Lord Kelvin Part 9

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 ma.s.s 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 ma.s.s, if it is zero at any one instant for every portion of the ma.s.s, 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 (?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 s.p.a.ce 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 ? be the angle between the direction of ds (taken positive in the sense from P to Q) and the direction of q: q cos ?.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 ?.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 s.p.a.ce-rates of variation of a function f (of x, y, z) parallel to the axes, or that q cos ? is the s.p.a.ce-rate of variation of f 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 f 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 pa.s.sing out of s.p.a.ce 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 u?, v?, ? 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 u?dx + v?dy + ?dz + udu + vdv + wdw or u?dx + v?dy + ?dz + qdq, where q has the meaning specified above. Thus if S be the flow for the whole path PQ, and ? its time-rate of variation, S' denote the sum of u?dx + v?dy + ?dz along the path from P to Q, and q1, q0 the resultant fluid velocities at Q and P, we get ? = S' + (q1 - q0).

This is Thomson's theorem. If the curve be closed, that is, if P and Q be coincident, q1 = q0 and ? = S'. But in certain circ.u.mstances S' is zero, and so therefore is also ?. Thus in the circ.u.mstances referred to, as the closed path moves with the fluid ? is continually zero, and it follows that if ? is zero at any instant it remains zero ever after. But ? is only zero if u, v, w are derivable from a potential, single valued in the s.p.a.ce in which the closed path is drawn, so that the path could be shrunk down to a point without ever pa.s.sing out of such s.p.a.ce. 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 circ.u.mstances in which S' is zero are these:--the external force, per unit ma.s.s, 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 s.p.a.ce, and the path may or may not be drawn so that S shall be zero; but what is necessary for irrotational motion within any s.p.a.ce is that S should vanish for all paths which are capable of being shrunk down to zero without pa.s.sing out of that s.p.a.ce. 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 pa.s.s into the other by change of position without ever pa.s.sing in whole or in part out of the s.p.a.ce. The potential is always single valued in fluid filling a singly continuous s.p.a.ce such as that within a spherical sh.e.l.l, or between two concentric sh.e.l.ls; 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 ?, 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 ?dS 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 ? 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 ?, will lengthen or shorten as dS contracts or expands.

Now sum the values of ?dS for the finite surface enclosed by the bounding curve. It follows from the fact that ?dS is equal to half the circulation round the edge of dS, that this sum, which is usually denoted by S?dS, 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 S?dS 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 2S?dS for the surface to the circulation round its edge was expressed by Thomson as an a.n.a.lytical 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 ? 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.cos?. 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 S?dS, 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 pa.s.s across to the other apparent termination, when it also lies on the surface, while the other lines pa.s.s 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 S?dS 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 a.n.a.logy 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.

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.

[Ill.u.s.tration: FIG. 13.]

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 part.i.tion 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 ma.s.s 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 s.p.a.ce, 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 a.n.a.lysis, which, it was pointed out by Helmholtz, was necessary to adapt it to a s.p.a.ce which is multiply continuous. The theorem connects a certain volume-integral taken throughout a closed s.p.a.ce with an integral taken over the bounding surface of the s.p.a.ce.

This arises from the fact noticed above that in multiply continuous s.p.a.ce (for example, the s.p.a.ce 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 a.s.sumed for the starting point, by a certain definite amount which is the same for every circuit of the s.p.a.ce. 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 s.p.a.ce 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 s.p.a.ce, which are not to be crossed by any closed path drawn within the s.p.a.ce, and the two surfaces of each of which are to be reckoned as part of the bounding surface of the s.p.a.ce. One such diaphragm is sufficient to convert a hollow anchor-ring into a singly continuous s.p.a.ce, 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 a.n.a.logous 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 pa.s.s 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 pa.s.sed 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 s.p.a.ce 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.

CHAPTER X

THE ENERGY THEORY OF ELECTROLYSIS--ELECTRICAL UNITS--ELECTRICAL OSCILLATIONS

ELECTROLYSIS AND ELECTRICAL UNITS

In December 1851 Thomson communicated an important paper to the _Philosophical Magazine_ on "The Mechanical Theory of Electrolysis," and "Applications of Mechanical Effect to the Measurement of Electromotive Forces, and of Galvanic Resistances, in Absolute Units."

In the first of these he supposed a machine of the kind imagined by Faraday, consisting of a metal disk, rotating uniformly with its plane at right angles to the lines of force of a uniform magnetic field, and touched at its centre and its circ.u.mference by fixed wires, to send a current through an electrochemical apparatus, to which the wires are connected. A certain amount of work W was supposed to be spent in a given time, during which a quant.i.ty of heat H was evolved in the circuit, and a certain amount of work M spent in the chemical apparatus in effecting chemical change. If H be taken in dynamical units, W = H + M.

The work done in driving the disk, if the intensity of the field is I, the current produced c, the radius of the disc r, and the angular velocity of turning w, is Ircw.

Thomson a.s.sumed that the work done in the electrochemical apparatus was equal to the heat of chemical combination of the substance or substances which underwent the chemical action, taken with the proper sign according to the change, if more compound substances than one were acted on. Hence M represented this resultant heat of combination.

The electrochemical apparatus was a voltameter containing a definite compound to be electrolysed, or a voltaic cell or battery. And by Faraday's experiments on electrolysis it was known that the amount of chemical action was proportional to the whole quant.i.ty of electricity pa.s.sed through the cell in a given time, so that the rate at which energy was being spent in the cell was at any instant proportional to the current at that instant.

The chemical change could be measured by considering only one of the elements set free, or made to combine, by the pa.s.sage of the current, and considering the quant.i.ty of heat ?, say, for the whole chemical change in the cell corresponding to the action on unit ma.s.s of that element. Thus if E denote the whole quant.i.ty of that element operated on the heat of combination in the vessel was ?E. If E be taken for unit of time, and e denote the quant.i.ty set free by the pa.s.sage of unit quant.i.ty of electricity, then E = ec, since a current conveys c units of electricity in one second. The number e is a definite quant.i.ty of the element, and is called its electrochemical equivalent. Again, from Joule's experiments, H = Rc, if R denote the resistance of the current, and so

Ircw = Rc + ?ec

and

c = (Irw - ?e)?R

The quant.i.ty Irw is the electromotive force due to the disk.