Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 332 / 3 / i

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- Solution i

Solution

Let

d (z, t)

be the local water-film thickness. To first order in the interface amplitudes,

d = h + (η_{2} - η_{1}) e^{i α z + σ t} .

The lubrication theory flux down the vertical surface, with a no-slip boundary condition at the ice and a stress-free boundary condition at the water-air interface, is

q = \frac{d ^{3}}{3 ν} (g - \frac{1}{ρ} p_{z}) .

For the unperturbed film

p_{z} = 0

, so

q = \frac{g h ^{3}}{3 ν}, h = (\frac{3 ν q}{g})^{1/3} .

The linearized Young–Laplace equation gives the capillary-pressure perturbation

p^{'} = γ α^{2} η_{2} e^{i α z + σ t}, p_{z}^{'} = iγ α^{3} η_{2} e^{i α z + σ t} .

Because the prescribed volume flux is uniform, its first-order perturbation must vanish. Linearization of the flux law gives

0 = \frac{g h ^{2}}{ν} (η_{2} - η_{1}) - \frac{i h ^{3} γ α ^{3}}{3 ρ ν} η_{2} .

Thus, with

Γ = γ / (3 ρ g)

,

η_{2} (1 - i Γ α^{3} h) = η_{1},

and hence

η_{2} = \frac{η _{1}}{1 - i Γ α ^{3} h} .

The complex amplitude ratio records the phase shift caused by surface tension in the long-wave approximation.

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