Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 344 / 2 / e

Solution

For an arrested domain size

Λ

, make the prescribed replacement

\dot{L} ⟶ \frac{Λ}{τ}, \ddot{L} ⟶ \frac{Λ}{τ ^{2}} .

Λ = L_{0} g (w), w = \frac{τ}{t _{0}},

gives the schematic algebraic equation

(α + β) \frac{g}{w ^{2}} = γ \frac{1}{w g} + \frac{δ}{g ^{2}},

(α + β) g^{3} = γ w g + δ w^{2} .

For

w ≪ 1

, viscous-capillary balance gives

g (w) \sim \frac{δ}{γ} w, Λ \sim \frac{σ τ}{η},

which is independent of

ρ

. For

w ≫ 1

, inertial-capillary balance gives

g (w) \sim (\frac{δ}{α + β})^{1/3} w^{2/3}, Λ \sim (\frac{σ τ ^{2}}{ρ})^{1/3},

which is independent of

η

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