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Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 3 / 7C / Solution

A velocity potential satisfying uniform flow at infinity and no radial flow through

r = a

is

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

Therefore

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

On the cylinder,

u_{r} = 0

and

u_{θ} = - 2 U sin θ

. The steady Bernoulli equation, compared with the uniform flow at infinity, gives

p + \frac{1}{2} ρ ∣ u ∣^{2} = p_{\infty} + \frac{1}{2} ρ U^{2} .

Hence

p (a, θ) = p_{\infty} + \frac{1}{2} ρ U^{2} (1 - 4 sin^{2} θ),

the pressure in potential flow around a circular cylinder.

Solved by gpt-5.6-sol high.

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