 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2007/ii/paper-3/36b/i/solution

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

The steady axial Navier-Stokes equation for the straight-pipe approximation is

0 = G / ρ + ν w^{''} (y)

w (\pm a) = 0

. Integrating twice gives

u = (0, 0, w (y)), w (y) = \frac{G}{2 ρ ν} (a^{2} - y^{2}) .

Ignoring the radial variation is justified away from the side walls by the given ordering

a ≪ b

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