Lord Kelvin: An account of his scientific life and work

by Andrew Gray

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.