CHAPTER IX
HYDRODYNAMICS—DYNAMICAL THEOREM OF MINIMUM ENERGY—VORTEX MOTION
Thomson devoted great attention from time to time to the science of hydrodynamics. This is perhaps the most abstruse subject in the domain of applied mathematics, and when viscosity (the frictional resistance to the relative motion of particles of the fluid) is taken into account, passes beyond the resources of mathematical science in its present state of development. But leaving viscosity entirely aside, and dealing only with so-called perfect fluids, the difficulties are often overwhelming. For a long time the only kind of fluid motion considered was, with the exception of a few simple cases, that which is called irrotational motion. This motion is characterised by the analytical peculiarity, that the velocity of an element of the fluid in any direction is the rate of variation per unit distance in that direction of a function of the coordinates (the distances which specify the position) of the particle. This condition very much simplifies the analysis; but when it does not hold we have much more serious difficulties to overcome. Then the elements of the fluid have what is generally, but quite improperly, called molecular rotation. For we know little of the molecules of a fluid; even when we deal with infinitesimal elements, in the analysis of fluid motion, we are considering the fluid in mass. But what is meant is elemental rotation, a rotation of the infinitesimal elements as they move. We have an example of such motion in the air when a ring of smoke escapes from the funnel of a locomotive or the lips of a tobacco-smoker, in the motion of part of the liquid when a cup of tea is stirred by drawing the spoon from one side to the other, or when the blade of an oar is moving through the water. In these last two cases the depressions seen in the surface are the ends of a vortex which extends between them and terminates on the surface. In all these examples what have been called vortices are formed, and hence the name vortex motion has been given to all those cases in which the condition of irrotationality is not satisfied.
The first great paper on vortex motion was published by von Helmholtz in 1858, and ten years later a memoir on the same subject by Thomson was published in the Transactions of the Royal Society of Edinburgh. In that memoir are given very much simpler proofs of von Helmholtz's main theorems, and, moreover, some new theorems of wide application to the motion of fluids. One of these is so comprehensive that it may be said with truth to contain the whole of the dynamics of a perfect fluid. We go on to indicate the contents of the principal papers, as far as that can be done without the introduction of analysis of a difficult description.
In Chapter VI reference has been made to the "Notes on Hydrodynamics" published by Thomson in the Cambridge and Dublin Mathematical Journal for 1848 and 1849. These Notes were not intended to be entirely original, but were composed for the use of students, like Airy's Tracts of fifteen years before.
The first Note dealt with the equation of continuity, that is to say, the mathematical expression of the obvious fact that if any region of space in a moving fluid be considered, the excess of rate of flow into the space across the bounding surface, above the rate of flow out, is equal to the rate of growth of the quantity of fluid within the space. The proof given is that now usually repeated in text-books of hydrodynamics.
The second Note discussed the condition fulfilled at the bounding surface of a moving fluid. The chief mathematical result is the equation which expresses the fact, also obvious without analysis, that there is no flow of the fluid across the surface. In other words, the component of the motion of a fluid particle in the immediate neighbourhood of the surface at any instant, taken in the direction perpendicular to the surface, must be equal to the motion of the surface in that direction at the same instant.
The third Note, published a year later (February 1849), is of considerable scientific importance. It is entitled, "On the Vis Viva of a Liquid in Motion." What used to be called the "vis viva" of a body is double what is now called the energy of motion, or kinetic energy, of the body. The term liquid is merely a brief expression for a fluid, the mass of which per unit volume is the same throughout, and suffers no variation. The fluid, moreover, is supposed devoid of friction, that is, the relative motions of its parts are unresisted by tangential force between them. The chief theorem proved and discussed may be described as follows.
The liquid is supposed to fill the space within a closed envelope, which fulfils the condition of being "simply continuous." The condition will be understood by imagining any two points A, B, within the space, to be joined by two lines ACB, ADB both lying within the space. These two lines will form a circuit ACBDA. If now this circuit, however it may be drawn, can be contracted down to a point, without any part of the circuit passing out of the space, the condition is fulfilled. Clearly the space within the surface of an anchor-ring, or a curtain-ring, would not fulfil this condition, for one part of the circuit might pass from A to B round the ring one way, and the other from A to B the other way. The circuit could not then be contracted towards a point without passing out of the ring.
