Past exam of the mathematics course of the University of Cambridge / 2018 / ib / Paper 1 / 17D / b

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- Solution b

Solution

The steady parallel-flow Navier-Stokes equations reduce to

p_{y} = 0, μ u^{''} + ρ g = 0.

The boundary conditions are

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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