Let X be the force which must be applied in a horizontal direction to either plate to keep it from approaching the other, then the forces acting on the first plate are T + X in the negative direction, and T sin α1+ ½gρh1² in the positive direction. Hence
X = ½gρh1² − T(1 − sin α1).
For the second plate
X = ½gρh2² − T(1 − sin α2).
Hence
X = ¼gρ(h1² + h2²) − T{1 − ½(sin α1+ sin α2)},
or, substituting the values of h1and h2,
the remaining terms being negligible when a is small. The force, therefore, with which the two plates are drawn together consists first of a positive part, or in other words an attraction, varying inversely as the square of the distance, and second, of a negative part of repulsion independent of the distance. Hence in all cases except that in which the angles α1and α2are supplementary to each other, the force is attractive when α is small enough, but when cos α1and cos α2are of different signs, as when the liquid is raised by one plate, and depressed by the other, the first term may be so small that the repulsion indicated by the second term comes into play. The fact that a pair of plates which repel one another at a certain distance may attract one another at a smaller distance was deduced by Laplace from theory, and verified by the observations of the abbé Haüy.
A Drop between Two Plates.—If a small quantity of a liquid which wets glass be introduced between two glass plates slightly inclined to each other, it will run towards that part where the glass plates are nearest together. When the liquid is in equilibrium it forms a thin film, the outer edge of which is all of the same thickness. If d is the distance between the plates at the edge of the film and Π the atmospheric pressure, the pressure of the liquid in the film is Π − (2T cos α)/d, and if A is the area of the film between the plates and B its circumference, the plates will be pressed together with a force
and this, whether the atmosphere exerts any pressure or not. The force thus produced by the introduction of a drop of water between two plates is enormous, and is often sufficient to press certain parts of the plates together so powerfully as to bruise them or break them. When two blocks of ice are placed loosely together so that the superfluous water which melts from them may drain away, the remaining water draws the blocks together with a force sufficient to cause the blocks to adhere by the process calledRegelation.
[An effect of an opposite character may be observed when the fluid is mercury in place of water. When two pieces of flat glass are pressed together under mercury with moderate force they cohere, the mercury leaving the narrow crevasses, even although the alternative is a vacuum. The course of events is more easily followed if one of the pieces of glass constitutes the bottom, or a side, of the vessel containing the mercury.]
In many experiments bodies are floated on the surface of water in order that they may be free to move under the action of slight horizontal forces. Thus Sir Isaac Newton placed a magnet in a floating vessel and a piece of iron in another in order to observe their mutual action, and A.M. Ampère floated a voltaic battery with a coil of wire in its circuit in order to observe the effects of the earth’s magnetism on the electric circuit. When such floating bodies come near the edge of the vessel they are drawn up to it, and are apt to stick fast to it. There are two ways of avoiding this inconvenience. One is to grease the float round its water-line so that the water is depressed round it. This, however, often produces a worse disturbing effect, because a thin film of grease spreads over the water and increases its surface-viscosity. The other method is to fill the vessel with water till the level of the water stands a little higher than the rim of the vessel. The float will then be repelled from the edge of the vessel. Such floats, however, should always be made so that the section taken at the level of the water is as small as possible.
[The Size of Drops.—The relation between the diameter of a tube and the weight of the drop which it delivers appears to have been first investigated by Thomas Tate (Phil. Mag.vol. xxvii. p. 176, 1864), whose experiments led him to the conclusion that “other things being the same, the weight of a drop of liquid is proportional to the diameter of the tube in which it is formed.” Sufficient time must of course be allowed for the formation of the drops; otherwise no simple results can be expected. In Tate’s experiments the period was never less than 40 seconds.
The magnitude of a drop delivered from a tube, even when the formation up to the phase of instability is infinitely slow, cannot be calculated a priori. The weight is sometimes equated to the product of the capillary tension (T) and the circumference of the tube (2πa), but with little justification. Even if the tension at the circumference of the tube acted vertically, and the whole of the liquid below this level passed into the drop, the calculation would still be vitiated by the assumption that the internal pressure at the level in question is atmospheric. It would be necessary to consider the curvatures of the fluid surface at the edge of attachment. If the surface could be treated as a cylindrical prolongation of the tube (radius a), the pressure would be T/a, and the resulting force acting downwards upon the drop would amount to one-half (πaT) of the direct upward pull of the tension along the circumference. At thisrate the drop would be but one-half of that above reckoned. But the truth is that a complete solution of the statical problem for all forms up to that at which instability sets in, would not suffice for the present purpose. The detachment of the drop is adynamicaleffect, and it is influenced by collateral circumstances. For example, the bore of the tube is no longer a matter of indifference, even though the attachment of the drop occurs entirely at the outer edge. It appears that when the external diameter exceeds a certain value, the weight of a drop of water is sensibly different in the two extreme cases of a very small and of a very large bore.
