PRINCIPLES INVOLVED
The reader's attention has now been directed to various features which,
with certain modifications, will be found in many of the splashes that
we shall examine; but so far the language used has been simply
descriptive and in no way explanatory. Instead of going on to describe
other splashes in the same way, and thus to accumulate a great mass of
uncooerdinated descriptive detail, it will be better to pause for a
moment in order to become acquainted with certain principles connected
with the behaviour of liquids, the application of which will go a long
way towards explaining what we see going on in any splash.
The first principle to be understood is that the surface layers of
any liquid behave like a uniformly stretched skin or membrane, which
is always endeavouring to contract and to diminish its area. If the
surface is flat, like the surface of still liquid in a bowl, this
surface-tension has only the effect of exerting a small inward pull on
the walls of the bowl. But if the surface is curved, with a convexity
outwards, then the surface layers, on account of their tension, press
the interior liquid back, and thus tend to check the growth of any
protuberance; while, on the other hand, if the surface is concave
outwards, then the surface-tension tends to pull the interior liquid
forward, and so to diminish the concavity.
Direct evidence of this surface-tension is easy to cite. We have it in
any pendent drop, such as any of those shown in the accompanying
figures.
[Illustration: WATER.
TURPENTINE.
Pendent drops (magnified 2-1/4 times).]
If we ask ourselves how it is that the liquid in the interior of one of
these drops does not flow out, pressed as it is by the liquid above it,
the answer is that everywhere the stretched skin presses it back. A
soap-bubble too presses on the air in its interior, both the outside
layers and the inside layers of the thin film being curved over the
interior space. This is the reason that a soap-bubble blown on the bowl
of a pipe will slowly collapse again if we remove the stem of the pipe
from our mouth. The bubble drives the interior air back through the
pipe. And it is easy to show that if two soap-bubbles be blown on the
ends of two tubes which can be connected together by opening a tap
between them, then the smaller will collapse and blow out the larger.
The reason of this is that in the bubble of smaller radius the surface
layers are more sharply curved, and therefore exert a greater pressure
on the air within. Thus if a strap be pulled at each end with a total
tension T and bent over a solid cylinder of small radius, as in Fig. 6,
it is easy to see that the pressure on the surface of the part of the
cylinder touched by the strap is less than if the strap be bent over an
equal area on a cylinder of larger radius (Fig. 7). The tension of the
surface layers of a liquid causes them to act on the liquid within,
exactly as does the stretched strap on the solid in these figures. If at
any place the liquid presents, as it generally does, not a cylindrical
surface, but one with curvature in two directions, then the pressure
corresponds to what would be produced by two straps crossing at right
angles, laid one over the other, each with the curvature of the surface
in its direction (Fig. 8).
[Illustration: FIG. 6]
[Illustration: FIG. 7]
[Illustration: FIG. 8]
[Illustration: FIG. 9]
We can now understand why the drop that has been lying on the
watch-glass should oscillate in its descent. The sharp curvature of the
edge AA of the drop (see Fig. 9) tells us that the liquid there is
pushed back by the pressure of the stretched surface layers, and when
the supporting glass is removed the sides of the drop move inwards,
driving the liquid into the lower part, the tendency being to make the
drop spherical, and so to equalize the pressure of the surface at all
points. But in the process the liquid overshoots the mark, and the drop
becomes elongated vertically and flattened at the sides. This causes the
curvature at top and bottom to be sharper than at the sides, and on this
account the back-pressure of the ends soon checks the elongation and
finally reverses the flow of liquid, and the drop flattens again. As an
example of the way in which a _concavity_ of the surface is pulled out
by the surface-tension may be cited the dimples made by the weight of an
aquatic insect, where its feet rest on the surface without penetrating
it.
This same surface-tension checks the rise of the crater, and would cause
it to subside again even without the action of gravity. Thus the
pressures of the sharply curved crater-edge on the liquid between the
crater walls are indicated by the dotted arrows in Fig. 10, and arise
from the surface-tension indicated by the full arrows. During the early
part of the splash the surface-tension is more important than gravity in
checking the rise of the walls. For, as the numbers show, the crater of
Series I is already at about its maximum height in No. 4, i.e. about
seven-thousandths of a second after first contact. In this time the fall
due to gravity would be only about 1/100 of an inch. Thus if gravity had
not acted the crater would only have risen about 1/100 of an inch
higher. The same reasoning applies to the rise of the central column,
but here the curvature at the summit is much less sharp. The numbers
show that the column reaches its maximum height in about 5/100 of a
second after its start in No. 10, and in this time the fall due to
gravity is about half an inch, so that gravity has reduced the height by
this amount.
[Illustration: FIG. 10]
The second principle which I will now mention enables us to explain the
occurrence of the jets and rays at the edge of the crater and their
splitting into drops.
