Math 2.0

Fluid mechanics Ideal fluids Theorem

Helmholtz's theorems on vortices

Builds on VorticityCirculationInviscid flowKelvin's circulation theorem

Statement

Let uu be a smooth velocity field on an open set Ω⊆ℝ3\Omega \subseteq \R^3, with vorticity ω=∇×u\omega = \nabla \times u. A vortex line is an integral curve of ω\omega. A vortex tube is the union of the vortex lines through a closed curve. Its strength is the flux of ω\omega through a cross-section, which equals the circulation of uu around a curve that goes once around the tube.

  1. The strength of a vortex tube is the same for every cross-section.
  2. A vortex tube cannot end inside the fluid: it either closes on itself or reaches the boundary of Ω\Omega (or extends to infinity).
  3. Assume moreover that the flow is inviscid and barotropic, with conservative body forces. Then vortex lines and vortex tubes move with the fluid, and the strength of each vortex tube is constant in time. In particular, a part of the fluid that is irrotational at one time stays irrotational.

Remarks

Parts 1 and 2 are kinematic. They follow from ∇⋅ω=0\nabla \cdot \omega = 0 and the divergence theorem, applied to a piece of a tube between two cross-sections. The flux through the side is 00, so the fluxes through the two ends are equal. Part 3 follows from Kelvin's circulation theorem. A material surface on which ω\omega is tangent has zero circulation around each small loop in it, and the flow preserves this property.

Example: a smoke ring is a vortex tube closed into a loop, which travels through the air with its vortex lines carried along.

Without part 3's hypotheses vortex lines need not move with the fluid; viscosity lets vortex lines reconnect.

Going deeper

For a student

The first two theorems hold for any smooth flow, since they only use that vorticity is divergence free. The third is the local form of Kelvin's circulation theorem. It is the basis of vortex dynamics: in all of ℝ3\R^3, for velocity decaying at infinity, the velocity is recovered from the vorticity by the Biot–Savart law. The vorticity is transported and stretched by the flow, as the vorticity equation shows.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

Show the path as a graph
Circulation, concept, no evidence levelCirculationconceptEuler equations, concept, no evidence levelEuler equationsconceptInviscid flow, concept, no evidence levelInviscid flowconceptMaterial derivative, concept, no evidence levelMaterial derivativeconceptVorticity, concept, no evidence levelVorticityconceptKelvin's circulation theorem, statement, no evidence levelKelvin's circulation …statementHelmholtz's theorems on vortices, statement, no evidence levelHelmholtz's theorems …statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Circulation concept
  2. Euler equations concept
  3. Inviscid flow concept
  4. Material derivative concept
  5. Vorticity concept
  6. Kelvin's circulation theorem statement
  7. Helmholtz's theorems on vortices statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Helmholtz%27s_theorems
About this page
TagD2KCKEP0N4
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Kindstatement, theorem
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MSC76B47
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AboutVorticity
UsesVorticity, Circulation, Inviscid flow, Kelvin's circulation theorem
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Changed3 November 2026

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