FOOTNOTES:[F]The information conveyed in this chapter was first published in a communication to the Mathematical and Physical Section of the British Association at Leicester in 1907.
[F]The information conveyed in this chapter was first published in a communication to the Mathematical and Physical Section of the British Association at Leicester in 1907.
[F]The information conveyed in this chapter was first published in a communication to the Mathematical and Physical Section of the British Association at Leicester in 1907.
In the present chapter will be described the splash that follows the entry of asolidsphere falling vertically into a liquid from a small height, and I should like to persuade the reader, if possible before he begins to read, or at any rate afterwards, to make a very simple experiment. Let a few child's marbles be taken—not glass "marbles," for these are seldom round enough or smooth enough, but what are sold in the toy-shops as "stone" marbles—and let one of these be well rubbed and polished with a dry handkerchief, and then dropped from a height of about 30 cm., or, say, 1 foot, into a deep bowl or basin of water, the bottom of which may be conveniently protected from breakage by a few folds of fine copper gauze.
If the polishing has been good, and the surface of the sphere has not been dimmed by subsequent handling with hot or greasy fingers, it will be observed that the splash is singularly insignificant, the sphereslipping noiselessly into the liquid with very little disturbance of the surface.
But if the same sphere be fished out of the water, and again let fall from the same height without being first dried, or, better still, if another marble be taken, which has been previously roughened with sand-paper, the resulting splash is totally different. There is now a noise of bubbles, which may be seen rising through the liquid, and a tall jet is seen to be tossed into the air.
(1) THE SPLASH OF A ROUGH SPHERE.
To understand the cause of this really surprising difference we must turn to the photographic record, and we will take first the case of a rough sphere falling into water to which milk has been added for the sake of clearness in the photographs. The diameter of the sphere was 1·5 cm. (or 3/5 inch), and the height of fall 15 cm., or just about 6 inches. The sphere on each occasion was fished out, redried, and re-roughened with sand- or emery-paper. Examination of the first photographs of Series V shows that the liquid, instead of flowing over and wetting the surface of the sphere, is driven violently away, so that as far as can be seen from above the upper portion is, at first at any rate, unwetted by the liquid. The crater that is subsequently formed is very similar to that which was thrown by the liquid drop in Series I, the main difference being that in the present-case the crater is thinner in the wall and more regular. This greater regularity is chiefly to be attributed to the fact thatthe solid sphere enters the liquid with a true spherical form, and is not distorted by the oscillations and tremors which disturb a falling drop. The gradual thickening of the wall and the corresponding reduction in the number of lobes as the subsidence proceeds is beautifully shown in Figs. 7, 8, 9, and 10, the last-mentioned figure being hardly distinguishable from the corresponding Fig. 9 of Series I,p. 17. This stage is in each case reached in about 58/1000 of a second.
SERIES V
Rough sphere. "Basket splash."
Diameter of sphere, 1·5 centim. Height of fall, 15 centim.
SERIES V
Rough sphere—(continued).
Now from the depths of the crater there rises with surprising velocity the exquisite jet of Fig. 11, which in obedience to the law of segmentation at once splits up in its upper portion into little drops, while at the same time it gathers volume from below, and rises ultimately as a tall, graceful column to a height which may be even greater than that from which the sphere fell. This is the emergent jet which one sees with the naked eye whenever a sufficiently rough sphere is dropped from a small height into water, but if we are to ascertain how this column originates, we must follow the sphere below the surface of the liquid. The arrangement already described onp. 69enables this to be done. We let the sphere fall into clear water contained in a narrow, flat-sided, inverted clock-shade and illuminate this from behind while the camera stands straight in front.
SERIES V
Rough sphere—(continued).
In this manner were obtained the photographs of Series VI, which require a little explanation. In the first figure we see the sphere just entering the liquid. The faint horizontal line shows the level of the surface. Above this line we see the internally reflected image of the part that has already entered, while still higher in the figure may be discerned the summit of the sphere itself. The slight lateral displacement of the part below the surface is due to refraction consequent on the camera having been set with its optic axis not quite perpendicular to the face of the vessel. In the subsequent figures it will be observed that the sphere, as it descends, drags with it the surface of the liquid in the form of a gradually deepening pocket or bag, the upper part of the sphere being for a long time quite unwetted by the liquid.
