FitzGerald's Spirality

The question of the stability of the turbulent motion remained undecided.

Back then, Thomson thought it likely that the motion would suffer diffusion.

But 2 years later[57], he showed that stability was ensured at any rate when space is filled with a set of approximately straight hollow vortex filaments.

FitzGerald[58] subsequently determined the energy per unit-volume in a turbulent liquid which is transmitting laminar waves. Writing for brevity

the equations are

If the quantity

is integrated throughout space, and the variations of the integral with respect to time are determined, it is found that

Integrating the second term under the integral by parts, and omitting the superficial terms (which may be at infinity, or wherever energy enters the space under consideration), we have

Hence it appears that the quantity Σ, which is of the dimensions of energy, must be proportional to the energy per unit-volume of the medium—a result which shows that there is a pronounced similarity between the dynamics of a vortex-sponge and of Maxwell’s elastic aether.

A definite vortex-sponge model of the aether was described by Hicks in his Presidential Address to the mathematical section of the British Association in 1895.[59]

In this the small motions whose function is to confer the quasi-rigidity were not completely chaotic, but were disposed systematically.

The medium was supposed to be constituted of cubical elements of fluid, each containing a rotational circulation complete in itself: in any element, the motion close to the central vertical diameter of the element is vertically upwards: the fluid which is thus carried to the upper part of the element flows outwards over the top, down the sides, and up the centre again.

In each of the six adjoining elements the motion is similar to this, but in the reverse direction.

The rotational motion in the elements confers on them the power of resisting distortion, so that waves may be propagated through the medium as through an elastic solid; but the rotations are without effect on irrotational motions of the fluid, provided the velocities in the irrotational motion are slow compared with the velocity of propagation of distortional vibrations.

A different model was described four years later by FitzGerald.[60]

The distribution of velocity of a fluid in the neighbourhood of a vortex filament is the same as the distribution of magnetic force around a wire of identical form carrying an electric current. Therefore, the fluid has more energy when the filament has the form of a helix than when it is straight.

If space were filled with vortices, whose axes were all parallel to a given direction, there would be an increase in the energy per unit volume when the vortices were bent into a spiral form.

This could be measured by the square of a vector—say, E—which may be supposed parallel to this direction.

If now a single spiral vortex is surrounded by parallel straight ones, the latter will not remain straight, but will be bent by the action of their spiral neighbour. The transference of spirality may be specified by a vector H, which will be distributed in circles round the spiral vortex.

Its magnitude will depend on the rate at which spirality is being lost by the original spiral, and can be taken such that its square is equal to the mean energy of this new motion. The vectors E and H will then represent the electric and magnetic vectors; the vortex spirals representing tubes of electric force.

FitzGerald’s spirality is essentially similar to the laminar motion investigated by Lord Kelvin, since it involves a flow in the direction of the axis of the spiral, and such a flow cannot take place along the direction of a vortex filament without a spiral deformation of a filament.

Other vortex analogues have been devised for electrostatical systems. Ono such, which was described in 1888 by W. M. Hicks,[61] depends on the circumstance that if two bodies in contact in an infinite fluid are separated from each other, and if there be a vortex filament which terminates on the bodies, there will be formed at the point where they separate a hollow vortex filament[62] stretching from one to the other, with rotation equal and opposite to that of the original filament.

As the bodies are moved apart, the hollow vortex may, through failure of stability, dissociate into a number of smaller ones; and if the resulting number be very large, they will ultimately take up a position of stable equilibrium. The two sets of filaments—the original filaments and their hollow companions—will be intermingled, and each will distribute itself according to the same law as the lines of force between the two bodies which are equally and oppositely electrified.

Since the pressure inside a hollow vortex is zero, the portion of the surface on which it abuts experiences a diminution of pressure, the two bodies are therefore attracted.

Moreover, as the two bodies separate further, the distribution of the filaments being the same as that of lines of electric force, the diminution of pressure for each line is the same at all distances, and therefore the force between the two bodies follows the same law as the force between two bodies equally and oppositely electrified.

The effect of the original filaments is similar, the diminution of pressure being half as large again as for the hollow vortices.

If another surface were brought into the presence of the others, those of the filaments which encounter it would break off and rearrange themselves so that each part of a broken filament terminates on the new body.

This analogy thus gives a complete account of electrostatic actions both quantitatively and qualitatively.

The electric charge on a body corresponds to the number of ends of filaments abutting on it, the sign being determined by the direction of rotation of the filament as viewed from the body.

A magnetic field may be supposed to be produced by the motion of the vortex filaments through the stationary aether, the magnetic force being at right angles to the filament and to its direction of motion. Electrostatic and magnetic fields thus correspond to states of motion in the medium, in which, however, there is no bodily flow; for the two kinds of filament produce circulation in opposite directions.

It is possible that hollow vortices are better adapted than ordinary vortex-filaments for the construction of models of the aether. Such, at any rate, was the opinion of Thomson (Kelvin) in his later years.[63]

The analytical difficulties of the subject are formidable. Progress is consequently slow.

But among the many mechanical schemes which have been devised to represent, electrical and optical phenomena, none possesses greater interest than that which pictures the aether as a vortex-sponge.