Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 3 / 38A / b

Table of contents
- Solution b

Solution

The definitions through the Stokes stream function give

\frac{1}{r} \partial_{r} (r u) + \partial_{z} w = - \frac{1}{r} \partial_{r} \partial_{z} Ψ + \frac{1}{r} \partial_{z} \partial_{r} Ψ = 0,

so incompressibility is automatic.

Balancing inertia

A^{2} / r^{5}

νU / δ^{2}

gives

δ (r) \sim (ν r^{3} / A)^{1/2} .

The streamfunction scale is

r U δ \sim (A ν r)^{1/2}

, motivating

Ψ = (A ν r)^{1/2} F (η), η = z (A / (ν r^{3}))^{1/2} .

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