The sides of this pocket or bag of air not being quite smooth, give a somewhat distorted appearance to the sphere within. Also, since the sides are sloping, their reflected image in the level surface slopes in the opposite direction and produces an angle where the two meet. This angle marks very clearly the level of the surface. Above the surface-line in Figs. 2 to 5 is seen the beaded lip of the crater which we have already viewed from above, but this is somewhat out of focus, for the camera had to be focused on the sphere as seen under water, and the effect of the water is to bring the sphere optically nearer. Hence only the nearer part of the crater, i.e. the middle part of the front edge, is distinctly shown.
SERIES VI
The splash of a rough sphere as seen below the surface.
Diameter, 1·5 centim. Height of fall, 15 centim.
Coming now to Fig. 6, we perceive that the long cylindrical hollow has begun to divide. In this spontaneous division we have another illustration of the law of instability which regulated the sub-division of the jets and columns of earlier series. This law is the same whether the cylinder be of air surrounded by liquid or of liquid surrounded by air. Hitherto we have only seen it operating in jets of liquid in air; now we have a jet of air in a liquid.
The lower part of the long cylinder of air splits off into a bubble just behind the sphere, and follows in its wake to the bottom of the vessel, and is only detached and rises to the surface when the sphere strikes the bottom. Many years ago, through the kindness of the curator of the Brighton Aquarium, I was enabled to watch this bubble of air following in the wake of the sphere to the bottom of the deepest tank.
Figs. 7, 8, and 9 show the two parts gradually separating.
SERIES VI—(continued)
Scale reduced to about 7/10.
Fig. 10 shows specially well the ripples on the surface of the descending bubble. These undulations sometimes become so accentuated that the upper part of this descending bubble is detached, and then the curious phenomenon may be seen of this detached part still following the rest downwards through the liquid with an unsteady, lurching motion.
Meanwhile the upper half of the divided air-column is seen in Fig. 9 to resemble a deep basin which now rapidly fills up by the influx of liquid from all sides. It is from the confluence of this inflowing liquid into channels which necessarily narrow as the centre is approached that the great velocity with which the liquid spirts upwards is obtained. In Fig. 11 the jet is just discernible above the surface, and in Fig. 13 it is well-established.
SERIES VI—(continued)
On increasing the height of fall of a rough sphere to 60 cm., we obtain a higher crater which closes and forms a bubble, exactly as when we increased the height of fall of a liquid drop. The process as viewed from above the surface is shown in Series VII. The first figure of this series shows very well how completely the liquid is driven away from the surface of the sphere the first moment of contact. The subsequent crater and bubble are of exquisite delicacy. This bubble, though it closes completely as in the last figure, is doomed to almost immediate destruction. For we see, on looking below the surface, that the proceedings there are of the same kind as in the case of the lower fall already described, and result in the formation of an upward-directed jet.
SERIES VII
Rough sphere falling 60 cm. Scale 3/4.
Thus the first three figures of Series VIII show the last moments of a bubble which has burst, spontaneously, and so has made way for the jet of Fig. 3. (These are taken from a splash into petroleum with 24·5 cm. fall.) But the last two figures, 4 and 5 (taken with a 32 cm. fall), show how a bubble which might otherwise have been permanent, is stabbed by the rising jet and destroyed. With water and 60 cm. fall the jet appears sometimes to rise quite unimpeded, and sometimes to be checked by the still closed bubble.
Before leaving the splash of a rough sphere, I desire to call the reader's attention to another point.
Such figures as 7, 9, and 10 of Series V,p. 77, show that the surface of the liquid beyond the walls of the crater is still flat and undisturbed; yet we now know from the corresponding Figs. 5, 6, and 7 of Series VI,p. 83, that a large volume of liquid has been displaced, much larger than the quantity required to form the crater wall. The inference is that the level of the surface has been slightly raised even at a great distance from the place of the splash. Figs. 7, 8, and 9 of Series VI themselves confirm the impression of the undisturbed flatness of the surface at even a small distance from the splash.
(2) THE SPLASH OF A SMOOTH SPHERE.
