Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 321 / 3 / b / ii

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- Solution ii

Solution

Differentiating the equilibrium velocity gives

2Ω + \frac{d u _{y}}{d x} = \frac{Ω}{2} + \frac{c _{s}^{2}}{2Ω} \frac{d ^{2} σ}{d x ^{2}} .

On the other hand, the definition of vortensity and

Σ = Σ_{0} e^{σ}

give

2Ω + \frac{d u _{y}}{d x} = ζ Σ_{0} e^{σ} .

Multiplying by

2/Ω

, and introducing the dimensionless variables

ξ = \frac{Ω}{c _{s}} x, q = \frac{2 Σ _{0}}{Ω} ζ,

yields

\frac{d ^{2} σ}{d ξ ^{2}} - q e^{σ} + 1 = 0 .

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