Helmholtz\'s theorems

For other uses, see Helmholtz theorem. There exist several Theorems named after Hermann von Helmholtz.

In fluid mechanics, Helmholtz's theorems describe the three-dimensional motion of fluid in the vicinity of vortex filaments. Fluid mechanics is the study of how Fluids move and the Forces on them V erification of the O rigins of R otation in T ornadoes Ex periment or VORTEX, is a field project that seeks to understand how a These theorems apply to inviscid flows and flows where the influence of viscous forces is small and can be ignored. In Fluid dynamics there are problems that are easily solved by using the simplifying assumption of an ideal Fluid that has no Viscosity. Viscosity is a measure of the resistance of a Fluid which is being deformed by either Shear stress or Extensional stress.

Helmholtz’s three theorems are as follows:^[1]
Helmholtz’s first theorem:

The strength of a vortex filament is constant along its length.

Helmholtz’s second theorem:

A vortex filament cannot end in a fluid; it must extend to the boundaries of the fluid or form a closed path.

Helmholtz’s third theorem:

In the absence of rotational external forces, a fluid 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 viscous forces. Viscosity is a measure of the resistance of a Fluid which is being deformed by either Shear stress or Extensional stress.

Alternative expressions of the three theorems are as follows:
1. The strength of a vortex tube does not vary with time. ^[2] 2. 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.
3. Fluid elements initially free of vorticity remain free of vorticity.

Helmholtz’s theorems have application in understanding:
Generation of lift on an airfoil
Starting vortex
Horseshoe vortex
Wingtip vortices

Helmholtz’s theorems are now generally proven with reference to Kelvin's circulation theorem. In the context of a Fluid flow relative to a body the lift force is the component of the Aerodynamic force that is Perpendicular to the flow An airfoil (in American English) or aerofoil (in British English) is the shape of a Wing or blade (of a Propeller, rotor The starting vortex is a Vortex which forms in the air adjacent to the trailing edge of an Airfoil as it is accelerated from rest in a fluid The horseshoe vortex model is a simplified representation of the Vortex system of a Wing. Wingtip vortices are tubes of circulating air which are left behind by the Wing as it generates lift. In Fluid mechanics, Kelvin's Circulation Theorem states " In an Inviscid, Barotropic flow with conservative body forces the circulation However the Helmholtz's theorems were published in 1858, nine years before the 1867 publication of Kelvin's theorem. Year 1858 ( MDCCCLVIII) was a Common year starting on Friday (link will display the full calendar of the Gregorian Calendar (or a Common Year 1867 ( MDCCCLXVII) was a Common year starting on Tuesday (link will display the full calendar of the Gregorian calendar (or a Common year starting 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. A smoke ring is a visible Vortex ring formed by expelling Smoke through an opening

References

M. J. Lighthill, An Informal Introduction to Theoretical Fluid Mechanics, Oxford University Press, 1986, ISBN 0-19-853630-5
P. G. Saffman, Vortex Dynamics, Cambridge University Press, 1995, ISBN 0-521-42058-X
G. K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press (1967, reprinted in 2000). George Keith Batchelor ( March 8 1920 - March 30 2000) was an Australian Applied mathematician and Fluid dynamicist
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, 2^nd edition. John Wiley & Sons, Inc. New York ISBN 0 471 50952 3

Notes

^ Kuethe and Schetzer, Foundations of Aerodynamics, Section 2. 14
^ The strength of a vortex tube (circulation), is defined as:

$\Gamma = \int_{A} \vec{\omega} \cdot \vec{n} dA = \oint_{c} \vec{u} \cdot d\vec{s}$

where $Γ$ is also the circulation, $\vec{\omega}$ is the vorticity vector, $\vec{n}$ is the normal vector to a surface A, formed by taking a cross-section of the vortex-tube with elemental area dA, $\vec{u}$ is the velocity vector on the closed curve C, which bounds the surface A. In Fluid dynamics, circulation is the Line integral around a closed curve of the Fluid Velocity. Vorticity is a mathematical concept used in Fluid dynamics. It can be related to the amount of " circulation " or "rotation" (or more strictly the In Physics, velocity is defined as the rate of change of Position. The convention for defining the sense of circulation and the normal to the surface A is given by the right-hand screw rule. For the related yet different principle relating to electromagnetic coils see Right hand grip rule. The third theorem states that this strength is the same for all cross-sections A of the tube and is independent of time. This is equivalent to saying

$\frac{D \Gamma}{Dt} = 0$ .

References

Notes

See also