Now let the liquid given at rest in such a space be set in motion by any arbitrarily specified variation of position of the envelope. The liquid within will be set in motion in a manner depending entirely on the motion of the envelope. It is possible to conceive of other motions of the liquid than that taken, which all agree in having the specified motion of the surface. Thomson's theorem asserts that the motion actually taken has less kinetic energy than that of any of the other motions which have the same motion of the bounding surface.
The motion produced has the property described by the word "irrotational," that is, the elements of the fluid have no spinning motion—they move without rotation. A small portion of a fluid may describe any path—may go round in a circle, for example—and yet have no rotation. The reader may imagine a ball carried round in a circle, but in such a way that no line in the body ever changes its direction. The body has translation, but no spin.
Irrotationality of a fluid is secured, as stated above, when the velocity of each element in any direction is the rate of variation per unit distance in that direction of a certain function of the coordinates, the distances, taken parallel to three lines perpendicular to one another and drawn from a point, which specify the position of the particle. In fact, what is called a velocity-potential exists, similar to the potential described in Chapter IV above, for an electric field. This condition, together with the specified motion of the surface, suffices to determine the motion of the fluid.
Two important particular consequences were pointed out by Thomson: (1) that the motion of the fluid at any instant depends solely on the form and motion of the bounding surface, and is therefore independent of the previous motion; and (2) that if the bounding surface be instantaneously brought to rest, the liquid throughout the vessel will also be instantly brought to rest.
This theorem was afterwards generalised by Thomson (Proc. R.S.E., 1863), and applied to any material system of connected particles set into motion by specified velocities simultaneously and suddenly imposed at selected points of the system. It was already known that the kinetic energy of a system of bodies connected in any manner, and set in motion by impulses applied at specified points, was either a maximum or a minimum, as compared with that for any other motion compatible with these impulses, and with the connections of the system. This was proved by Lagrange in the Mécanique Analytique as a generalisation of a theorem given by Euler for a rigid body set into rotation by an impulse.
Bertrand proved in 1842 that when the impulses applied are given in amount, and are applied at specified points, the system starts off with kinetic energy greater than that of any other motion which is consistent with the given impulses and the connections of the system. This other motion must be such as could be produced in the system by the given impulses, together with any other set of impulses capable of doing no work on the whole.
Thomson's theorem is curiously complementary to Bertrand's. Let the system be acted on by impulses applied at certain specified points, and by no other impulses of any kind; and let the impulses be such as to start those selected points with any prescribed velocities. The system will start off with kinetic energy which is less than that of any other motion which the system could have consistently with the prescribed velocities, and which it could be constrained to take by impulses which do no work on the whole. In each case the difference of energies is the energy of the motion which must be compounded with one motion to give the other which is compared with it.
A simple example, such as might be taken of the particular case considered by Euler, may help to make these theorems clear. Imagine a straight uniform rod to lie on a horizontal table, between which and the rod there is no friction. Let the rod be struck a blow at one end in a horizontal direction at right angles to the length of the rod. If no other impulse acts, the end of the rod will move off with a certain definite velocity, and the other parts of the rod (which is supposed perfectly unbending) will be started by the connections of the system. It is obvious that any number of other motions of the rod can be imagined, all of which give the same motion of the extremity struck. But the actual motion taken is one of turning about that point of the rod which is two-thirds of the length from the end struck. If the reader will consider the kinetic energy for any other horizontal turning motion consistent with the same motion of the end, he will find that the kinetic energy is greater than that of the motion just specified. This motion could be produced by applying at the point about which the rod turns the impulse required to keep that point at rest. The impulse so applied would do no work. The actual value is 1⁄8mv2, where m denotes the mass of the rod and v the velocity of the end. If the motion taken were one of rotation about a point of the rod at distance x from the end struck, the kinetic energy would be m (4l2 − 6lx + 3x2) v2 ⁄ 6x2, where 2l is the length of the rod, and this has its least value 1⁄8mv2 for x = 4l ⁄ 3. For example, x = 2l gives 1⁄6mv2, which is greater than the value just found.
