Superphysics

# The Vibrations in the Lucretius atom

##### 12 minutes  • 2505 words

Helmholtz made a very remarkable discovery: the simple vortex ring always moves relatively to the distant parts of the fluid.

This movement is:

• in a direction perpendicular to its plane
• towards the side towards which the rotatory motion carries the inner parts of the ring.

The determination of the velocity of this motion for rings of which the sectional radius is small compared with the radius of the circular axis, has presented mathematical difficulties which have not yet been over-come [2].

The observed smoke-rings seem to be always smaller than the velocity of the fluid along the straight axis through the centre of the ring.

This is because the observer standing besides the line of motion of the ring sees, as its plane passes through the position of his eye, a convex [3] outline of an atmosphere of smoke in front of the ring.

This convex outline indicates the bounding surface between the quantity of smoke which is carried forward with the ring in its motion and the surrounding air which yields to let it pass. It is not so easy to distinguish the corresponding convex outline behind the ring, because a confused trail of smoke is generally left in the rear.

In a perfect fluid the bounding surface of the portion carried forward would necessarily be quite symmetrical on the anterior and posterior sides of the middle plane of the ring.

The motion of the surrounding fluid must be precisely the same as it would be if the space within this surface were occupied by a smooth solid; but in reality the air within it is in a state of rapid motion, circulating round the circular axis of the ring with increasing velocity on the circuits nearer and nearer to the ring itself.

The circumstances of the actual motion may be imagined thus:—Let a solid column of india-rubber, of circular section, with a diameter small in propotion to its length, be bent into a circle, and its two ends properly spliced together so that it may keep the circular shape when left to itself.

Let the aperture of the ring be closed by an infinitely thin film; let an impulsive pressure be applied all over this film, of intensity so distributed as to produce the definite motion of the fluid, specified as follows, and instantly thereafter let the film be all liquified.

This motion is, in accordance with one of Helmholtz’s laws, to be along those curves which would be the lines of force, if, in place of the india-rubber circle, were substituted a ring electromagnet [4]; and the velocities at different points are to be in proportion to the intensities of the magnetic forces in the corresponding points of the magnetic field.

The motion, as has long been known, will fulfil this definition. It will continue fulfilling it, if the initiating velocities at every point of the film perpendicular to its own plane be in proportion to the intensities of the magnetic force in the corresponding points of the magnetic field.

Let now the ring be moved perpendicular to its own plane in the direction with the motion of the fluid through the middle of the ring, with a velocity very small in comparison with that of the fluid at the centre of the ring.

A large approximately globular portion of the fluid will be carried forward with the ring.

Let the velocity of the ring be increased; the volume of fluid carried forward will be diminished in every diameter, but most in the axial or fore-and-aft diameter, and its shape will thus become sensibly oblate.

By increasing the velocity of the ring forward more and more, this oblateness will increase, until, instead of being wholly convex, it will be concave before and behind, round the two ends of the axis.

If the forward velocity of the ring be increased until it is just equal to the velocity of the fluid through the centre of the ring, the axial section of the outline of the portion of the fluid carried forward will become a lemniscate.

If the ring be carried still faster forward, the portion of it carried with the india-rubber ring will be itself annular; and, relatively to the ring, the motion of the fluid will be backwards through the centre. In all cases the figure of the portion of fluid carried forward and the lines of motion will be symmetrical, both relatively to the axis and relatively to the two sides of the equatorial plane.

Any one of the states of motion thus described might of course be produced either in the order described, or by first giving a velocity to the ring and then setting the fluid in motion by aid of an instantaneous film, or by applying the two initiative actions simultaneously.

The whole amount of the impulse required, or, as we may call it, the effective momentum of the motion, or simply the momentum of the motion, is the sum of the integral values of the impulses on the ring and on the film required to produce one or other of the two components of the whole motion.

Now it is obvious that as the diameter of the ring is very small in comparison with the diameter of the circular axis, the impulse on the ring must be very small in comparison with the impulse on the film, unless the velocity given to the ring is much greater than that given to the central parts of the film.

Hence, unless the velocity given to the ring is so very great as to reduce the volume of the fluid carried forward with it to something not incomparably greater than the volume of the solid ring itself, the momenta ofthe several configurations of motions we have been considering will exceed by but insensible quantities the momentum when the ring is fixed. The value of this momentum is easily found by a proper application of Green’s formulæ.

Thus the actual momentum of the portion of fluid carried forward (being the same as that of a solid of the same density moving with the same velocity), together with an equivalent for the inertia of the fluid yielding to let it pass, is approximately the same in all these cases. and is equal to a Green’s integral expressing the whole initial impulse on the film. The equality of the effective momentum for the different velocities of the ring is easily verified without analysis for velocities not so great as to cause sensible deviations from spherical figure in the portion of the fluid carried forward. Thus in every case the length of the axis of the portion of the fluid carried forward is determined by finding the point in the axis of the ring at which the velocity is equal to the velocity of the ring.

At great distances from the plane of the ring that velocity varies, as does the magnetic force of an infinitesimal magnet on a point in its axis, inversely as the cube of the distance from the centre. Hence the cube of the radius of the approximately globular portion carried forward is in simple inverse proportion to the velocity of the ring, and therefore its momentum is constant for different velocities of the ring.

To this must be added, as was proved by Poisson, a quantity equal to half its own amount, as an equivalent for the inertia of the external fluid; and the sum is the whole effective momentum of the motion. Hence we see not only that the whole effective momentum is independent of the velocity of the ring, but that its amount is the same as the magnetic moment in the corresponding ring electromagnet. The same result is of course obtained by the Green’s integral referred to above.

