p = T(1/R1 + 1/R2) (14).
This equation, which gives the pressure in terms of the principal radii
of curvature, though here proved only in the case of a surface of
revolution, must be true of all surfaces. For the curvature of any
surface at a given point may be completely defined in terms of the
positions of its principal normal sections and their radii of curvature.
Before going further we may deduce from equation 9 the nature of all the
figures of revolution which a liquid film can assume. Let us first
determine the nature of a curve, such that if it is rolled on the axis
its origin will trace out the meridian section of the bubble. Since at
any instant the rolling curve is rotating about the point of contact
with the axis, the line drawn from this point of contact to the tracing
point must be normal to the direction of motion of the tracing point.
Hence if N is the point of contact, NP must be normal to the traced
curve. Also, since the axis is a tangent to the rolling curve, the
ordinate PR is the perpendicular from the tracing point P on the
tangent. Hence the relation between the radius vector and the
perpendicular on the tangent of the rolling curve must be identical with
the relation between the normal PN and the ordinate PR of the traced
curve. If we write r for PN, then y = r cos[alpha], and equation 9
becomes
/ T \ F
y^2( 2 -- -1 ) = -----.
\ pr / [pi]p
This relation between y and r is identical with the relation between the
perpendicular from the focus of a conic section on the tangent at a
given point and the focal distance of that point, provided the
transverse and conjugate axes of the conic are 2a and 2b respectively,
where
T F
a = ---, and b^2 = -----.
p [pi]p
Hence the meridian section of the film may be traced by the focus of
such a conic, if the conic is made to roll on the axis.
_On the different Forms of the Meridian Line._--1. When the conic is an
ellipse the meridian line is in the form of a series of waves, and the
film itself has a series of alternate swellings and contractions as
represented in figs. 9 and 10. This form of the film is called the
unduloid.
1a. When the ellipse becomes a circle, the meridian line becomes a
straight line parallel to the axis, and the film passes into the form of
a cylinder of revolution.
1b. As the ellipse degenerates into the straight line joining its foci,
the contracted parts of the unduloid become narrower, till at last the
figure becomes a series of spheres in contact.
In all these cases the internal pressure exceeds the external by 2T/a
where a is the semi-transverse axis of the conic. The resultant of the
internal pressure and the surface-tension is equivalent to a tension
along the axis, and the numerical value of this tension is equal to the
force due to the action of this pressure on a circle whose diameter is
equal to the conjugate axis of the ellipse.
2. When the conic is a parabola the meridian line is a catenary (fig.
11); the internal pressure is equal to the external pressure, and the
tension along the axis is equal to 2[pi]Tm where m is the distance of
the vertex from the focus.
3. When the conic is a hyperbola the meridian line is in the form of a
looped curve (fig. 12). The corresponding figure of the film is called
the nodoid. The resultant of the internal pressure and the
surface-tension is equivalent to a pressure along the axis equal to that
due to a pressure p acting on a circle whose diameter is the conjugate
axis of the hyperbola.
When the conjugate axis of the hyperbola is made smaller and smaller,
the nodoid approximates more and more to the series of spheres touching
each other along the axis. When the conjugate axis of the hyperbola
increases without limit, the loops of the nodoid are crowded on one
another, and each becomes more nearly a ring of circular section,
without, however, ever 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,[3] Goldschmidt,[4] L.L. Lindelof and F.M.N. Moigno,[5] C.E.
Delaunay,[6] A.H.E. Lamarle,[7] A. Beer,[8] and V.M.A. Mannheim,[9] and
have been produced experimentally by Plateau[10] in the two different
ways already described.
[Illustration: FIG. 10.--Unduloid.]
[Illustration: FIG. 11.--Catenoid.]
[Illustration: FIG. 12.--Noboid.]
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.
[Illustration: FIG. 13.]
_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(x/a) 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 = 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. 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 A
and 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 T/r as before. If A1, C1 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 T/r. If, on the other
hand, the disks are at A2 and C2, 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 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. A and B, and let a third disk, C, be placed midway between. Let C
be slightly displaced towards A. If AC and CB are each less than half
the circumference of a disk the pressure on C will increase on the side
of A and diminish on the side of B. The resultant force on C will
therefore tend to oppose the displacement and to bring C back to its
original position. The equilibrium of C is therefore stable. It is easy
to show that if 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 C to be placed midway between A and B, and to be moved towards A,
the pressure on the side next A will diminish, and that on the side next
B will increase, so that the resultant force will tend to increase the
displacement, and the equilibrium of the disk 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 ("Memoire sur
la constitution des veines liquides lancees 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. 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
/ / T \
q = / ( -------- ).F(ka), (2)
\/ \[rho]a^3/
where, as before, T is the superficial tension, [rho] the density, and F
is given by the following table: --
+--------+--------+--------+--------+
| k^2a^2.| F(ka). | k^2a^2.| F(ka). |
+--------+--------+--------+--------+
| .05 | .1536 | .4 | .3382 |
| .1 | .2108 | .5 | .3432 |
| .2 | .2794 | .6 | .3344 |
| .3 | .3182 | .8 | .2701 |
| | | .9 | .2015 |
+--------+--------+--------+--------+
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 X 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
a^(3/2)
q_(-1) = --------- = .115d^(3/2) (3),
81 X .343
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 = 3log_e 10 = 6.9, so that t = .79d^(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 a1e^(q1t),
a2e^(q2t), in which q1 exceeds q2, their ratio is (a1/a2) e^{-(q1-q2)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^(3/2). Again, when the fluid is changed, the time
varies as [rho]^1/2 T^(-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
____
\/(2gh)
frequency = ------- (4)
4.508d
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
of it is propagated through the air. With respect to the limits of
pitch, Savart found that the note might be a fifth above, and more than
an octave below, that proper to the jet. According to theory, there
would be no well-defined lower limit; on the other side, the external
vibration cannot be efficient if it tends to produce divisions whose
length is less than the circumference of the jet. This would give for
the interval defining the upper limit [pi] : 4.508, which is very nearly
a fifth. In the case of Plateau's numbers ([pi] : 4.38) the discrepancy
is a little greater.
The detached masses into which a jet is resolved do not at once assume
and retain a spherical form, but execute a series of vibrations, being
alternately compressed and elongated in the direction of the axis of
symmetry. When the resolution is effected in a perfectly periodic
manner, each drop is in the same phase of its vibration as it passes
through a given point of space; and thence arises the remarkable
appearance of alternate swellings and contractions described by Savart.
The interval from one swelling to the next is the space described by the
drop during one complete vibration, and is therefore (as Plateau shows)
proportional _ceteris paribus_ to the square root of the head.
The time of vibration is of course itself a function of the nature of
the fluid and of the size of the drop. By the method of dimensions alone
it may be seen that the time of infinitely small vibrations varies
directly as the square root of the mass of the sphere and inversely as
the square root of the capillary tension; and it may be proved that its
expression is
/ / 3[pi][rho]V \
r = / ( ----------- ), (5)
\/ \ 8T /