Bertrand's theorem applied to this case of motion is not quite so easy, perhaps, to understand. The motion which is said to have maximum energy is one given by a specified impulse at the end struck, and this, in the absence of any other impulses, would be a motion of minimum energy. But let the alternative motion, which is to be compared with that actually taken, be one constrained by additional impulses such as can together effect no work, and the existence of the maximum is accounted for. The kinetic energy produced is one-half the product of the impulse into the velocity of the point struck, that is ½Iv, and it has just been seen that this is the product of 1⁄6mv2 by the factor (4l2 − 6lx + 3x2) ⁄ x2. This factor is 3I ⁄ mv, and is a minimum when x = 4l ⁄ 3. Thus for a given I, v will have its maximum value when the factor referred to is least, and ½Iv will then be a maximum.
The bar can be constrained to turn about another point by a fixed pivot there situated. An impulse will be applied to the rod by the pivot, simultaneously with the blow; and it is obvious that this impulse does no work, since there is no displacement of the point to which it is applied.
The two theorems are consequences of one principle. The constraint in each case increases what may be called the effective inertia, which may be taken as I ⁄ v. Thus when v is given, I is increased by any constraint compelling the rod to rotate about a particular axis, and so ½Iv, or the kinetic energy, is increased. On the other hand, when I is given the same constraint diminishes v, and so ½Iv is diminished.
A short paper published in the B. A. Report for 1852 points out that the lines of force near a small magnet, placed with its axis along the lines of force in a uniform magnetic field, as it would rest under the action of the field, are at corresponding points similar to those of the field of an insulated spherical conductor, under the inductive influence of a distant electric change. Further, the fact is noted that, if the magnet be oppositely directed to the field, the lines of force are curved outwards, just as the lines of flow of a uniform stream would be by a spherical obstacle, at the surface of which no eddies were caused. This is one of those instructive analogies between the theory of fluid motion and other theories involving perfectly analogous fundamental ideas, which Thomson was fond of pointing out, and which helped him in his repeated attempts to imagine mechanical representations of physical phenomena of different kinds.
With these may be placed another, which in lectures he frequently dwelt on—a simple doublet, as it is called, consisting of a point-source of fluid and an equal and closely adjacent point-sink. A short tube in an infinite mass of liquid, which is continually flowing in at one end and out at the other, may serve as a realisation of this arrangement. The lines of flow outside the tube are exactly analogous to the lines of force of a small magnet; and if at the same time there exist a uniform flow of the liquid in the direction of the length of the tube, the field of flow will be an exact picture of the field of force of the small magnet, when it is placed with its length along the lines of a previously existing uniform field. The flow in the doublet will be with or against the general flow according as the magnet is directed with or against the field.
The paper on vortex-motion has been referred to above, and an indication given of the nature of the fluid-motion described by this title. There are, however, two cases of fluid-motion which are referred to as vortices, though the fundamental criterion of vortex-motion—the non-existence of a velocity-potential—is satisfied in only one of them. The exhibition of one of these was a favourite experiment in Thomson's ordinary lectures, as his old students will remember. If water in a large bowl is stirred rapidly with a teaspoon carried round and round in a circle about the axis of the bowl, the surface will become concave, and the form of the central part will be a paraboloid of revolution about the vertical through the lowest point, that is to say, any section of that part of the surface made by a vertical plane containing the axis will be a parabola symmetrical about the axis. The motion can be better produced by mounting the vessel on a whirling-table, and rotating it about the vertical axis coinciding with its axis of figure; but the phenomenon can be quite well seen without this machinery. In this case the velocity of each particle of the water is proportional to its distance from the axis, and the whole mass, when relative equilibrium is set up, turns, as if it were rigid, about the axis of the vessel. Each element of the fluid in this "forced vortex," as it is called, is in rotation, and, like the moon, makes one turn in one revolution about the centre of its path. This is, therefore, a true, though very simple, case of vortex-motion.
On the other hand, what may be called a "free vortex" may exist, and is approximated to sometimes when water in a vessel is allowed to run off through an escape pipe at the bottom. The velocity of an element in this "vortex" is inversely proportional to its distance from the centre, and the form of the free surface is quite different from that in the other case. The name "free vortex" is often given to this case of motion, but there is no vortex-motion about it whatever.
Thomson's great paper on vortex-motion was read before the Royal Society of Edinburgh in 1867, and was recast and augmented in the following year. It will be possible to give here only a sketch of its scope and main results.