It was shown in 1873 by the blind Belgian philosopher, Plateau,[D] that
a cylinder of liquid is not a figure of stable equilibrium if its length
exceeds about 3-1/7 times its diameter. Thus a long cylindrical rod of
liquid, such as Fig. 11, if it could be obtained and left for a moment
to itself, would at once topple into a row of sensibly equal,
equidistant drops, the number of which is expressed by a very simple
law, viz. that for every 3-1/7 times the diameter there is a drop, or
that the distance between the centres of the drops is equal to the
circumference of the cylinder.
[Illustration: FIG. 11]
The cause of this instability is the action of the same skin-tension
that we have already spoken of. Calculation shows, and Plateau was able
to confirm the calculation by experiment, that if through chance
agitations lobes are formed at a nearer distance apart than 3-1/7 times
the radius, with hollows between as in the accompanying Fig. 12, then
the curvatures will be such as to make the skin-tension push the
protuberances back and pull the hollows out. But if the protuberances
occur at any greater distances apart than the length of the perimeter,
then the sharper curvature of the narrower parts will drive the liquid
there into the parts already wider, thus any such an initial accidental
inequality of diameter will go on increasing, or the whole will topple
into drops.
[Illustration: FIG. 12]
At the last moment the drops are joined by narrow necks of liquid (Fig.
13), which themselves split up into secondary droplets (Fig. 14).
[Illustration: FIG. 13]
[Illustration: FIG. 14]
What we have said of a straight liquid cylinder applies also to an
annulus of liquid made by bending such a cylinder into a ring. This also
will spontaneously segment or topple into drops according to the same
law.[E] Now the edge of the crater is practically such a ring, and it
topples into a more or less regular set of protuberances, the liquid
being driven from the parts between into the protuberances.
Now while the crater is rising the liquid is flowing up from below
towards the rim, and the spontaneous segmentation of the rim means that
channels of easier flow are created, whereby the liquid is driven into
the protuberances, which thus become a series of jets. These are the
jets or arms which we see at the edge of the crater. Examination with a
lens of some of the craters will show that the lines of easier flow
leading to a jet are often marked by streaks of lamp-black in Series I,
or by streaks of milk in Series II. This explanation of the formation of
the jets applies also to a similar phenomenon on a much larger scale,
with which the reader will be already familiar. If he has ever watched
on a still day, on a straight, slightly shelving sandy shore, the waves
that have just impetus enough to curl over and break, he will have
noticed that up to a certain moment the wave presents a long, smooth,
horizontal cylindrical edge (see Fig. 15-a) from which, at a given
instant, are shot out an immense array of little jets which speedily
break into foam, and at the same moment the back of the wave, hitherto
smooth, is seen to be furrowed or combed (see Fig. 15-b). The jets are
due to the segmentation of the cylindrical rim according to Plateau's
law, and the ridges between the furrows mark the lines of easier flow
determined by the position of the jets.
[Illustration: FIG. 15-b
FIG. 15-a
Diagrams of a breaking wave.]
The tendency of the central column of Series I to separate into two
parts is only another illustration of the same instability of a liquid
cylinder. The column, however, is much thicker than the jets, and its
surface is therefore less sharply curved, and consequently the inward
pressure of the stretched curved surface is relatively slight and the
segmentation proceeds only slowly. Since this segmentation must
originate in some accidental tremor, we see how it is that the summit
of the column may succeed in separating off on some occasions and not on
others. As a matter of fact, the height of fall for this particular
splash was purposely selected, so that the column thrown up should
_just_ not succeed in dividing in order that the formation of the
subsequent ripples might not be disturbed by the falling in of the drops
split off. But, as the reader will have perceived, the margin allowed
was not quite sufficient.
The two principles that I have now explained, viz. the principle of the
skin-tension, and the principle of the instability and spontaneous
segmentation of a liquid cylinder, jet, or annulus, will go far to
explain much that we shall see in any splash, but it is well that the
reader should realize how much has been left unexplained. Why, for
example, should the crater rise so suddenly and vertically immediately
round the drop as it enters? Why should the drop spread itself out as a
lining over the inside of the crater, turning itself inside out, as it
were, and making an inverted umbrella of itself? Why when the crater
subsides should it flow inwards rather than outwards, so as to throw up
such a remarkable central column?
These questions, which demand that we should trace the motion of every
particle of the water back to the original impulse given by the impact
of the drop, are much more difficult to answer, and can only be
satisfactorily dealt with by a complicated mathematical analysis.
Something, however, in the way of a general explanation will be given in
a later chapter.
FOOTNOTES:
[D] _Statique Experimentale et Theorique des Liquides._
[E] See Worthington on the "Segmentation of a Liquid Annulus," _Proc.
Roy. Soc._, No. 200, 1879.