In consequence of the rapidity of the motion some optical device is
necessary to render apparent the phenomena attending the disintegration
of a jet. Magnus employed a rotating mirror, and also a rotating disk
from which a fine slit was cut out. The readiest method of obtaining
instantaneous illumination is the electric spark, but with this Magnus
was not successful. The electric spark had, however, been used
successfully for this purpose some years before by H. Buff (_Liebigs
Ann._ lxxviii. 1851), who observed the _shadow_ of the jet on a white
screen. Preferable to an opaque screen is a piece of ground glass, which
allows the shadow to be examined from the farther side (Lord Rayleigh).
Further, the jet may be very well observed directly, if the illumination
is properly managed. For this purpose it is necessary to place it
between the source of light and the eye. The best effect is obtained
when the light of the spark is somewhat diffused by being passed (for
example) through a piece of ground glass.
The spark may be obtained from the secondary of an induction coil, whose
terminals are in connexion with the coatings of a Leyden jar. By
adjustment of the contact breaker the series of sparks may be made to
fit more or less perfectly with the formation of the drops. A still
greater improvement may be effected by using an electrically maintained
fork, which performs the double office of controlling the resolution of
the jet and of interrupting the primary current of the induction coil.
In this form the experiment is one of remarkable beauty. The jet,
illuminated only in one phase of transformation, appears almost
perfectly steady, and may be examined at leisure. In one experiment the
jet issued horizontally from an orifice of about half a centimetre in
diameter, and almost immediately assumed a rippled outline. The
gradually increasing amplitude of the disturbance, the formation of the
elongated ligament, and the subsequent transformation of the ligament
into a spherule, could be examined with ease. In consequence of the
transformation being in a more advanced stage at the forward than at the
hinder end, the ligament remains for a moment connected with the mass
behind, when it has freed itself from the mass in front, and thus the
resulting spherule acquires a backwards relative velocity, which of
necessity leads to a collision. Under ordinary circumstances the
spherule rebounds, and may be thus reflected backwards and forwards
several times between the adjacent masses. Magnus showed that the stream
of spherules may be diverted into another path by the attraction of a
powerfully electrified rod, held a little below the place of resolution.
Very interesting modifications of these phenomena are observed when a
jet from an orifice in a thin plate (Tyndall has shown that a pinhole
gas burner may also be used with advantage) is directed obliquely
upwards. In this case drops which break away with different velocities
are carried under the action of gravity into different paths; and thus
under ordinary circumstances a jet is apparently resolved into a
"sheaf," or bundle of jets all lying in one vertical plane. Under the
action of a vibrator of suitable periodic time the resolution is
regularized, and then each drop, breaking away under like conditions, is
projected with the same velocity, and therefore follows the same path.
The apparent gathering together of the sheaf into a fine and
well-defined stream is an effect of singular beauty.
In certain cases where the tremor to which the jet is subjected is
compound, the single path is replaced by two, three or even more paths,
which the drops follow in a regular cycle. The explanation has been
given with remarkable insight by Plateau. If, for example, besides the
principal disturbance, which determines the size of the drops, there be
another of twice the period, it is clear that the alternate drops break
away under different conditions and therefore with different velocities.
Complete periodicity is only attained after the passage of a _pair_ of
drops; and thus the odd series of drops pursues one path, and the even
series another.
Electricity, as has long been known, has an extraordinary influence upon
the appearance of a fine jet of water ascending in a nearly
perpendicular direction. In its normal state the jet resolves itself
into drops, which even before passing the summit, and still more after
passing it, are scattered through a considerable width. When a feebly
electrified body (such as a stick of sealing-wax gently rubbed upon the
coat sleeve) is brought into its neighbourhood, the jet undergoes a
remarkable transformation and appears to become coherent; but under more
powerful electrical action the scattering becomes even greater than at
first. The second effect is readily attributed to the mutual repulsion
of the electrified drops, but the action of feeble electricity in
producing apparent coherence was long unexplained.