The fluid is supposed contained in a closed fixed vessel which is either simply or multiply continuous (see p. [156]), and may contain immersed in it simply or multiply continuous solids. When these solids exist their surfaces are part of the boundary of the liquid; they are surrounded by the liquid unless they are anywhere in contact with the containing vessel, and their density is supposed to be the same as that of the liquid. They may be acted on by forces from without, and they act on the liquid with pressure-forces, and either directly or through the liquid on one another.
The first result obtained is fairly obvious. The centre of mass of the whole system must remain at rest whatever external forces act on the solids, since the density is the same everywhere within the vessel, and the vessel is fixed; that is to say, there is no momentum of the contents of the vessel in any direction. For whatever motion of the solids is set up by the external forces, must be accompanied by a motion of the liquid, equal and opposite in the sense here indicated.
After a discussion of what he calls the impulse of the motion, which is the system of impulsive forces on the movable solids which would generate the motion from rest, Thomson proceeds to prove the important proposition that the rotational motion of every portion of the liquid mass, if it is zero at any one instant for every portion of the mass, remains always zero. This is done by considering the angular momentum of any small spherical portion of the liquid relatively to an axis through the centre of the sphere, and proving that in order that it may vanish, for every axis, the component velocities of the fluid at the centre must be derivable from a velocity-potential. The angular momentum of a particle about an axis is the product of the component of the particle's momentum, at right angles to the plane through the particle and the axis, by the distance of the particle from the axis. The sum of all such products for the particles making up the body (when proper account is taken of the signs according to the direction of turning round the axis) is the angular momentum. The proof of this result adopted is due to Stokes. The angular velocities of an element of fluid at a point x, y, z, about the axes of x, y, z are shown to be ½ (∂w ⁄ ∂y − ∂v ⁄ ∂z), etc.
The condition was therefore shown to be necessary; it remained to prove that it was sufficient. This is obvious at once from the definition of the velocity-potential, which must now be supposed to exist in order that its sufficiency may be proved. If any diameter of the spherical portion be taken as the axis, and any plane through that axis be considered, the velocity of a particle at right angles to that plane can be at once expressed as the rate at which the velocity-potential varies per unit distance along the circle, symmetrical about the axis, on which the particle lies. The integral of the velocity-potential round this circle vanishes, and so the angular momentum for any thin uniform ring of particles about the axis also vanishes, and as the sphere is made up of such rings, the whole angular momentum is zero. Thus the condition is sufficient.
Thomson then proves that if the angular momentum thus considered be zero for every portion of the liquid at any one instant, it remains zero at every subsequent instant; that is, no physical action whatsoever could set up angular momentum within the fluid, which, it is to be remembered, is supposed to be frictionless. The proof here given cannot be sketched because it depends on the differential equation of continuity satisfied by the velocity-potential throughout the fluid (the same differential equation, in fact, that is satisfied by the distribution of temperature in a uniform conducting medium in the stationary state), and the consequent expression of this function for any spherical space in the fluid as a series of spherical harmonic functions. To a reader to whom the properties of these functions are known the process can present no difficulty.
An entirely different proof of this proposition is given subsequently in the paper, and depends on a new and very general theorem, which has been described as containing almost the whole theory of the motion of a fluid. This depends on what Thomson called the flow along any path joining any two points P, Q in the fluid. Let q be the velocity of the fluid at any element of length ds of such a path, and θ be the angle between the direction of ds (taken positive in the sense from P to Q) and the direction of q: q cos θ . ds is the flow along ds. If u, v, w be the components of q at ds, parallel to the axes, and dx, dy, dz be the projections of ds on the axes, udx + vdy + wdz is the same thing as q cos θ . ds. The sum of the values of either of these expressions for all the elements of the path between P and Q is the flow along the path. The statement that u, v, w are the space-rates of variation of a function φ (of x, y, z) parallel to the axes, or that q cos θ is the space-rate of variation of φ along ds, merely means that this sum is the same for whatever path may be drawn from P to Q. This, however, is only the case when the paths are so taken that in each case the value of φ returns after variation along a closed path to the value which it had at the starting point, that is, the closed path must be capable of being contracted to a point without passing out of space occupied by irrotationally moving fluid.
Since the flow from P to Q is the same for any two paths which fulfil this condition, the flow from P to Q by any one path and from Q to P by any other must be zero. The flow round such a closed path is not zero if the condition is not fulfilled, and its value was called by Thomson the circulation round the path.
