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ID: past-exam-of-the-mphil-in-micro-and-nanotechnology-enterprise-course-of-the-university-of-cambridge/2020/paper-1/section-b/10/i/c/solution

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Past exam of the MPhil in micro- and nanotechnology enterprise course of the University of Cambridge / 2020 / Paper 1 / Section B / 10 / i / c / Solution by

Codex  0 2026-10-20

Treat the two film surfaces as an effective interfacial tension

Γ = 2 γ

. The relevant Bond number for the gas-density difference is

Bo = Δ ρ g r^{2} /Γ

. Substituting the Young–Laplace equation and the preceding density relation gives

Bo = \frac{( 4 ρ _{0} γ / ( r p _{0} )) g r ^{2}}{2 γ} = \frac{2 r}{h _{0}}, h_{0} = \frac{p _{0}}{ρ _{0} g} .

(1)

Thus the formal crossover

Bo > 1

gives $r > h_{0} /2 ≃ 4.25 km$ . This compares gravity on the excess gas with capillary restoration and neglects film weight. It does not say gravity-driven drainage of a real water film is negligible below kilometre scales. At this enormous radius, treating ambient pressure and density as uniform is itself no longer quantitatively reliable.

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