fluid mechanics Fluid mechanics is the branch of physics concerned with the mechanics of fluids ( liquids, gases, and plasmas) and the forces on them. It has applications in a wide range of disciplines, including mechanical, aerospace, civil, chemical and ...

, Helmholtz's theorems, named after

Hermann von Helmholtz Hermann Ludwig Ferdinand von Helmholtz (31 August 1821 – 8 September 1894) was a German physicist and physician who made significant contributions in several scientific fields, particularly hydrodynamic stability. The Helmholtz Associatio ...

, describe the three-dimensional motion of fluid in the vicinity of

vortex In fluid dynamics, a vortex ( : vortices or vortexes) is a region in a fluid in which the flow revolves around an axis line, which may be straight or curved. Vortices form in stirred fluids, and may be observed in smoke rings, whirlpools in ...

lines. These theorems apply to

inviscid flow In fluid dynamics, inviscid flow is the flow of an inviscid (zero-viscosity) fluid, also known as a superfluid. The Reynolds number of inviscid flow approaches infinity as the viscosity approaches zero. When viscous forces are neglected, suc ...

s and flows where the influence of

viscous forces The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water. Viscosity quantifies the inter ...

are small and can be ignored. Helmholtz's three theorems are as follows: ;Helmholtz's first theorem: :The strength of a vortex line is constant along its length. ;Helmholtz's second theorem: :A vortex line cannot end in a fluid; it must extend to the boundaries of the fluid or form a closed path. ;Helmholtz's third theorem: :A fluid element that is initially irrotational remains irrotational. Helmholtz's theorems apply to inviscid flows. In observations of vortices in real fluids the strength of the vortices always decays gradually due to the dissipative effect of

. Alternative expressions of the three theorems are as follows:
# The strength of a vortex tube does not vary with time. # 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. Also vortex lines and tubes must appear as a closed loop, extend to infinity or start/end at solid boundaries. # Fluid elements initially free of

vorticity In continuum mechanics, vorticity is a pseudovector field that describes the local spinning motion of a continuum near some point (the tendency of something to rotate), as would be seen by an observer located at that point and traveling along wi ...

remain free of vorticity. Helmholtz's theorems have application in understanding: * Generation of lift on an airfoil * Starting vortex * Horseshoe vortex *

Wingtip vortices Wingtip vortices are circular patterns of rotating air left behind a wing as it generates lift.Clancy, L.J., ''Aerodynamics'', section 5.14 One wingtip vortex trails from the tip of each wing. Wingtip vortices are sometimes named ''trailing ...

. Helmholtz's theorems are now generally proven with reference to

Kelvin's circulation theorem In fluid mechanics, Kelvin's circulation theorem (named after William Thomson, 1st Baron Kelvin William Thomson, 1st Baron Kelvin, (26 June 182417 December 1907) was a British mathematician, Mathematical physics, mathematical physicist and ...

. However Helmholtz's theorems were published in 1858, nine years before the 1867 publication of Kelvin's theorem.

Notes

References

* M. J. Lighthill, ''An Informal Introduction to Theoretical Fluid Mechanics'', Oxford University Press, 1986, * P. G. Saffman, ''Vortex Dynamics'', Cambridge University Press, 1995, * G. K. Batchelor, ''An Introduction to Fluid Dynamics'', Cambridge University Press (1967, reprinted in 2000). * Kundu, P and Cohen, I, ''Fluid Mechanics'', 2nd edition, Academic Press 2002. * George B. Arfken and Hans J. Weber, ''Mathematical Methods for Physicists'', 4th edition, Academic Press: San Diego (1995) pp. 92–93 * A.M. Kuethe and J.D. Schetzer (1959), ''Foundations of Aerodynamics'', 2nd edition. John Wiley & Sons, Inc. New York {{DEFAULTSORT:Helmholtz's Theorems Aerodynamics Fluid dynamics Theorems in mathematical physics Hermann von Helmholtz