The general theorem which he established may now be stated. Consider any path joining PQ, and moving with the fluid, so that the line contains always the same fluid particles. Let u̇, v̇, ẇ be the time-rates of change of u, v, w at an element ds of the path, at any instant, and du, dv, dw the excesses of the values of u, v, w at the terminal extremity of ds above the values at the other extremity; then the time-rate of variation of udx + vdy + wdz is u̇dx + v̇dy + ẇdz + udu + vdv + wdw or u̇dx + v̇dy + ẇdz + qdq, where q has the meaning specified above. Thus if S be the flow for the whole path PQ, and Ṡ its time-rate of variation, S' denote the sum of u̇dx + v̇dy + ẇdz along the path from P to Q, and q1, q0 the resultant fluid velocities at Q and P, we get Ṡ = S' + ½(q12 − q02). This is Thomson's theorem. If the curve be closed, that is, if P and Q be coincident, q1 = q0 and Ṡ = S'. But in certain circumstances S' is zero, and so therefore is also Ṡ. Thus in the circumstances referred to, as the closed path moves with the fluid Ṡ is continually zero, and it follows that if Ṡ is zero at any instant it remains zero ever after. But Ṡ is only zero if u, v, w are derivable from a potential, single valued in the space in which the closed path is drawn, so that the path could be shrunk down to a point without ever passing out of such space. In a perfect fluid if this condition is once fulfilled for a closed curve moving with the fluid, it is fulfilled for this curve ever after.
The circumstances in which S' is zero are these:—the external force, per unit mass, acting on the fluid at any point is to be derivable from a potential-function, and the density of the fluid is to be a function of the pressure (also a function of the coordinates); and these functions must be such as to render S' always zero for the closed path. This condition is manifestly fulfilled in many important cases; for example, the forces are derivable from a potential due to actions, such as gravity, the origin of which is external to the fluid; and the density is a function of the pressure (in the present case it is a constant), such that the part of S' which depends on pressure and density vanishes for the circuit.
It is to be clearly understood that the motion of a fluid may be irrotational although the value of S does not vanish for every closed path that can be drawn in it. The fluid may occupy multiply continuous space, and the path may or may not be drawn so that S shall be zero; but what is necessary for irrotational motion within any space is that S should vanish for all paths which are capable of being shrunk down to zero without passing out of that space. S need not vanish for a path which cannot be so shrunk down, but it must, if the condition just stated is fulfilled, have the same value for any two paths, one of which can be made to pass into the other by change of position without ever passing in whole or in part out of the space. The potential is always single valued in fluid filling a singly continuous space such as that within a spherical shell, or between two concentric shells; within a hollow anchor-ring the potential, though it exist, and the motion be irrotational, is not single valued. In the latter case the motion is said to be cyclic, in the former acyclic.
A number of consequences are deduced from this theorem; and from these the properties of vortices, which had previously been discovered by von Helmholtz, immediately follow. First take any surface whatever which has for bounding edge a closed curve drawn in the fluid, and draw from any element of this surface, of area dS, a line perpendicular to the surface towards the side chosen as the positive side, and calculate the angular velocity ω, say, of the fluid about that normal from the components of angular velocity determined in the manner explained at p. [164]. This Thomson called the rotation of the element. Now take the product ωdS for the surface element. It is easy to see that this is equal to half the circulation round the bounding edge of the element. As the fluid composing the element moves the area dS may change, but the circulation round its edge by Thomson's theorem remains unaltered. Thus ω alters in the inverse ratio of dS, and the line drawn at right angles to the surface at dS, if kept of length proportional to ω, will lengthen or shorten as dS contracts or expands.
Now sum the values of ωdS for the finite surface enclosed by the bounding curve. It follows from the fact that ωdS is equal to half the circulation round the edge of dS, that this sum, which is usually denoted by ΣωdS, is equal to half the circulation round the closed curve which forms the edge of the surface. Also as the fluid moves the circulation round the edge remains unaltered, and therefore so does also ΣωdS for the elements enclosed by it. It is important to notice that this sum being determined by the circulation in the bounding curve is the same for all surfaces which have the same boundary.