It was shown by W. von Beetz that the coherence is apparent only, and
that the place where the jet breaks into drops is not perceptibly
shifted by the electricity. By screening the various parts with metallic
plates in connexion with earth, Beetz further proved that, contrary to
the opinion of earlier observers, the seat of sensitiveness is not at
the root of the jet where it leaves the orifice, but at the place of
resolution into drops. An easy way of testing this conclusion is to
excite the extreme tip of a glass rod, which is then held in succession
to the root of the jet, and to the place of resolution. An effect is
observed in the latter, and not in the former position.
The normal scattering of a nearly vertical jet is due to the _rebound_
of the drops when they come into collision with one another. Such
collisions are inevitable in consequence of the different velocities
acquired by the drops under the action of the capillary force, as they
break away irregularly from the continuous portion of the jet. Even when
the resolution is regularized by the action of external vibrations of
suitable frequency, as in the beautiful experiments of Savart and
Plateau, the drops must still come into contact before they reach the
summit of their parabolic path. In the case of a continuous jet, the
equation of continuity shows that as the jet loses velocity in
ascending, it must increase in section. When the stream consists of
drops following one another in single file, no such increase of section
is possible; and then the constancy of the total stream requires a
gradual approximation of the drops, which in the case of a nearly
vertical direction of motion cannot stop short of actual contact.
Regular vibration has, however, the effect of postponing the collisions
and consequent scattering of the drops, and in the case of a direction
of motion less nearly vertical, may prevent them altogether.
Under moderate electrical influence there is no material change in the
resolution into drops, nor in the subsequent motion of the drops up to
the moment of collision. The difference begins here. Instead of
rebounding after collision, as the unelectrified drops of clean water
generally, or always, do, the electrified drops _coalesce_, and then the
jet is no longer scattered about. When the electrical influence is more
powerful, the repulsion between the drops is sufficient to prevent
actual contact, and then, of course, there is no opportunity for
amalgamation.
These experiments may be repeated with extreme ease, and with hardly any
apparatus. The diameter of the jet may be about 1/20 in., and it may
issue from a glass nozzle. The pressure may be such as to give a
fountain about 2 ft. high. The change in the sound due to the falling
drops as they strike the bottom of the sink should be noticed, as well
as that in the appearance of the jet.
The actual behaviour of the colliding drops becomes apparent under
instantaneous illumination, e.g. by sparks from a Leyden jar. The jet
should be situated between the sparks and the eye, and the observation
is facilitated by a piece of ground glass held a little beyond the jet,
so as to diffuse the light; or the _shadow_ of the jet may be received
on the ground glass, which is then held as close as possible on the side
towards the observer.
In another form of the experiment, which, though perhaps less striking
to the eye, lends itself better to investigation, the collision takes
place between two still unresolved jets issuing horizontally from glass
nozzles in communication with reservoirs containing water. One at least
of the reservoirs must be insulated. In the absence of dust and greasy
contamination, the obliquely colliding jets may rebound from one another
without coalescence for a considerable time. In this condition there is
complete electrical insulation between the jets, as may be proved by the
inclusion in the circuit of a delicate galvanometer, and a low
electromotive force. But if the difference of potential exceed a small
amount (1 or 2 volts), the jets instantaneously coalesce. There is no
reason to doubt that in the case of the fountain also, coalescence is
due to _differences_ of potential between colliding drops.
If the water be soapy, and especially if it contain a small proportion
of milk, coalescence ensues without the help of electricity. In the case
of the fountain the experiment may be made by leading tap-water through
a Woulfe's bottle in which a little milk has been placed. As the milk is
cleared out, the scattering of the drops is gradually re-established.
