Page:EB1911 - Volume 05.djvu/282

reaching this form. The only closed surface belonging to the series is the sphere.

These figures of revolution have been studied mathematically by C. W. B. Poisson,^[1] Goldschmidt,^[2] L. L. Lindelöf and F. M. N. Moigno,^[3] C. E. Delaunay,^[4] A. H. E. Lamarle,^[5] A. Beer,^[6] and V. M. A. Mannheim,^[7] and have been produced experimentally by Plateau^[8] in the two different ways already described.

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Fig. 10.—Unduloid.Fig. 11.—Catenoid.Fig. 12.—Nodoid

The limiting conditions of the stability of these figures have been studied both mathematically and experimentally. We shall notice only two of them, the cylinder and the catenoid.

Stability of the Cylinder.—The cylinder is the limiting form of the unduloid when the rolling ellipse becomes a circle. When the ellipse differs infinitely little from a circle, the equation of the meridian line becomes approximately $y=a=c\sin(xa)$ where $c$ is small. This is a simple harmonic wave-line, whose mean distance from the axis is a, whose wave-length is $2\pi a$ and whose amplitude is $c$. The internal pressure corresponding to this unduloid is as before $p={\mbox{T}}/a$. Now consider a portion of a cylindric film of length $x$ terminated by two equal disks of radius $r$ and containing a certain volume of air.
Fig. 13.Let one of these disks be made to approach the other by a small quantity $dx$. The film will swell out into the convex part of an unduloid, having its largest section midway between the disks, and we have to determine whether the internal pressure will be greater or less than before. If ${\mbox{A}}$ and ${\mbox{C}}$ (fig. 13) are the disks, and if $x$ the distance between the disks is equal to $\pi r$ half the wave-length of the harmonic curve, the disks will be at the points where the curve is at its mean distance from the axis, and the pressure will therefore be ${\mbox{T}}/r$ as before. If ${\mbox{A}}_{1}$, ${\mbox{C}}_{1}$ are the disks, so that the distance between them is less than $\pi r$, the curve must be produced beyond the disks before it is at its mean distance from the axis. Hence in this case the mean distance is less than $r$, and the pressure will be greater than ${\mbox{T}}/r$. If, on the other hand, the disks are at ${\mbox{A}}_{2}$ and ${\mbox{C}}_{2}$, so that the distance between them is greater than $\pi r$, the curve will reach its mean distance from the axis before it reaches the disks. The mean distance will therefore be greater than $r$, and the pressure will be less than ${\mbox{T}}/r$. Hence if one of the disks be made to approach the other, the internal pressure will be increased if the distance between the disks is less than half the circumference of either, and the pressure will be diminished if the distance is greater than this quantity. In the same way we may show that if the distance between the disks is increased, the pressure will be diminished or increased according as the distance is less or more than half the circumference of either.

Now let us consider a cylindric film contained between two equal fixed disks. ${\mbox{A}}$ and ${\mbox{B}}$, and let a third disk, ${\mbox{C}}$, be placed midway between. Let ${\mbox{C}}$ be slightly displaced towards ${\mbox{A}}$. If ${\mbox{AC}}$ and ${\mbox{CB}}$ are each less than half the circumference of a disk the pressure on ${\mbox{C}}$ will increase on the side of ${\mbox{A}}$ and diminish on the side of ${\mbox{B}}$. The resultant force on ${\mbox{C}}$ will therefore tend to oppose the displacement and to bring ${\mbox{C}}$ back to its original position. The equilibrium of ${\mbox{C}}$ is therefore stable. It is easy to show that if ${\mbox{C}}$ had been placed in any other position than the middle, its equilibrium would have been stable. Hence the film is stable as regards longitudinal displacements. It is also stable as regards displacements transverse to the axis, for the film is in a state of tension, and any lateral displacement of its middle parts would produce a resultant force tending to restore the film to its original position. Hence if the length of the cylindric film is less than its circumference, it is in stable equilibrium. But if the length of the cylindric film is greater than its circumference, and if we suppose the disk ${\mbox{C}}$ to be placed midway between ${\mbox{A}}$ and ${\mbox{B}}$, and to be moved towards ${\mbox{A}}$, the pressure on the side next ${\mbox{A}}$ will diminish, and that on the side next ${\mbox{B}}$ will increase, so that the resultant force will tend to increase the displacement, and the equilibrium of the disk ${\mbox{C}}$ is therefore unstable. Hence the equilibrium of a cylindric film whose length is greater than its circumference is unstable. Such a film, if ever so little disturbed, will begin to contract at one secton and to expand at another, till its form ceases to resemble a cylinder, if it does not break up into two parts which become ultimately portions of spheres.