The equality of 2ΣωdS for the surface to the circulation round its edge was expressed by Thomson as an analytical theorem of integration, which was first given by Stokes in a Smith's Prize paper set in 1854. It is here stated, apparently by an oversight, that it was first given in Thomson and Tait's Natural Philosophy, § 190. In the second edition of the Natural Philosophy the theorem is attributed to Stokes. It is now well known as Stokes's theorem connecting a certain surface integral with a line integral, and has many applications both in physics and in geometry.
Now consider the resultant angular velocity at any point of the fluid, and draw a short line through that point in the direction of the axis of rotation. That line may be continued from point to point, and will coincide at every one of its points with the direction of the axis of rotation there. Such an axial curve, as it may be called, it is clear moves with the fluid. For take any infinitesimal area containing an element of the line; the circulation round the edge of this area is zero, since there is no rotation about a line perpendicular to the area. Hence the circulation along the axial curve is zero, and the axial curves move with the fluid.
Take now any small plane area dS moving with the fluid, and draw axial lines through every point of its boundary. These will form an axial tube enclosing dS. If θ be the angle between the direction of resultant rotation and a perpendicular to dS, the cross-section of the tube at right angles to the normal, and to the axial lines which bound it, is dS.cosθ. Let these axial lines be continued in both directions from the element dS. They will enclose a tube of varying normal cross-section; but the product of rotation and area of normal cross-section has everywhere the same value. A vortex-tube with the fluid within it is called a vortex-filament.
It will be seen that this vortex-tube must be endless, that is, it must either return into itself, or be infinitely long in one or both directions. For if it were terminated anywhere within the fluid, it would be possible to form a surface, starting from a closed circuit round the tube, continued along the surface of the tube to the termination, and then closed by a cap situated beyond the termination. At no part of this surface would there be any rotation, and ΣωdS, which is equal to the circulation, would be zero for it; and of course this cannot be the case. Thus the tube cannot terminate within the fluid. It can, however, have both of its ends on the surface, or one on the bounding surface and the other at infinity, if the fluid is infinitely extended in one direction, but in that case the termination is only apparent. The section is widened out at the surface; some of the bounding lines pass across to the other apparent termination, when it also lies on the surface, while the other lines pass off to infinity along the surface, and correspond to other lines coming in from infinity to the other termination. Whether the surface is infinite or not, the vortex is spread out into what is called a vortex-sheet, that is, in a surface on the two sides of which the fluid moves with different tangential velocities.
Through a vortex-ring or tube, the fluid circulates in closed lines of flow, each one of which is laced through the tube. The circulation along every line of flow which encloses the same system of vortex-tubes has the same value.
If any surface be drawn cutting a vortex-tube, it is clear from the definition of the tube that the value of ΣωdS for every such surface must be the same. This Thomson calls the "rotation of the tube."
As was pointed out first by von Helmholtz, vortex-filaments correspond to circuits carrying currents and the velocity in the surrounding fluid to magnetic field-intensity. The "rotation of the tube" corresponds to the strength of the current, and sources and sinks to positive and negative magnetic poles. Thomson made great use of this analogy in his papers on electromagnetism.
Examples of vortex-tubes are indicated on p. [154]; and the reader may experiment with vortices in liquids with water in a tea-cup, or in a river or pond, at pleasure. Air vortices may be experimentally studied by means of a simple apparatus devised by Professor Tait, which may be constructed by anyone.
Fig. 13.
In one end of a packing-box, about 2ft. long by 18in. wide and 18in. deep, a circular hole is cut, and the edges of the hole are thinned down to a blunt edge. This can be closed at pleasure by a piece of board. The opposite end is removed, and a sheet of canvas stretched tightly in its place, and tacked to the ends of the sides. Through two holes bored in one of the sides the mouths of two flasks with bent necks protrude into the box. One of these flasks contains ammonia, the other hydrochloric acid. When the hole at one end is closed up by a slip of tinplate, and the liquids are heated with a spirit-lamp, the vapours form a cloud of sal-ammoniac within the box, which is retained during its formation. The hole is then opened, and the canvas struck smartly with the palm of the open hand. Immediately a beautiful ring of smoke emerges, clear-cut and definite as a solid, and moves across the room. (See Fig. [13].) Of course, it is a ring of air, made visible by the smoke carried with it. By varying the shape of the aperture—for example, by using instead of the hole cut in the wood, a slide of tinplate with an elliptic hole cut in it—the vortex-rings can be set in vibration as they are created, and the vibrations studied as the vortex moves.