In attempting to explain these curious phenomena, it is well to consider
what occurs during a collision. As the liquid masses approach one
another, the intervening air has to be squeezed out. In the earlier
stages of approximation the obstacle thus arising may not be important;
but when the thickness of the layer of air is reduced to the point at
which the colours of thin plates are visible, the approximation must be
sensibly resisted by the viscosity of the air which still remains to be
got rid of. No change in the capillary conditions can arise until the
interval is reduced to a small fraction of a wave-length of light; but
such a reduction, unless extremely local, is strongly opposed by the
remaining air. It is true that this opposition is temporary. The
question is whether the air can everywhere be squeezed out during the
short time over which the collision extends.
It would seem that the forces of electrical attraction act with peculiar
advantage. If we suppose that upon the whole the air cannot be removed,
so that the mean distance between the opposed surfaces remains constant,
the electric attractions tend to produce an instability whereby the
smaller intervals are diminished while the larger are increased.
Extremely local contacts of the liquids, while opposed by capillary
tension which tends to keep the surfaces flat, are thus favoured by the
electrical forces, which moreover at the small distances in question act
with exaggerated power.
A question arises as to the mode of action of milk or soap turbidity.
The observation that it is possible for soap to be in excess may here
have significance. It would seem that the surfaces, coming into
collision within a fraction of a second of their birth, would still be
subject to further contamination from the interior. A particle of soap
rising accidentally to the surface would spread itself with rapidity.
Now such an outward movement of the liquid is just what is required to
hasten the removal of intervening air. It is obvious that the effect
would fail if the contamination of the surface had proceeded too far
previously to the collision.
This view is confirmed by experiments in which other gases are
substituted for air as the environment of colliding jets. Oxygen and
coal-gas were found to be without effect. On the other hand, the more
soluble gases, carbon dioxide, nitrous oxide, sulphur dioxide, and
steam, at once caused union.]
_Stability of the Catenoid._--When the internal pressure is equal to the
external, the film forms a surface of which the mean curvature at every
point is zero. The only surface of revolution having this property is
the catenoid formed by the revolution of a catenary about its directrix.
This catenoid, however, is in stable equilibrium only when the portion
considered is such that the tangents to the catenary at its extremities
intersect before they reach the directrix.
To prove this, let us consider the catenary as the form of equilibrium
of a chain suspended between two fixed points A and B. Suppose the chain
hanging between A and B to be of very great length, then the tension at
A or B will be very great. Let the chain be hauled in over a peg at A.
At first the tension will diminish, but if the process be continued the
tension will reach a minimum value and will afterwards increase to
infinity as the chain between A and B approaches to the form of a
straight line. Hence for every tension greater than the minimum tension
there are two catenaries passing through A and B. Since the tension is
measured by the height above the directrix these two catenaries have the
same directrix. Every catenary lying between them has its directrix
higher, and every catenary lying beyond them has its directrix lower
than that of the two catenaries.
Now let us consider the surfaces of revolution formed by this system of
catenaries revolving about the directrix of the two catenaries of equal
tension. We know that the radius of curvature of a surface of revolution
in the plane normal to the meridian plane is the portion of the normal
intercepted by the axis of revolution.
The radius of curvature of a catenary is equal and opposite to the
portion of the normal intercepted by the directrix of the catenary.
Hence a catenoid whose directrix coincides with the axis of revolution
has at every point its principal radii of curvature equal and opposite,
so that the mean curvature of the surface is zero.
The catenaries which lie between the two whose direction coincides with
the axis of revolution generate surfaces whose radius of curvature
convex towards the axis in the meridian plane is less than the radius of
concave curvature. The mean curvature of these surfaces is therefore
convex towards the axis. The catenaries which lie beyond the two
generate surfaces whose radius of curvature convex towards the axis in
the meridian plane is greater than the radius of concave curvature. The
mean curvature of these surfaces is, therefore, concave towards the
axis.
Now if the pressure is equal on both sides of a liquid film, and if its
mean curvature is zero, it will be in equilibrium. This is the case with
the two catenoids. If the mean curvature is convex towards the axis the
film will move from the axis. Hence if a film in the form of the
catenoid which is nearest the axis is ever so slightly displaced from
the axis it will move farther from the axis till it reaches the other
catenoid.
