Lord Kelvin: An account of his scientific life and work

by Andrew Gray

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.