 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/ib/paper-3/16c/b/iv/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

Per unit depth, the bubble volume is its area

V = π a^{2}

. At equilibrium its pressure equals

p_{\infty}

, so

p_{0} V = p_{\infty} π a_{0}^{2}

. Hence

p_{0} = p_{\infty} \frac{a _{0}^{2}}{a ^{2}} = p_{\infty} (1 + ϵ)^{- 2} = p_{\infty} (1 - 2 ϵ) + O (ϵ^{2}) .

Continuity of fluid pressure at the interface gives

p (a, t) = p_{0} (t)

. Consequently

p (a, t) - p_{\infty} = - 2 p_{\infty} ϵ + O (ϵ^{2}) .

 New to topics? Read the docs here!