The reader who has been sufficiently interested to make for himself the simple experiment suggested at the beginning of this chapter, will have already realized that the splash of a smooth sphere is totally different from that of a rough one. The photographsof Series IX show that the difference is quite pronounced from the first instant of contact. In this series the sphere was of polished stone 3·2 cm. in diameter and fell 14 cm. The scale of magnification is 3/4. The second figure shows that the liquid, instead of being driven away from the surface as was the case with a rough sphere, now rises up in a thin, closely-fitting sheath which (see Fig. 3) completely envelops the sphere even before its summit has reached the water-level. Figs. 4 and 5 show the comparatively insignificant column that remains to mark the spot where the sphere has entered. Fig. 6 was the result of a lucky accident, which left the sphere rough on the right-hand side, smooth on the left. Nothing could show better than this photograph the essential difference between the two splashes.
SERIES VIII
Rough sphere. Splashes viewed below the surface.
The bursting of the bubble.
From a splash into Petroleum
24·5 cm. fall.
From a splash into Petroleum
32 cm. fall.
The reader's attention is directed to the remarkably deep furrows which characterize the whole sheath in Fig. 3 and the left-hand (smooth splash) part in Fig. 5. About these furrows we shall have something to say later.
A better idea of the extreme thinness of the enveloping sheath is obtained when the illumination is from behind as in Series X, in which the sphere was of highly polished serpentine stone 2·57 cm. (or just over 1 inch) in diameter, the fall being 14 cm. (or not quite 6 inches).
SERIES IX
The "sheath" splash of a smooth sphere.
Examination of either Series IX or Series X shows that with the smooth sphere as with the rough the amount of water lifted above the surface in theimmediate neighbourhood of the splash is much less than the whole volume displaced, so that we are again driven to the conclusion that the surface at even a considerable distance must be bodily lifted without its flatness being sensibly disturbed. This conclusion was confirmed by a direct experiment. The not very wide vessel of Fig. A was taken and filled brimful with milk, and the lower edge of a card millimetre scale was placed just in contact with the liquid surface at one side. The reader should notice that the liquid is not quite up to the level of the spout on the right-hand side of this figure. Then the sphere was dropped in and the photograph of Fig. B was taken when the sphere was about two-thirds immersed. The rise at the edge of the scale is about 3 millimetres, and there is an apparently equal rise at the spout, where, however, the surface appears quite flat.
Fig. A
Fig. B
It seems probable, then, that whenever a stone is thrown into a lake the impulse accompanying its entry travels with the velocity of a compressional wave (i.e. with the velocity of sound) through theliquid, and is therefore almost instantly felt and produces a minute rise of level even in remote parts of the lake long before the arrival of any ripple or surface disturbance.
SERIES X
Polished serpentine sphere falling 14 cm. into water.
It may here be observed that whether the sphere be rough or smooth, its size makes little or no difference in the character of the splash, within a range of diameter from 12 to 32 millimetres—i.e. from about 1/2 inch to about 1-1/3 inches. No doubt with a very large sphere, taking a long time to enter, the splash would be controlled more by gravity than by surface-tension, but so long as the sphere is within the limits mentioned this is not the case unless the height of fall be made very small indeed.
THE INFLUENCE OF VELOCITY.
If we gradually increase the velocity with which a well-polished sphere enters the liquid we find that there is a gradual transition from the silent "smooth" or "sheath" splash taking down no air and giving rise to only an insignificant column, to the noisy, "rough," "basket" splash taking down much air and throwing up a tall and conspicuous jet. Thus in the fourth figure of Series XI, in which the height of fall has been increased from 15 to 60 cm. (i.e. from 6 inches to 2 feet), the sphere being of polished ivory, we see that the enveloping sheath has in many places broken away from the surface before the summit has been covered. It is well known that a sphere moving through a liquid pushes away the liquid in front of it, which flowing round closes in at the back of the sphere.
Although the surface round the column of Fig. 6 is still very flat, the liquid just below it must be streaming inwards,[G]as is indicated by the radial striæ. To the meeting of these converging streams we must attribute the large access of liquid that is forced up into the column, whose subsequent toppling into drops is accompanied by the curious, characteristic, lop-sided curvature of the later figures.
SERIES XI
Polished ivory sphere, 1·9 centim. in diameter, falling 60 cm. into water mixed with milk.
