Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 344 / 3 / d / Solution

Integrate the diffusion-controlled droplet growth equation: $d(R^3)/dt=-3M\gamma /(2{\varphi }_B^2)$. ThusSetting the radius to zero givesThis is the lifetime predicted by the sharp-interface quasi-static model. The final stage at a radius comparable to the interface width falls outside that approximation, while the leading lifetime for a large initial droplet is dominated by the larger radii.