Page:EB1911 - Volume 05.djvu/283

 The wave-length of the disturbance may be called $$\lambda$$, and is connected with $$k$$ by the equation $$k=2\pi/\lambda$$. The capillary tension endeavours to contract the surface of the fluid; so that the stability, or instability, of the cylindrical form of equilibrium depends upon whether the surface (enclosing a given volume) be greater or less respectively after the displacement than before. It has been proved by Plateau (vide supra) that the surface is greater than before displacement if $$ka < 1$$, that is, if $$\lambda > 2\pi a$$; but less if $$ka < 1$$, or $$\lambda > 2\pi a$$. Accordingly, the equilibrium is stable if $$\lambda$$ be less than the circumference; but unstable if $$\lambda$$ be greater than the circumference of the cylinder. Disturbances of the former kind lead to vibrations of harmonic type, whose amplitudes always remain small; but disturbances, whose wave-length exceeds the circumference, result in a greater and greater departure from the cylindrical figure. The analytical expression for the motion in the latter case involves exponential terms, one of which (except in case of a particular relation between the initial displacements and velocities) increases rapidly, being equally multiplied in equal times. The coefficient ($$q$$) of the time in the exponential term ($$e^{qt}$$) may be considered to measure the degree of dynamical instability; its reciprocal $$1/q$$ is the time in which the disturbance is multiplied in the ratio $$1:e$$.

The degree of instability, as measured by $$q$$, is not to be determined from statical considerations only; otherwise there would be no limit to the increasing efficiency of the longer wave-lengths. The joint operation of superficial tension and inertia in fixing the wave-length of maximum instability was first considered by Lord Rayleigh in a paper (Math. Soc. Proc., November 1878) on the “Instability of Jets.” It appears that the value of $$q$$ may be expressed in the form

where, as before, $$\mbox{T}$$ is the superficial tension, $$\rho$$ the density, and $$\mbox{F}$$ is given by the following table:—

The greatest value of F thus corresponds, not to a zero value of $$k^2a^2$$, but approximately to $$k^2a^2 = .4858$$, or to $$\lambda = 4.508 \times 2a$$. Hence the maximum instability occurs when the wave-length of disturbance is about half as great again as that at which instability first commences.

Taking for water, in C.G.S. units, $$T = 81$$, $$\rho = 1$$, we get for the case of maximum instability

if $$d$$ be the diameter of the cylinder. Thus, if $$d=1$$, $$q^{-1}=.115$$; or for a diameter of one centimetre the disturbance is multiplied 2.7 times in about one-ninth of a second. If the disturbance be multiplied 1000 fold in time, $$t$$, $$qt=3\log_e 10=6.9$$, so that $$t=.79d^{\tfrac{3}{2}}$$. For example, if the diameter be one millimetre, the disturbance is multiplied 1000 fold in about one-fortieth of a second. In view of these estimates the rapid disintegration of a fine jet of water will not cause surprise.

The relative importance of two harmonic disturbances depends upon their initial magnitudes, and upon the rate at which they grow. When the initial values are very small, the latter consideration is much the more important; for, if the disturbances be represented by $$\alpha_1e^{q_1t},\,a_2e^{q_2t}$$, in which $$q_1$$ exceeds $$q_2$$, their ratio is $$(\alpha_2/\alpha_1)e^{-(q_1-q_2)t}$$; and this ratio decreases without limit with the time, whatever be the initial (finite) ratio $$\alpha_2;\,\alpha_1$$. If the initial disturbances are small enough, that one is ultimately preponderant for which the measure of instability is greatest. The smaller the causes by which the original equilibrium is upset, the more will the cylindrical mass tend to divide itself regularly into portions whose length is equal to 4.5 times the diameter. But a disturbance of less favourable wave-length may gain the preponderance in case its magnitude be sufficient to produce disintegration in a less time than that required by the other disturbances present.

The application of these results to actual jets presents no great difficulty. The disturbances by which equilibrium is upset are impressed upon the fluid as it leaves the aperture, and the continuous portion of the jet represents the distance travelled during the time necessary to produce disintegration. Thus the length of the continuous portion necessarily depends upon the character of the disturbances in respect of amplitude and wave-length. It may be increased considerably, as F. Savart showed, by a suitable isolation of the reservoir from tremors, whether due to external sources or to the impact of the jet itself in the vessel placed to receive it. Nevertheless it does not appear to be possible to carry the prolongation very far. Whether the residuary disturbances are of external origin, or are due to friction, or to some peculiarity of the fluid motion within the reservoir, has not been satisfactorily determined. On this point Plateau’s explanations are not very clear, and he sometimes expresses himself as if the time of disintegration depended only upon the capillary tension, without reference to initial disturbances at all.

Two laws were formulated by Savart with respect to the length of the continuous portion of a jet, and have been to a certain extent explained by Plateau. For a given fluid and a given orifice the length is approximately proportional to the square root of the head. This follows at once from theory, if it can be assumed that the disturbances remain always of the same character, so that the time of disintegration is constant. When the head is given, Savart found the length to be proportional to the diameter of the orifice. From (3) it appears that the time in which a disturbance is multiplied in a given ratio varies, not as $$d$$, but as $$d^{\tfrac{3}{2}}$$. Again, when the fluid is changed, the time varies as $$\rho^{\tfrac{1}{2}}\mbox{T}^{-\tfrac{1}{2}}$$. But it may be doubted whether the length of the continuous portion obeys any very simple laws, even when external disturbances are avoided as far as possible.

When the circumstances of the experiment are such that the reservoir is influenced by the shocks due to the impact of the jet, the disintegration usually establishes itself with complete regularity, and is attended by a musical note (Savart). The impact of the regular series of drops which is at any moment striking the sink (or vessel receiving the water), determines the rupture into similar drops of the portion of the jet at the same moment passing the orifice. The pitch of the note, though not absolutely definite, cannot differ much from that which corresponds to the division of the jet into wave-lengths of maximum instability; and, in fact, Savart found that the frequency was directly as the square root of the head, inversely as the diameter of the orifice, and independent of the nature of the fluid—laws which follow immediately from Plateau’s theory.

From the pitch of the note due to a jet of given diameter, and issuing under a given head, the wave-length of the nascent divisions can be at once deduced. Reasoning from some observations of Savart, Plateau finds in this way 4.38 as the ratio of the length of a division to the diameter of the jet. The diameter of the orifice was 3 millims., from which that of the jet is deduced by the introduction of the coefficient .8. Now that the length of a division has been estimated a priori, it is perhaps preferable to reverse Plateau’s calculation, and to exhibit the frequency of vibration in terms of the other data of the problem. Thus

But the most certain method of obtaining complete regularity of resolution is to bring the reservoir under the influence of an external vibrator, whose pitch is approximately the same as that proper to the jet. H. G. Magnus (Pogg. Ann. cvi., 1859) employed a Neef’s hammer, attached to the wooden frame which supported the reservoir. Perhaps an electrically maintained tuning-fork is still better. Magnus showed that the most important part of the effect is due to the forced vibration of that side of the vessel which contains the orifice, and that but little