Helmholtz's theorems
Encyclopedia
In fluid mechanics
, Helmholtz's theorems, named after Hermann von Helmholtz
, describe the three-dimensional motion of fluid in the vicinity of vortex filaments
. These theorems apply to inviscid flow
s and flows where the influence of viscous forces
is small and can be ignored.
Helmholtz’s three theorems are as follows:
Helmholtz’s first theorem:
Helmholtz’s second theorem:
Helmholtz’s third theorem:
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
.
Alternative expressions of the three theorems are as follows:
1. The strength of a vortex tube does not vary with time.
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:
Helmholtz’s theorems are now generally proven with reference to Kelvin's circulation theorem
. However the Helmholtz's theorems were published in 1858, 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 ring
s.
Fluid mechanics
Fluid mechanics is the study of fluids and the forces on them. Fluid mechanics can be divided into fluid statics, the study of fluids at rest; fluid kinematics, the study of fluids in motion; and fluid dynamics, the study of the effect of forces on fluid motion...
, Helmholtz's theorems, named after Hermann von Helmholtz
Hermann von Helmholtz
Hermann Ludwig Ferdinand von Helmholtz was a German physician and physicist who made significant contributions to several widely varied areas of modern science...
, describe the three-dimensional motion of fluid in the vicinity of vortex filaments
Vortex
A vortex is a spinning, often turbulent,flow of fluid. Any spiral motion with closed streamlines is vortex flow. The motion of the fluid swirling rapidly around a center is called a vortex...
. These theorems apply to inviscid flow
Inviscid flow
In fluid dynamics there are problems that are easily solved by using the simplifying assumption of an ideal fluid that has no viscosity. The flow of a fluid that is assumed to have no viscosity is called inviscid flow....
s and flows where the influence of viscous forces
Viscosity
Viscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher viscosity...
is small and can be ignored.
Helmholtz’s three theorems are as follows:
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
Viscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher viscosity...
.
Alternative expressions of the three theorems are as follows:
1. The strength of a vortex tube does not vary with time.
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 airfoilAirfoilAn airfoil or aerofoil is the shape of a wing or blade or sail as seen in cross-section....
- Starting vortexStarting vortexThe 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. It leaves the airfoil , and remains stationary in the flow...
- Horseshoe vortexHorseshoe vortexThe horseshoe vortex model is a simplified representation of the vortex system of a wing. In this model the wing vorticity is modelled by a bound vortex of constant circulation, travelling with the wing, and two trailing vortices, therefore having a shape vaguely reminiscent of a horseshoe...
- Wingtip vorticesWingtip vorticesWingtip vortices are tubes of circulating air that are left behind a wing as it generates lift. One wingtip vortex trails from the tip of each wing. The cores of vortices spin at very high speed and are regions of very low pressure...
.
Helmholtz’s theorems are now generally proven with reference to Kelvin's circulation theorem
Kelvin's circulation theorem
In fluid mechanics, Kelvin's circulation theorem states In an inviscid, barotropic flow with conservative body forces, the circulation around a closed curve moving with the fluid remains constant with time. The theorem was developed by William Thomson, 1st Baron Kelvin...
. However the Helmholtz's theorems were published in 1858, 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 ring
Smoke ring
A smoke ring is a visible vortex ring formed by sudden release of smoke. It can be created by blowing smoke from the mouth, quickly lighting a cigarette lighter and putting it out or holding a burning incense stick or a cigarette vertically, pushing it with the burning side up and suddenly pulling...
s.