial velocity and velocity potential areChoose the gauge in the Unsteady Bernoulli equation by evaluating it at infinity, where $u=0$, $\varphi =0$, and $p=p_0$:At the cavity surface the vacuum pressure is zero, and the kinematic boundary condition gives $u(a,t)=\dot{a}$. SinceBernoulli's equation becomesTherefore

At the moving boundary,Substitution into the result of part (b) gives the Rayleigh collapse equationTreat $v=\dot{a}$ as a function of $a$, so that $\ddot{a}=vdv/da$. With $w=v^2$, the equation becomesMultiplication by the integrating factor $a^3$ after division by $a$ givesUsing $w(a_0)=0$,The collapsing branch has negative radial velocity, so

Since $dt=da/\dot{a}$ on the collapsing branch, the time for $a$ to decrease from $a_0$ to zero isEquivalently, scaling $a=a_{0}x$ gives