Instability of a Jet of Liquid.—When a liquid flows out of a vessel through a circular opening in the bottom of the vessel, the form of the stream is at first nearly cylindrical though its diameter gradually diminishes from the orifice downwards on account of the increasing velocity of the liquid. But the liquid after it leaves the vessel is subject to no forces except gravity, the pressure of the air, and its own surface-tension. Of these gravity has no effect on the form of the stream except in drawing asunder its parts in a vertical direction, because the lower parts are moving faster than the upper parts. The resistance of the air produces little disturbance until the velocity becomes very great. But the surface-tension, acting on a cylindric column of liquid whose length exceeds the limit of stability, begins to produce enlargements and contractions in the stream as soon as the liquid has left the orifice, and these inequalities in the figure of the column go on increasing till it is broken up into elongated fragments. These fragments as they are falling through the air continue to be acted on by surface-tension. They therefore shorten themselves, and after a series of oscillations in which they become alternately elongated and flattened, settle down into the form of spherical drops.

This process, which we have followed as it takes place on an individual portion of the falling liquid, goes through its several phases at different distances from the orifice, so that if we examine different portions of the stream as it descends, we shall find next the orifice the unbroken column, then a series of contractions and enlargements, then elongated drops, then flattened drops, and so on till the drops become spherical.

[The circumstances attending the resolution of a cylindrical jet into drops were admirably examined and described by F. Savart (“Mémoire sur la constitution des veines liquides lancées par des orifices circulaires en minces parois,” Ann. d. Chim. t. liii., 1833) and for the most part explained with great sagacity by Plateau. Let us conceive an infinitely long circular cylinder of liquid, at rest (a motion common to every part of the fluid is necessarily without influence upon the stability, and may therefore be left out of account for convenience of conception and expression), and inquire under what circumstances it is stable or unstable, for small displacements, symmetrical about the axis of figure.

Whatever the deformation of the originally straight boundary of the axial section may be, it can be resolved by Fourier’s theorem into deformations of the harmonic type. These component deformations are in general infinite in number, of very wave-length and of arbitrary phase; but in the first stages of the motion, with which alone we are at present concerned, each produces its effect independently of every other, and may be considered by itself. Suppose, therefore, that the equation of the boundary is

$r=a+a\cos kz,$

(1)

where $a$ is a small quantity, the axis of $z$ being that of symmetry.

1. Nouvelle théorie de l’action capillaire (1831).
2. Determinatio superficiei minimae rotatione curvae data duo puncta jungentis circa datum axem ortae (Göttingen, 1831).
3. Leçons de calcul des variations (Paris, 1861).
4. “Sur la surface de révolution dont la courbure moyenne est constante,” Liouville’s Journal, vi.
5. “Théorie géometrique des rayons et centres de courbure,” Bullet, de l’Acad. de Belgique, 1857.
6. Tractatus de Theoria Mathematica Phaenomenorum in Liquidis actioni gravitatis detractis observatorum (Bonn, 1857).
7. Journal de l’Institut, No. 1260.
8. Statique expérimental et théorique des liquides, 1873.