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

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 .

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 .

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}

.

Solved by gpt-5.6-sol high.

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