Series XII shows how even with a very highly polished metal sphere falling into water from the still greater height of 100 cm. the characteristic sheath of the "smooth" splash is no longer so closely fitting even at an early stage, but is beginning to resemble the earlier stages of the basket-shaped crater of the "rough" splash; yet no air was taken down at this height.
The transition was also watched by means of photographs taken below the water-line.
It may be well here to guard the reader against a possible misconception. The curved outline of the liquid in these photographs does not represent the path followed by the particles. Each particle must have travelled in a nearly straight line from the moment it left the surface of the sphere, and must still be moving upwards and outwards. Gravity has not had time to produce any sensible displacement. This applies also to the curved outlines in many other early figures.
INFLUENCE OF THE CONDITION OF THE SURFACE.
By very careful rubbing of such a polished, steel sphere, it was found possible to increase the height of fall to 162·5 cm. (well over 5 feet) and yet to secure a perfectly "airless," "smooth" splash. But the equilibrium of the splash, if I may use the phrase, is, at this high velocity of entry (564 cm. per sec., or about 18 feet per sec.), very unstable, and was found to depend on minute differences in the condition of the surface.How minute this difference may be, which yet makes the whole difference in the character of the splash, may be gathered from the following extract from the original paper:—
SERIES XII
Smooth sphere of polished serpentine falling 100 centim. into water. Scale 3/4.
"A polished steel sphere 15·9 cm. in diameter was found (by naked-eye observation) to give an airless splash when falling into water from a height of 132·5 cm.; at 137·5 cm., there was much air taken down. This observation at 137·5 cm. was repeated three times, observer C. doing the polishing. Then observer W. polished, and the splash was firstnearlyairless and thenquiteairless. Then, by persevering in the rubbing, the height of fall was gradually raised to 162·5 cm., and a perfectly airless splash was secured, and even at 172·5 cm. the record was 'very little air indeed.'
"Again, a polished marble sphere 2·57 cm. in diameter falling into water from a height of 112 cm. was found to take down 'much air' when rubbed with a certain clean handkerchief A, and 'none at all, or only very little,' when rubbed with clean handkerchief B. This result was confirmed four times with B and five with A. These handkerchiefs were subsequently examined under the microscope, but were found to be extremely similar, and the cause of the difference remained for the time beyond conjecture.
"On another occasion, of two similar nickel-plated steel spheres, each 19 millimetres in diameter, and each treated in exactly the same way, falling 22 cm. into paraffin oil, one would always take down much air and the other little or none, and againmicroscopic examination showed only a very slight difference in the surfaces."
By wetting the surface of a smooth sphere we can always convert a smooth or "sheath" splash into a rough or "basket" splash. Thus when the ivory sphere (which when dry and well-polished gave, with a fall of 60 cm., the splash of Series XI,p. 97), was allowed to fallwetinto the liquid, all other circumstances remaining the same, the splash of Series XIII,p. 103, was obtained, which is entirely different from the first. The wetting was effected by dipping the sphere into the bowl of milky water into which it was to fall, and then shaking off as much as possible of the adherent liquid, but in all cases the splash quickly became unsymmetrical, probably through the liquid, during the fall, drifting to one side of the sphere.
INFLUENCE OF THE NATURE OF THE LIQUID.
The nature of the liquid employed has a great influence in determining whether at a given height the splash shall be "rough" or "smooth."
Thus with paraffin oil the maximum height that could be reached with an airless splash with highly polished nickel-plated spheres, well rubbed on a selvyt cloth, was found to be only 24·7 cm. (about 10 inches), but, with water, a fall of 160 cm. (over 5 feet) could be reached. The paraffin oil used in these experiments had, at a temperature of 12°·5 centigrade, a specific gravity ·840 and a surface-tension about ·39 of that of water. Since only a small increase of height was required with this liquid to make a smooth sphere givethe same splash as a rough one, this liquid was found much more convenient than water in investigating the transition.
When water is made more viscid by the gradual addition of glycerine,[H]the surface-tension and the specific gravity are but little altered though the viscosity is steadily and sensibly increased. An admixture of two parts of glycerine to fifty-one of water produced no perceptible change in the splashes observed. When the glycerine was increased to six volumes in fifty-one of water, though this made the viscosity half as great again, the change was noticeable but still slight, the chief difference being, with a smooth sphere, the greater salience of the ribs or flutings in some of the earlier stages of the glycerine splash, and the much greater reluctance of the subsequent jets to topple into droplets. This latter feature is well seen in the first figure onpage 105, showing the entry of a smooth sphere of polished serpentine stone into this glycerine mixture from a height of 50 cm.
