 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/ib/paper-1/17d/b/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

The steady parallel-flow Navier-Stokes equations reduce to

p_{y} = 0, μ u^{''} + ρ g = 0.

p (h) = p_{0}, u (0) = 0, μ u^{'} (h) = - k u (h) .

The last condition balances the liquid's tangential traction with the opposing air stress. Hence

p = p_{0}

and

u (y) = - \frac{ρ g}{2 μ} y^{2} + \frac{ρ g h}{μ + kh} (1 + \frac{kh}{2 μ}) y .

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