 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/ii/paper-4/38a/b/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

The only vorticity component is

ω = (1/ r) (r v)'

. Its steady equation is

- α r ω' = 2 α ω + ν (ω'' + r^{- 1} ω')

, solved by

ω = ω_{0} e^{- α r^{2} / (2 ν)}

. Regularity at zero gives

v (r) = \frac{ω _{0} ν}{α r} (1 - e^{- α r^{2} / (2 ν)}) .

Its maximum satisfies

e^{s} = 1 + 2 s

s = α r^{* 2} / (2 ν)

, so

r^{*} = O (ν / α)

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