But although a complete solution of the dynamical problem is impracticable, much interesting information may be obtained from the principle of dynamical similarity. The argument has already been applied by Dupré (Théorie mécanique de la chaleur, Paris, 1869, p. 328), but his presentation of it is rather obscure. We will assume that when, as in most cases, viscosity may be neglected, the mass (M) of a drop depends only upon the density (σ), the capillary tension (T), the acceleration of gravity (g), and the linear dimension of the tube (a). In order to justify this assumption, the formation of the drop must be sufficiently slow, and certain restrictions must be imposed upon the shape of the tube. For example, in the case of water delivered from a glass tube, which is cut off square and held vertically, a will be the external radius; and it will be necessary to suppose that the ratio of the internal radius to a is constant, the cases of a ratio infinitely small, or infinitely near unity, being included. But if the fluid be mercury, the flat end of the tube remains unwetted, and the formation of the drop depends upon the internal diameter only.
The “dimensions” of the quantities on which M depends are:—
of which M, a mass, is to be expressed as a function. If we assume
M ∝ Tx·gy·σz·au,
we have, considering in turn length, time and mass,
y − 3z + u = 0, 2x + 2y = 0, x + z = 1;
so that
y = −x, z = 1 − x, u = 3 − 2x.
Accordingly
Since x is undetermined, all that we can conclude is that M is of the form
where F denotes an arbitrary function.
Dynamical similarity requires that T/gσa² be constant; or, if g be supposed to be so, that a² varies as T/σ. If this condition be satisfied, the mass (or weight) of the drop is proportional to T and to a.
If Tate’s law be true, thatceteris paribusM varies as a, it follows from (1) that F is constant. For all fluids and for all similar tubes similarly wetted, the weight of a drop would then be proportional not only to the diameter of the tube, but also to the superficial tension, and it would be independent of the density.
Careful observations with special precautions to ensure the cleanliness of the water have shown that over a considerable range, the departure from Tate’s law is not great. The results give material for the determination of the function F in (1).
In the preceding table, applicable to thin-walled tubes, the first column gives the values of T/gσa², and the second column those of gM/Ta, all the quantities concerned being in C.G.S. measure, or other consistent system. From this the weight of a drop of any liquid of which the density and surface tension are known, can be calculated. For many purposes it may suffice to treat F as a constant, say 3.8. The formula for the weight of a drop is then simply
Mg = 3.8Ta, (2)
in which 3.8 replaces the 2π of the faulty theory alluded to earlier (see Rayleigh,Phil. Mag., Oct. 1899).]
Phenomena arising from the Variation of the Surface-tension.—Pure water has a higher surface-tension than that of any other substance liquid at ordinary temperatures except mercury. Hence any other liquid if mixed with water diminishes its surface-tension. For example, if a drop of alcohol be placed on the surface of water, the surface-tension will be diminished from 80, the value for pure water, to 25, the value for pure alcohol. The surface of the liquid will therefore no longer be in equilibrium, and a current will be formed at and near the surface from the alcohol to the surrounding water, and this current will go on as long as there is more alcohol at one part of the surface than at another. If the vessel is deep, these currents will be balanced by counter currents below them, but if the depth of the water is only two or three millimetres, the surface-current will sweep away the whole of the water, leaving a dry spot where the alcohol was dropped in. This phenomenon was first described and explained by James Thomson, who also explained a phenomenon, the converse of this, called the “tears of strong wine.”
If a wine-glass be half-filled with port wine the liquid rises a little up the side of the glass as other liquids do. The wine, however, contains alcohol and water, both of which evaporate, but the alcohol faster than the water, so that the superficial layer becomes more watery. In the middle of the vessel the superficial layer recovers its strength by diffusion from below, but the film adhering to the side of the glass becomes more watery, and therefore has a higher surface-tension than the surface of the stronger wine. It therefore creeps up the side of the glass dragging the strong wine after it, and this goes on till the quantity of fluid dragged up collects into a drop and runs down the side of the glass.