If the mean curvature is concave towards the axis the film will tend to
approach the axis. Hence if a film in the form of the catenoid which is
nearest the axis be displaced towards the axis, it will tend to move
farther towards the axis and will collapse. Hence the film in the form
of the catenoid which is nearest the axis is in unstable equilibrium
under the condition that it is exposed to equal pressures within and
without. If, however, the circular ends of the catenoid are closed with
solid disks, so that the volume of air contained between these disks and
the film is determinate, the film will be in stable equilibrium however
large a portion of the catenary it may consist of.
The criterion as to whether any given catenoid is stable or not may be
obtained as follows:--
[Illustration: FIG. 14.]
Let PABQ and ApqB (fig. 14) be two catenaries having the same directrix
and intersecting in A and B. Draw Pp and Qq touching both catenaries, Pp
and Qq will intersect at T, a point in the directrix; for since any
catenary with its directrix is a similar figure to any other catenary
with its directrix, if the directrix of the one coincides with that of
the other the centre of similitude must lie on the common directrix.
Also, since the curves at P and p are equally inclined to the directrix,
P and p are corresponding points and the line Pp must pass through the
centre of similitude. Similarly Qq must pass through the centre of
similitude. Hence T, the point of intersection of Pp and Qq, must be the
centre of similitude and must be on the common directrix. Hence the
tangents at A and B to the upper catenary must intersect above the
directrix, and the tangents at A and B to the lower catenary must
intersect below the directrix. The condition of stability of a catenoid
is therefore that the tangents at the extremities of its generating
catenary must intersect before they reach the directrix.
_Stability of a Plane Surface._--We shall next consider the limiting
conditions of stability of the horizontal surface which separates a
heavier fluid above from a lighter fluid below. Thus, in an experiment
of F. Duprez ("Sur un cas particulier de l'equilibre des liquides,"
_Nouveaux Mem. del' Acad. de Belgique, 1851 et 1853_), a vessel
containing olive oil is placed with its mouth downwards in a vessel
containing a mixture of alcohol and water, the mixture being denser than
the oil. The surface of separation is in this case horizontal and
stable, so that the equilibrium is established of itself. Alcohol is
then added very gradually to the mixture till it becomes lighter than
the oil. The equilibrium of the fluids would now be unstable if it were
not for the tension of the surface which separates them, and which, when
the orifice of the vessel is not too large, continues to preserve the
stability of the equilibrium.
When the equilibrium at last becomes unstable, the destruction of
equilibrium takes place by the lighter fluid ascending in one part of
the orifice and the heavier descending in the other. Hence the
displacement of the surface to which we must direct our attention is one
which does not alter the volume of the liquid in the vessel, and which
therefore is upward in one part of the surface and downward in another.
The simplest case is that of a rectangular orifice in a horizontal
plane, the sides being a and b.
Let the surface of separation be originally in the plane of the
orifice, and let the co-ordinates x and y be measured from one corner
parallel to the sides a and b respectively, and let z be measured
upwards. Then if [rho] be the density of the upper liquid, and [sigma]
that of the lower liquid, and P the original pressure at the surface
of separation, then when the surface receives an upward displacement
z, the pressure above it will be P - [rho]gz, and that below it will
be P - [sigma]gz, so that the surface will be acted on by an upward
pressure ([rho] - [sigma])gz. Now if the displacement z be everywhere
very small, the curvature in the planes parallel to xz and yz will be
d^2z/dx^2 and d^2z/dy^2 respectively, and if T is the surface-tension
the whole upward force will be
/ d^2z d^2z \
T ( ---- + ---- ) + ([rho] - [sigma])gz.
\ dx^2 dy^2 /
If this quantity is of the same sign as z, the displacement will be
increased, and the equilibrium will be unstable. If it is of the
opposite sign from z, the equilibrium will be stable. The limiting
condition may be found by putting it equal to zero. One form of the
solution of the equation, and that which is applicable to the case of
a rectangular orifice, is
z = C sin px sin qy.