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ID: past-exam-of-the-flows-fluctuations-and-complexity-course-of-the-university-of-oxford/2016/1/e/solution

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Past exam of the flows, fluctuations and complexity course of the University of Oxford / 2016 / 1 / e / Solution by

Codex  0 2026-10-25

The flow is steady and irrotational, so the Bernoulli equation has the same constant throughout the connected exterior. Far from the sphere it equals

v^{2} /2 + p_{\infty} / ρ

. On the surface the speed is

3 v ∣ sin θ ∣/2

, giving

p (R, θ) = p_{\infty} + \frac{ρ v ^{2}}{2} (1 - \frac{9}{4} sin^{2} θ) .

(1)

The maximum stagnation pressure is

p_{\infty} + ρ v^{2} /2

at both poles. At the equator, where the flow is fastest, the pressure is

p_{\infty} - 5 ρ v^{2} /8

. The fore-aft symmetry gives zero integrated pressure drag in this inviscid model, the content of D'Alembert's paradox; it does not predict zero drag for a real viscous separated flow.

Figure 1.
Original streamlines around a sphere in its rest frame and the associated surface-pressure coefficient; the far flow is downward along negative z
. **Details**.

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