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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/ii/paper-2/39c/c/i/solution

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Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 2 / 39C / c / i / Solution by

Codex  0 Created 2026-09-24 Updated 2026-10-03

Work in the frame translating with the bubble. The bubble is the fixed circle

r = a

, while the velocity tends to

- U e_{x}

at infinity. The potential flow around a circular cylinder has velocity potential

ϕ = - U (r + \frac{a ^{2}}{r}) cos θ .

(1)

It is harmonic for

r > a

and gives

u_{r} = - U (1 - \frac{a ^{2}}{r ^{2}}) cos θ, u_{θ} = U (1 + \frac{a ^{2}}{r ^{2}}) sin θ .

(2)

Thus $u_{r} = 0$ on $r = a$ , satisfying the kinematic boundary condition, and $u \to - U e_{x}$ as $r \to \infty$ . The laboratory-frame flow is obtained by adding $U e_{x}$ .

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