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

Balancing the downslope gravitational force density

ρ g sin α

against the viscous term

μ u_{yy}

gives the velocity scale

U \sim \frac{ρ g h ^{2} sin α}{μ} .

If the streamwise variation scale is

L \sim h /∣ h_{x} ∣

, the lubrication theory requirement is first

∣ h_{x} ∣ ≪ 1 .

The ratio of streamwise inertia

ρ U^{2} / L

to transverse viscous stress

μU / h^{2}

is

\frac{ρ U h ^{2}}{μL} \sim \frac{ρ ^{2} g h ^{3} s i n α}{μ ^{2}} ∣ h_{x} ∣.

Thus self-consistency also requires

\frac{ρ ^{2} g h ^{3} sin α}{μ ^{2}} ∣ h_{x} ∣ ≪ 1 .

Solved by gpt-5.6-sol high.

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