The motion of small pieces of camphor floating on water arises from the gradual solution of the camphor. If this takes place more rapidly on one side of the piece of camphor than on the other side, the surface-tension becomes weaker where there is most camphor in solution, and the lump, being pulled unequally by the surface-tensions, moves off in the direction of the strongest tension, namely, towards the side on which least camphor is dissolved.
If a drop of ether is held near the surface of water the vapour of ether condenses on the surface of the water, and surface-currents are formed flowing in every direction away from under the drop of ether.
If we place a small floating body in a shallow vessel of water and wet one side of it with alcohol or ether, it will move off with great velocity and skim about on the surface of the water, the part wet with alcohol being always the stern.
The surface-tension of mercury is greatly altered by slight changes in the state of the surface. The surface-tension of pure mercury is so great that it is very difficult to keep it clean, for every kind of oil or grease spreads over it at once.
But the most remarkable effects of change of surface-tension are those produced by what is called the electric polarization of the surface. The tension of the surface of contact of mercury and dilute sulphuric acid depends on the electromotive force acting between the mercury and the acid. If the electromotive force is from the acid to the mercury the surface-tension increases; if it is from the mercury to the acid, it diminishes. Faraday observed that a large drop of mercury, resting on the flat bottom of a vessel containing dilute acid, changes its form in a remarkable way when connected with one of the electrodes of a battery, the other electrode being placed in the acid. When the mercuryis made positive it becomes dull and spreads itself out; when it is made negative it gathers itself together and becomes bright again. G. Lippmann, who has made a careful investigation of the subject, finds that exceedingly small variations of the electromotive force produce sensible changes in the surface-tension. The effect of one of a Daniell’s cell is to increase the tension from 30.4 to 40.6. He has constructed a capillary electrometer by which differences of electric potential less than 0.01 of that of a Daniell’s cell can be detected by the difference of the pressure required to force the mercury to a given point of a fine capillary tube. He has also constructed an apparatus in which this variation in the surface-tension is made to do work and drive a machine. He has also found that this action is reversible, for when the area of the surface of contact of the acid and mercury is made to increase, an electric current passes from the mercury to the acid, the amount of electricity which passes while the surface increases by one square centimetre being sufficient to decompose .000013 gramme of water.
[The movements of camphor scrapings referred to above afford a useful test of the condition of a water surface. If the contamination exceed a certain limit, the scrapings remain quite dead. In a striking form of the experiment, the water is contained, to the depth of perhaps one inch, in a large flat dish, and the operative part of the surface is limited by a flexible hoop of thin sheet brass lying in the dish and rising above the water-level. If the hoop enclose an area of (say) one-third of the maximum, and if the water be clean, camphor fragments floating on the interior enter with vigorous movements. A touch of the finger will then often reduce them to quiet; but if the hoop be expanded, the included grease is so far attenuated as to lose its effect. Another method of removing grease is to immerse and remove strips of paper by which the surface available for the contamination is in effect increased.
The thickness of the film of oil adequate to check the camphor movements can be determined with fair accuracy by depositing a weighed amount of oil (such as .8 mg.) upon the surface of water in a large bath. Calculated as if the density were the same as in a normal state, the thickness of the film is found to be about two millionths of a millimetre.
Small as is the above amount of oil, the camphor test is a comparatively coarse one. Conditions of a contaminated surface may easily be distinguished, upon all of which camphor fragments spin vigorously. Thus, a shallow tin vessel, such as the lid of a biscuit box, may be levelled and filled with tap-water through a rubber hose. Upon the surface of the water a little sulphur is dusted. An application of the finger for 20 or 30 seconds to the under surface of the vessel will then generate enough heat to lower appreciably the surface-tension, as is evidenced by the opening out of the dust and the formation of a bare spot perhaps 1½ in. in diameter. When, however, the surface is but very slightly greased, a spot can no longer be cleared by the warmth of the finger, or even of a spirit lamp, held underneath. And yet the greasing may be so slight that camphor fragments move with apparently unabated vigour.
