Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 4 / 38C / d

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- Solution d

Solution

The volume flux per unit transverse width is

q (x, t) = \int_{0}^{h} u d y = \frac{ρ g h ^{3}}{3 μ} (sin α - cos α h_{x}) .

Integrating incompressibility across the layer and using the wall and free-surface kinematic conditions gives local mass conservation

h_{t} + q_{x} = 0

. Hence the gravity-driven thin film on an incline obeys

h_{t} + \frac{ρ g}{3 μ} \partial_{x} [h^{3} (sin α - cos α h_{x})] = 0 .

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