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Cylindrical vortex sheet Created 2026-09-28 Updated 2026-11-05

A cylindrical vortex sheet is a vortex sheet supported on a cylindrical surface. Azimuthal Fourier disturbances have harmonic velocity potentials proportional to

r^{m}

inside and

r^{- m}

outside.

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Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 331 / 1 / iii / c / Solution 2026-09-28

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Away from the cylindrical vortex sheet, the disturbance is both incompressible and irrotational, so its potential is harmonic. Regularity at the axis and decay at infinity select

ϕ_{1} = A r^{m} e^{σ t + im θ}, ϕ_{2} = B r^{- m} e^{σ t + im θ}

(1)

for a positive integer

m

. The kinematic conditions give their values on the sheet:

ϕ_{1} (R_{0}) = \frac{R _{0} σ}{m} ξ, ϕ_{2} (R_{0}) = - \frac{R _{0}}{m} (σ + im Ω_{0}) ξ .

(2)

Substitution into the dynamical condition gives the dispersion relation

σ^{2} + im Ω_{0} σ + \frac{m ( 1 - m )}{2} Ω_{0}^{2} = 0.

(3)

Thus

σ = \frac{Ω _{0}}{2} [- im \pm m^{2} - 2 m] .

(4)

For every

m \geq 3

, one root has positive real part. The cylindrical sheet is therefore subject to a Kelvin-Helmholtz instability.

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