Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 3 / 16C / b / i / Solution

A radially symmetric velocity potential has $u_r={\partial }_r\varphi$. The incompressible flow condition in two-dimensional polar coordinates isso $r{u}_r=A(t)$. The kinematic boundary condition at the bubble surface gives $u_r(a,t)=\dot{a}$, henceAt the outer interface, $\dot{b}=u_r(b,t)=a\dot{a}/b$. ThereforeandThe area $\pi (b^2-a^2)$ of the incompressible liquid is conserved. These formulas are the radially symmetric incompressible flow in a planar annulus.