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

The unperturbed temperature is linear in each material. Continuity of the conductive heat flux at

x = h

gives

k_{w} \frac{T _{m} - T _{h}}{h} = k_{a} \frac{T _{h} - T _{a}}{δ} .

On defining

ϵ = \frac{k _{a}}{k _{w}} \frac{h}{δ},

this becomes

T_{m} - T_{h} = ϵ (T_{h} - T_{a})

. Therefore

T_{m} - T_{h} = \frac{ϵ}{1 + ϵ} (T_{m} - T_{a}) .

Let

ρ_{i}

L_{f}

its latent heat of fusion per unit mass. The ice is isothermal in this model, so the Stefan condition equates latent-heat production to the heat flux conducted through the water:

ρ_{i} L_{f} V = k_{w} \frac{T _{m} - T _{h}}{h} .

Consequently the unperturbed lateral solidification speed is

V = \frac{k _{w}}{ρ _{i} L _{f} h} \frac{ϵ}{1 + ϵ} (T_{m} - T_{a}) = \frac{k _{a}}{ρ _{i} L _{f} δ} \frac{T _{m} - T _{a}}{1 + ϵ} .

k_{w}, k_{a}

ρ_{i}

L_{f}

; the geometric thermal length is

δ

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