SERIES XIII
Splash of a smooth wet sphere.
With pure glycerine, which is much more viscous, the splash of the same polished serpentine sphere falling from 75 cm. (about 2-1/2 feet), is shown in Series XIV. In the original photographs the radial furrows on the right-hand side of Fig. 2 are very pronounced, and even in Fig. 1 the fluting of the film is seen to be already well developed on the left-hand side; butthese details have proved rather too delicate for reproduction in the plate. Two photographs taken of stage 2 had each of them an isolated jet, owing probably to the fact that when working with so sticky a liquid it was difficult to avoid contaminating the cloth on which the sphere was each time repolished after washing in water, with the result that the spheres behaved as if locally rough. The relatively great length and height of this jet brings out well the part played by viscosity, both in delaying segmentation into droplets and also in hindering the flow of the rest of the liquid sheath which has remained in contact with the sphere.
With a rough sphere falling into pure glycerine from the same height of 75 cm., except for an occasional jet that may escape as in Fig. 4 of Series XV, the proceedings are uneventful, as a glance at the series will show. With the same height of fall into water we should have had an exquisite crater fringed with a multitude of fine jets, and ultimately closing to form a bubble. We thus see how little play is given to the action of the surface-tension in a very viscous liquid.
Polished stone sphere falling 15 centim. into water mixed with glycerine.
SERIES XIV
Polished stone sphere falling 75 centim. into pure glycerine. Scale 9/10.
SERIES XV
Rough sphere falling 75 centim. into pure glycerine. Scale 1/1.
THE INFLUENCE OF TEMPERATURE.
It was found that if a polished sphere was heated in boiling water, quickly rubbed dry, and let fall while still hot, a very marked difference was produced. With the sphere hot, the height of fall can be much increased before the splash becomes "rough." Thus with paraffin oil, the height with a nickel-plated sphere rose from 22·2 cm. to 29·3 cm., and with water from 157 cm. to 234 cm.
THE REMARKABLE INFLUENCE OF A FLAME HELD NEAR THE LIQUID, AND TRAVERSED BY THE SPHERE IN ITS FALL.
In our search for the explanation of the difference between the rough and the smooth splash, it occurred to us to let the smooth sphere drop through a flame held near the liquid, and the result was very remarkable. With paraffin oil (and the sphere hot) the airless height now rose from 29·3 cm. to 45·3 cm., and with water and a cold sphere, it rose from 157 cm. to over 258 cm., which was the greatest height that the laboratory would permit. Either the luminous flame of a bat's-wing burner or the flame of a Bunsen burner held nearly horizontal produces the effect, provided the flame is held near enough to the surface of the liquid, and it is a very striking experiment to let the polished sphere fall several times from a height which gives a large volume of bubbles rising with much noise to the surface, and then to let it fall through the flame, and to observe the completechange in the phenomenon. On a sphere already roughened the flame has no observable effect.
THE SUPPOSITION OF ELECTRIFICATION TESTED AND REJECTED.
The behaviour with a flame led at first to the supposition that we had to deal with an electrical phenomenon, for a flame would certainly discharge completely any electrified sphere passing through it, and it appeared reasonable to suppose that the sphere might become electrified by friction with the air through which it fell.
It required a long series of experiments, into the details of which I need not now ask my readers to enter, to prove that this tempting explanation was untenable, and that there was no reason to believe that electrification had anything to do with the matter.
EXPERIMENTS IN VACUO.
It remained to examine what part was played by the air in the whole transaction. This could only be settled by removing the air and letting the spheres, whether rough or smooth, fall through a vacuum into the liquid, or rather through a space occupied only by the vapour of the liquid in use.
Instantaneous photographs obtained under these conditions showed that the presence of the air has no material influence on the early course of the splash, and that a sphere which gives a "smooth" splash in air will give a "smooth" splash in vacuo, while if the splash is "rough" in air, it will also be "rough" in vacuo.