Lord Kelvin: An account of his scientific life and work

by Andrew Gray

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.