The varying degrees of contamination to which a water surface is subject are the cause of many curious phenomena. Among these is thesuperficial viscosityof Plateau. In his experiments a long compass needle is mounted so as to swing in the surface of the liquid under investigation. The cases of ordinary clean water and alcohol are strongly contrasted, the motion of the needle upon the former being comparatively sluggish. Moreover, a different behaviour is observed when the surfaces are slightly dusted over. In the case of water the whole of the surface in front of the needle moves with it, while on the other hand the dust floating on alcohol is scarcely disturbed until the needle actually strikes it. Plateau attributed these differences to a special quality of the liquids, named by him “superficial viscosity.” It has been proved, however, that the question is one of contamination, and that a water surface may be prepared so as to behave in the same manner as alcohol.
Another consequence of the tendency of a moderate contamination to distribute itself uniformly is the calming effect of oil, investigated by B. Franklin. On pure water the propagation of waves would be attended by temporary extensions and contractions of the surface, but these, as was shown by O. Reynolds, are resisted when the surface is contaminated.
Indeed the possibility of the continued existence of films, such as constitute foam, depends upon the properties now under consideration. If, as is sometimes stated, the tension of a vertical film were absolutely the same throughout, the middle parts would of necessity fall with the acceleration of gravity. In reality, the tension adjusts itself automatically to the weight to be supported at the various levels.
Although throughout a certain range the surface-tension varies rapidly with the degree of contamination, it is remarkable that, as was first fully indicated by Miss Pockels, the earlier stages of contamination have little or no effect upon surface-tension. Lord Rayleigh has shown that the fall of surface-tensionbeginswhen the quantity of oil is about the half of that required to stop the camphor movements, and he suggests that this stage may correspond with a complete coating of the surface with a single layer of molecules.]
On the Forms of Liquid Films which are Figures of Revolution.— A soap bubble is simply a small quantity of soap-suds spread out so as to expose a large surface to the air. The bubble, in fact, has two surfaces, an outer and an innerSpherical soap-bubble.surface, both exposed to air. It has, therefore, a certain amount of surface-energy depending on the area of these two surfaces. Since in the case of thin films the outer and inner surfaces are approximately equal, we shall consider the area of the film as representing either of them, and shall use the symbol T to denote the energy of unit of area of the film, both surfaces being taken together. If T′ is the energy of a single surface of the liquid, T the energy of the film is 2T′. When by means of a tube we blow air into the inside of the bubble we increase its volume and therefore its surface, and at the same time we do work in forcing air into it, and thus increase the energy of the bubble.
That the bubble has energy may be shown by leaving the end of the tube open. The bubble will contract, forcing the air out, and the current of air blown through the tube may be made to deflect the flame of a candle. If the bubble is in the form of a sphere of radius r this material surface will have an area
S = 4πr². (1)
If T be the energy corresponding to unit of area of the film the surface-energy of the whole bubble will be
ST = 4πr²T. (2)
The increment of this energy corresponding to an increase of the radius from r to r + dr is therefore
TdS = 8πTdr. (3)
Now this increase of energy was obtained by forcing in air at a pressure greater than the atmospheric pressure, and thus increasing the volume of the bubble.
Let Π be the atmospheric pressure and Π + p the pressure of the air within the bubble. The volume of the sphere is
V =4⁄3πr3, (4)
and the increment of volume is
dV = 4πr²dr. (5)
Now if we suppose a quantity of air already at the pressure Π + p, the work done in forcing it into the bubble pdV. Hence the equation of work and energy is
p dV = Tds (6)
or
4πpr²dr = 8πrdrT (7)
or
p = 2T/r. (8)
This, therefore, is the excess of the pressure of the air within the bubble over that of the external air, and it is due to the action of the inner and outer surfaces of the bubble. We may conceive this pressure to arise from the tendency which the bubble has to contract, or in other words from the surface-tension of the bubble.
If to increase the area of the surface requires the expenditureof work, the surface must resist extension, and if the bubble in contracting can do work, the surface must tend to contract. The surface must therefore act like a sheet of india-rubber when extended both in length and breadth, that is, it must exert surface-tension. The tension of the sheet of india-rubber, however, depends on the extent to which it is stretched, and may be different in different directions, whereas the tension of the surface of a liquid remains the same however much the film is extended, and the tension at any point is the same in all directions.