The synthetical method just explained is not confined to the case of a single circular ring specially referred to, but is equally applicable to a number of rings of any form, detached from one another, or linked through one another in any way, or to a single line knotted to any degree and quality of “multiple continuity,” and joined continuously so as to have no end.

In every possible such case the motion of the fluid at every point, whether of the vortex core or of the fluid filling all space round it, is perfectly determined by Helmholtz’s formulæ when the shape of the core is given. And the synthetic investigation now explained proves that the effective momentum of the whole fluid motion agrees in magnitude and direction with the magnetic moment of the corresponding electromagnet.

Hence, still considering for simplicity only an infinitely thin line of core, let this line be projected on each of three planes at right angles to one another.

The areas of the plane circuit thus obtained (to be reckoned according to De Morgan’s rule when autotomic, as the will generally be) are the components of momentum perpendicular to these three planes.

The verification of this result will be a good exercise on “multiple continuity.” The author is not yet sufficiently acquainted with Riemann’s remarkable researches on this branch of analytical geometry to know whether of not all the kinds of “multiple continuity” now suggested are included in his classification and nomenclature.

That part ofthe synthetical investigation in which a thin solid wire ring is supposed to be moving in any direction through a fluid with the free vortex motion previously excited in it, requires the diameter of the wire at every point to be infinitely small in comparison with the radius of curvature of its axis and with the distance of the nearest of any other part of the circuit from that point of the wire.

But when the effective moment of the whole fluid motion has been found for a vortex with infinitely thing core, we may suppose any number of such vortices, however near one another, to be excited simultaneously; and the whole effective momentum in magnitude and direction will be the resultant of the momenta of the different component vortices each estimated separately.

Hence we have the remarkable proposition that the effective momentum of any possible motion in an infinite incompressible fluid agrees in direction and magnitude with the magnetic moment of the corresponding electromagnet in Helmholtz’s theory. The author hopes to give the mathematical formulæ expressing and proving this statement in the more detailed paper, which he expects soon to be able to lay before the Royal Society.

The question early occurs to any one either observing the phenomena of smoke-rings of investigating the theory,—What conditions determine the size of the ring in any case? Helmholtz’s investigation proves that the angular vortex velocity of the core varies directly as its length, or inversely as its sectional area.

Hence the strength of the electric current in the electromagnet, corresponding to an infinitely thin vortex core, remains constant, however much its length may be altered in the course of the transformations which it experiences by the motion of the fluid.

Hence it is obvious that the larger the diameter of the ring for the same volume and strength of vortex motions in an ordinary Helmholtz ring, the greater is the whole kinetic energy of the fluid, and the greater is the momentum; and we therefore see that the dimensions of a Helmholtz ring are determined when the volume and strength of the vortex motion are given, and, besides, either the kinetic energy or the momentum of the whole fluid motion due to it.

Hence if, after any number of collisions or influences, a Helmholtz ring escapes to a great distance from others and is then free, or nearly free, from vibrations, its diameter will have been increased or diminished according as it has taken energy from, or given energy to, the others. A full theory of the swelling of vortex atoms by elevation of temperature is to be worked out from this principle.

Professor Tait’s plan of exhibiting smoke-rings is as follows:—A large rectangular box, open at one side, has a circular hole of 6 or 8 inches diameter cut in the opposite side.

A common rough packing-box of 2 feet cube, or thereabout, will answer the purpose very well. The open side of the box is closed by a stout towel or piece of cloth, or by a sheet of india-rubber stretched across it.

A blow on this flexible side causes a circular vortex ring to shoot out from the hole on the other side. The vortex rings thus generated are visible if the box is filled with smoke.

One of the most convenient ways of doing this is to use two retorts with their necks thrust into holes made for the purpose in one of the sides of the box.

A small quantity of muriatic acid is put into one of these retorts, and of strong liquid ammonia into the other.

By a spirit-lamp applied from time to time to one or the other of these retorts, a thick cloud of sal-ammoniac is readily maintained in the inside of the box.

A curious and interesting experiment may be made with two boxes thus arranged, and placed either side by side close to one another or facing one another so as to project smoke-rings meeting from opposite directions—or in various relative positions, so as to give smoke-rings proceeding in paths inclined to one another at any angle, and passing one another at various distances.

An interesting variation of the experiment may be made by using clear air without smoke in one of the boxes. The invisible vortex rings projected from it render their existence startingly sensible when they come near any of the smoke-rings proceeding from the other box.

## Footnotes

[1] April 26, 1867.—The author has seen reason for believing that the sodium characteristic might be realised by a certain configuration of a single line of vortex core, to be described in the mathematical paper which he intends to communicate to the Society.

[2] See, however, note added to Professor Tait’s translation of Helmholtz’s paper (Phil. Mag. 1867, vol. xxxiii, Suppl.), where the result [see infra, p.67] of a mathematical investigation which the author of the present communication has recently succeeded in executing is given.

[3] The diagram represents precisely the convex outline referred to, and the lines of motion of the interior fluid carried along by the vortex, for the case of a double vortex consisting of two infinitely long, parallel, straight vortices of equal rotations in opposite directions. The curves have been drawn by Mr D. M`Farlane, from calculations which he has performed by means of the equation of the system of curves, which is

y²/a = 2x/a * (N+1)/(N-1) - (1+(x²/a²)), where logeN = (x+b)/a. [ED. Appearance altered from original due to HTML limitations]

The proof will be given in the mathematical paper which the author intends to communicate in a short time to the Royal Society of Edinburgh.

[4] That is to say, a circular conductor with a current of electricity maintained circulating through it.

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