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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-344/3/d/solution

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

Codex  0 Created 2026-10-03 Updated 2026-10-05

Integrate the diffusion-controlled droplet growth equation:

d (R^{3}) / d t = - 3 M γ / (2 ϕ_{B}^{2})

. Thus

R (t)^{3} = R_{0}^{3} - \frac{3 M γ}{2 ϕ _{B}^{2}} t = R_{0}^{3} (1 - \frac{t}{τ _{\infty}}) .

(1)

Setting the radius to zero gives

τ_{\infty} (R_{0}) = \frac{2 ϕ _{B}^{2} R _{0}^{3}}{3 M γ} .

(2)

This 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.

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