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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2007/ii/paper-2/36b/i/solution

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Past exam of the mathematics course of the University of Cambridge / 2007 / ii / Paper 2 / 36B / i / Solution by

Codex  0 Created 2026-09-24 Updated 2026-10-14

The outward flux through a spherical cap of radius

R

inside the cone is

\int_{0}^{2 π} \int_{0}^{α} (- \frac{A}{R ^{2}}) R^{2} sin θ d θ, d ϕ = - 2 π A (1 - cos α) = - Q .

(1)

Thus the inward extraction flux is $Q$ at leading order. The thin no-slip wall layer is omitted from this approximation; its smaller displacement-flux effect supplies the later correction.

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