The intensity of this surface-tension is measured by the stress which it exerts across a line of unit length. Let us measure it in the case of the spherical soap-bubble by considering the stress exerted by one hemisphere of the bubble on the other, across the circumference of a great circle. This stress is balanced by the pressure p acting over the area of the same great circle: it is therefore equal to πr²p. To determine the intensity of the surface-tension we have to divide this quantity by the length of the line across which it acts, which is in this case the circumference of a great circle 2πr. Dividing πr²p by this length we obtain ½pr as the value of the intensity of the surface-tension, and it is plain from equation 8 that this is equal to T. Hence the numerical value of the intensity of the surface-tension is equal to the numerical value of the surface-energy per unit of surface. We must remember that since the film has two surfaces the surface-tension of the film is double the tension of the surface of the liquid of which it is formed.
To determine the relation between the surface-tension and the pressure which balances it when the form of the surface is not spherical, let us consider the following case:—
Let fig. 9 represent a section through the axis Cc of a soap-bubble in the form of a figure of revolution bounded by two circular disks AB and ab, and having the meridian section APa. Let PQNon-spherical soap bubble.be an imaginary section normal to the axis. Let the radius of this section PRbey, and let PT, the tangent at P, make an angle a with the axis.
Let us consider the stresses which are exerted across this imaginary section by the lower part on the upper part. If the internal pressure exceeds the external pressure by p, there is in the first place a force πy²p acting upwards arising from the pressure p over the area of the section. In the next place, there is the surface-tension acting downwards, but at an angle a with the vertical, across the circular section of the bubble itself, whose circumference is 2πy, and the downward force is therefore 2πyT cos a.
Now these forces are balanced by the external force which acts on the disk ACB, which we may call F. Hence equating the forces which act on the portion included between ACB and PRQ
πy²p − 2πyT cos α = −F (9).
If we make CR = z, and suppose z to vary, the shape of the bubble of course remaining the same, the values of y and of a will change, but the other quantities will be constant. In studying these variations we may if we please take as our independent variable the length s of the meridian section AP reckoned from A. Differentiating equation 9 with respect to s we obtain, after dividing by 2π as a common factor,
Now
The radius of curvature of the meridian section is
The radius of curvature of a normal section of the surface at right angles to the meridian section is equal to the part of the normal cut off by the axis, which is
R2= PN = y/cos α (13).
Hence dividing equation 10 by y sin α, we find
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 α, and equation 9 becomes
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
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 2πTm 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, everreaching 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,3Goldschmidt,4L.L. Lindelöf and F.M.N. Moigno,5C.E. Delaunay,6A.H.E. Lamarle,7A. Beer,8and V.M.A. Mannheim,9and have been produced experimentally by Plateau10in the two different ways already described.
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.
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 2πa 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 πr 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, C1are the disks, so that the distance between them is less than πr, 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 A2and C2, so that the distance between them is greater than πr, 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 (“Mémoire sur la constitution des veines liquides lancées 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 λ, and is connected with k by the equation k = 2π/λ. 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 λ < 2πa; but less if ka < 1, or λ > 2πa. Accordingly, the equilibrium is stable if λ be less than the circumference; but unstable if λ be greater than the circumference of the cylinder. Disturbances of the former kind lead tovibrationsof 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 (eqt) 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 andinertiain 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, T is the superficial tension, ρ the density, and F is given by the following table: —
The greatest value of F thus corresponds, not to a zero value of k²a², but approximately to k²a² = .4858, or to λ = 4.508 × 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, ρ = 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 = 3loge10 = 6.9, so that t = .79d3/2. 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 α1eq1t, α2eq2t, in which q1exceeds q2, their ratio is (α1/α2)e−(q1 − q2)t; and this ratio decreases without limit with the time, whatever be the initial (finite) ratio α2: α1. 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 thetimeof 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 d3/2. Again, when the fluid is changed, the time varies as ρ1/2T−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 littleof 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 π : 4.508, which is very nearly a fifth. In the case of Plateau’s numbers (π : 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) proportionalceteris paribusto 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
V being the volume of the vibrating mass.
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 theshadowof 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 apairof 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 thereboundof 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 materialchange 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 dropscoalesce, 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 about1⁄20in., 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 theshadowof 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 todifferencesof 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:—
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’équilibre des liquides,”Nouveaux Mém. 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 ρ be the density of the upper liquid, and σ 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 − ρgz, and that below it will be P − σgz, so that the surface will be acted on by an upward pressure (ρ − σ)gz. Now if the displacement z be everywhere very small, the curvature in the planes parallel to xz and yz will be d²z/dx² and d²z/dy² respectively, and if T is the surface-tension the whole upward force will be