Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 344 / 2 / c / Solution

The curved interface requires exterior composition $-{\varphi }_B+\delta$, but the far field supplies only $-{\varphi }_B$. Material therefore diffuses away from the droplet, which evaporates. Write $\varphi =-{\varphi }_B+g$ outside. Linearization gives $\mu =\alpha g$, and the quasistatic condition $\dot{\varphi }=M{\nabla }^2\mu \simeq 0$ givesSpherical symmetry yields $g(r)=\delta R/r$. Taking the normal outward from the droplet,The interface converts positive-phase material into negative-phase material. Integrating the continuity equation through the moving interface gives the Stefan conditionwhere $[\varphi ]=2{\varphi }_B$, and hence