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Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 38D / b / Solution

Let

U (x)

and

Δ (x)

denote the characteristic vertical speed and width. Integrating the result of part (a), with

b (x) = B x^{- 1/5}

, gives the scaling

ρ U^{2} Δ \sim B x^{4/5} .

The boundary-layer scaling in the momentum equation balances vertical inertia against transverse viscosity:

ρ \frac{U ^{2}}{x} \sim μ \frac{U}{Δ ^{2}}, U \sim \frac{μx}{ρ Δ ^{2}} .

Substituting this into the momentum-flux relation gives

\frac{μ ^{2} x ^{2}}{ρ Δ ^{3}} \sim B x^{4/5},

and therefore

Δ \sim (\frac{μ ^{2}}{ρB})^{1/3} x^{2/5} .

Returning to the viscous-inertial balance then gives

U \sim (\frac{B ^{2}}{ρ μ})^{1/3} x^{1/5} .

These are the similarity scaling of a two-dimensional laminar plume.

Solved by gpt-5.6-sol high.

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