XIII. On the motion of two spheres in a fluid

The general theory of the motion of a single rigid body through an infinite incompressible fluid is well known, chiefly through the work of Thomson and Tait and Kirchhoff, and we are able to calculate numerically the results in the case of the sphere, the ellipsoid, and a large number of cylindrical surfaces. The theory of the motion of two or more bodies in a fluid has naturally not made the same progress, and we are unable to determine the form of the expressions involved for the general motion of any particular solids. So far as I am aware, the first attempt was made by Stokes, in a paper read before the Cambridge Philosophical Society in 1843, entitled “On some cases of Fluid Motion.” In this paper, amongst other problems, he considers the case of two spheres. He determines the instantaneous velocity potential for two concentric spheres and for two concentric cylinders with fluid between them, and finds that the effect of the fluid is to increase the inertia of the inner sphere by a mass = ½. a 3 + 2 b 3 / b 3 - a 3 of the mass of the fluid displaced, and that of the inner cylinder by a mass b 2 + a 2 / b 2 - a 2 of the displaced, a, b , being the radii of the spheres or cylinders. He also approximates to the cases where one sphere is moving in the presence of another in an infinite fluid; and also in the presence of a plane, the method used being first to calculate the velocity potential for any motion of the points of the plane, and then suppose them actually animated with velocities equal and opposite to the normal velocities of the fluid motion at those points if the plane had been removed. He applies the same method also to the consideration of the motion of two spheres. In a note in the Report of the British Association at Oxford, 1847, he states the theorem given by me in § 4 relating to the image of a doublet whose axis passes through the centre, and mentions that this will easily serve to determine the motion. In 1863 Herr Bjerknes communicated a paper to the Society of Sciences at Christiana, on the motion of a sphere which changes its volume, and in which he approximates for the motion of two spheres. I have not been able to see this paper, nor some others which he presented to the same Society at some later periods; but he has given an account of his researches in the 'Comptes Rendus,' together with some historical notices on the development of the theory. He does not seem, however, to have been acquainted with the important paper of Stokes above referred to.† In 1867 Thomson and Tait’s ‘Natural Philosophy’ appeared, containing general theorems on the motion of a sphere in a fluid bounded by an infinite plane, viz.: that a sphere moving perpendicularly to the plane moves as if repelled by it, whilst if it moves parallel to it it is attracted. In a paper on vortex motion in the same year (Edin. Trans., vol. xxv.), Thomson proved that a body or system of bodies passing on one side of a fixed obstacle move as if attracted or repelled by it, according as the translation is in the direction of the resultant impulse or opposite to it. In the ‘Philosophical Magazine’ for June, 1871, Professor Guthrie publishes some letters from Sir W. Thomson on the apparent attraction or repulsion between two spheres, one of which is vibrating in the line of centres. Results only are given, and he states that if the density of the free globe is less than that of the fluid, there is a “critical” distance beyond which it is attracted, and within which it is repelled. The problem of two small spheres is also considered by Kirchhoff in his ‘Vorles. ü. Math. Phys.,’ pp. 229, 248. In his later papers Bjerknes takes up the question of “pulsations” as well as vibrations. Of solutions for other cases than spheres, Kirchhoff has considered‡ the case of two thin rigid rings, the axes of the rings being any closed curves and the sections by planes perpendicular to the axis being small circles of constant radii, and he arrives at the result that their action on one another may be represented by supposing electric currents to flow round them; and I have recently solved the problem of the motion of two cylinders in any manner with their axes always parallel. The velocity potentials for the motion of the two cylinders are found in general as definite integrals, which, when the cylinders move as a rigid body, are expressed in a simple finite form as elliptic functions of bipolar coordinates. The functions involved in the coefficients of the velocities in the expression for the energy have a close analogy with those for spheres arrived at in the following investigation.