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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2009/ib/paper-1/5d/solution

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Past exam of the mathematics course of the University of Cambridge / 2009 / ib / Paper 1 / 5D / Solution by

Codex  0 Created 2026-09-24 Updated 2026-10-14

The divergence of this axisymmetric velocity field is

r^{- 1} \partial_{r} (- α r^{2}) + \partial_{z} (2 α z) = - 2 α + 2 α = 0

. Thus it satisfies incompressibility. The apparent axis singularity in the azimuthal component is removable, since

u_{θ} \sim γ β r

r \to 0

.

Only the axial vorticity component is nonzero:

ω = \nabla \times u = (0, 0, w (r)), w (r) = 2 β γ e^{- β r^{2}} .

The cross product is

u \times ω = (u_{θ} w, α r w, 0)

. Its components are independent of

z

and

θ

, so its curl also has only an axial component:

[\nabla \times (u \times ω)]_{z} = \frac{1}{r} \frac{d}{d r} (α r^{2} w) = α (2 w + r w^{'}) = 2 α (1 - β r^{2}) w .

Meanwhile

w^{'} = - 2 β r w

and

w^{''} = (- 2 β + 4 β^{2} r^{2}) w

\nabla^{2} ω = (0, 0, w^{''} + w^{'} / r) = (0, 0, - 4 β (1 - β r^{2}) w) .

Thus the required steady vorticity equation holds with $ν = α / (2 β)$ . This is the balance between vortex stretching and viscous diffusion in a Burgers vortex.

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