The stability of an infinite system of circular vortices

Abstract

A system of equal circular vortex filaments have their centres evenly spaced along a straight line, and their planes at right angles to this line. The present investigation is concerned with the stability or instability of such an arrangement. The corresponding problem in two dimensions has been dealt with by Kármán who considered the case of two infinite trails of parallel rectilinear vortices, with the object of applying his results to the resistance of an infinite cylinder moving in a fluid and to the state of motion in the rear of the cylinder. The infinite system of circular vortex filaments, on the other hand, may be supposed, in certain circumstances to be discussed in a later paper, to be generated in the rear of a three-dimensional body in motion in a fluid. The present investigation may therefore be regarded as a first step towards an examination of the three-dimensional problem analogous to that treated by Kármán for two dimensions. A special difficulty arises in an investigation of this type from the fact that the system of vortex rings, possessing an infinite number of degrees of freedom, are capable of adopting an infinite number of possible configurations about the position of equilibrium.