Helmholtz's theorems
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Template:Otheruses3 In fluid mechanics, Helmholtz's theorems describe the behaviour of vortex lines in a fluid. The theorems apply to fluids that are inviscid (ie without viscosity), incompressible, of constant density and under the influence of a conservative body force (such as gravity). The theorems were published by Hermann von Helmholtz in 1858.
The two theorems state:
- Fluid elements lying on a vortex line at some instant continue to lie on that vortex line. More simply, vortex lines move with the fluid.
- The strength of a 'vortex tube', (whose walls are vortex lines), is defined as <math>\Gamma = \int_{S} \mathbf{\omega}\cdot\mathbf{n} \, dS</math>—here ω is the vorticity vector, n is the normal vector to a surface S, formed by taking a cross-section of the vortex-tube, (ie bounded by a closed curve in the fluid) and dS is an element of its area or cross-section— this strength is the same for all cross-sections S of the tube. Furthermore, Γ is independent of time.
The theorems are now generally proven with reference to Kelvin's circulation theorem, however the Helmholtz's theorems were published nine years before the 1867 publication of Kelvin's theorem. There was much communication between the two men on the subject of vortex lines, with many references to the application of their theorems to the study of smoke rings.
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References
- George B. Arfken and Hans J. Weber, Mathematical Methods for Physicists, 4th edition, Academic Press: San Diego (1995) pp. 92-93.
- G. K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press (1967, reprinted in 2000).