Still more beautiful vortices can be formed in water by using a long tank of clear water to replace the air in which the vortex moves, and a compartment at one end filled with water coloured with aniline, instead of the smoke-box. A hole in the dividing partition enables the vortex to be formed, and a piston arrangement fitted to the opposite side enables the impulse to the water to be given from without.
From the account of the nature of vortex-motion given above, it will be clear that vortices in a perfect fluid once existent must be ever existent. To create a vortex within a mass of irrotationally moving perfect fluid is physically impossible. It occurred to Thomson, therefore, that ordinary matter might be portions of a perfect fluid, filling all space, differentiated from the surrounding fluid by the rotation which they possess. Such matter would fulfil the law of conservation, as it could neither be created nor destroyed by any physical act.
The results of such experiments led Thomson to frame his famous vortex-atom theory of matter, a theory, however, which he felt ultimately was beset with so many difficulties as to be unworkable.
The paper on vortex-motion also deals with the modification of Green's celebrated theorem of analysis, which, it was pointed out by Helmholtz, was necessary to adapt it to a space which is multiply continuous. The theorem connects a certain volume-integral taken throughout a closed space with an integral taken over the bounding surface of the space. This arises from the fact noticed above that in multiply continuous space (for example, the space within an endless tube) the functions which are the subject of integration may not be single valued. Such a function would be the velocity-potential for fluid circulating round the tube—cyclic motion, as it was called by Thomson. If a closed path of any form be drawn in such a tube, starting from a point P, and doubling back so as to return to P without making the circuit of the tube, the velocity-potential will vary along the tube, but will finally return to its original value when the starting point is reached. And the circulation round this circuit will be zero. But if the closed path make the circuit of the tube, the velocity-potential will continuously vary along the path, until finally, when P is reached again, the value of the function is greater (or less) than the value assumed for the starting point, by a certain definite amount which is the same for every circuit of the space. If the path be carried twice round in the same direction, the change of the function will be twice this amount, and so on. The space within a single endless tube such as an anchor-ring is doubly continuous; but much more complicated cases can be imagined. For example, an anchor-ring with a cross-connecting tube from one side to the other would be triply continuous.
Thomson showed that the proper modification of the theorem is obtained by imagining diaphragms placed across the space, which are not to be crossed by any closed path drawn within the space, and the two surfaces of each of which are to be reckoned as part of the bounding surface of the space. One such diaphragm is sufficient to convert a hollow anchor-ring into a singly continuous space, two would be required for the hollow anchor-ring with cross-connection, and so on. The number of diaphragms required is always one less than the degree of multiplicity of the continuity.
The paper also deals with the motion of solids in the fluid and the analogous motions of vortex-rings and their attraction by ordinary matter. These can be studied with vortex-rings in air produced by the apparatus described above. Such a ring made to pass the re-entrant corner of a wall—the edge of a window recess, for example—will appear to be attracted. A large sphere such as a large terrestrial globe serves also very well as an attracting body.
Two vortex-rings projected one after the other also act on one another in a very curious manner. Their planes are perpendicular to the direction of motion, and the fluid is moving round the circular core of the ring. There is irrotational cyclic motion of the fluid through the ring in one direction and back outside, as shown in Fig. [13], which can be detected by placing a candle flame in the path of the centre. The first ring, in consequence of the existence of that which follows it, moves more slowly, and opens out more widely, the following ring hastens its motion and diminishes in diameter, until finally it overtakes the former and penetrates it. As soon as it has passed through it moves ahead more and more slowly, until the one which has been left behind begins to catch it up, and the changes which took place before are repeated. The one penetrating becomes in its turn the penetrated, and so on in alternation. Great care and skill are, however, necessary to make this interesting experiment succeed.
We have not space to deal here with other hydrodynamical investigations, such as the contributions which Thomson made to the discussion of the many difficult problems of the motion of solids through a liquid, or to his very numerous and important contributions to the theory of waves. The number and importance of his hydrodynamical papers may be judged from the fact that there are no less than fifty-two references to his papers, and thirty-five to Thomson and Tait's Natural Philosophy in the latest edition of Lamb's Hydrodynamics, and that many of these are concerned with general theorems and results of great value.