VII. On the steady motion and small vibrations a hollow vortex

Abstract

The following pages form a continuation of some researches commenced about three years ago, but which the author was compelled by other engagements to lay aside until the beginning of the present year. The general theory of the functions employed was published in the Transactions of this Society (Part III., 1881), under the title of “Toroidal Functions.” These and analogous functions are employed in the present communication, and references in square brackets, with the letters T. F., refer to this paper. Since it was written I have found that Carl Neumann had already given the general transformation [T. F. §1] by means of conjugate functions, in a pamphlet published at Halle in 1864, with the title ‘Theorie der Elektricitäts- and Wärme-Vertheilung in einem Ringe.' The theory of the motion of vortices is interesting, not only from the mathematical difficulties encountered in its treatment, but also from its connexion with Sir W. Thomson’s theory of the vortex atom constitution of matter. In an abstract of the present paper intended for the Proceedings of this Society, I have given some physical speculations which induced me to take up the question of the motion of a hollow vortex—that is, where cyclic motion exists in a fluid without the presence of any actual rotational filaments—in which case there must be a ring-shaped hollow in the fluid, however great the pressure may be, so long as it is finite. The essential quality of all vortex motion is the cyclic motion existing in the fluid outside the filament, and not the rotational motion of the filament itself. Whether the filament be present or not, it is often possible to get some general idea of the motion that ensues in many cases without recourse to actual calculation. Thus, for instance, the treatment by Sir W. Thomson of the action of two vortices on one another, and of the form of the axis of a ring, along which waves of displacement are running, may be cited. The same course of general reasoning, which was applied in a paper on the steady motion of two cylinders in a fluid, will also apply to illustrate the mechanism , so to speak, which causes a single vortex ring to move with a motion of translation. Thus suppose a single vortex ring, which is for a moment at rest. It is clear that the velocity of the fluid just inside the aperture is greater than outside, and therefore the pressure less inside than outside, whilst the pressure is the same at corresponding points in the front and hinder portions. The consequence of this is that the ring begins to contract without a general motion of translation. But the effect of this contraction of aperture itself produces velocities in the surrounding fluid, which, combined with the cyclic motion, increase the velocities in front of the ring, and decrease them behind. The consequence of this is a difference of pressures, which urges the ring in the direction of the cyclic motion through the ring, and it begins to move forward with increasing velocity. After a time this translatory motion would increase so much as to make the velocity within the aperture approach to that with out; the state of motion will therefore be one in which the translatory velocity tends continually to a limit.