On the stability of a plain hollow vortex: PMM vol. 36, n≗1, 1972, pp. 60–64

Abstract

Proof is given of the idifferent stability with respect to small perturbations of two flows: a hollow vortex bounded on the outside by a circular wall, and a free hollow vortex. A method of analyzing the stability of plane potential flows of a perfect incompressible fluid with respect to small perturbations was suggested in [1] by which the difficulties arising in the determination of eigenfunctions of two-dimensional hydrodynamic flows. The method proposed there for the analysis of stability consists of the linearization of equations of hydrodynamics by conformal mapping of the unperturbed flow region onto that of the perturbed flow. It is applicable to fairly simple regions of the unperturbed flow, otherwise the feasability of conformal mapping becomes problematic. This aspect was not touched upon in [1]; some of the flows considered by the Authors cannot be analyzed in this way, since for these conformal mapping is impossible. Neither the question of completeness of the system of eigenfunctions in cases in which mapping is possible was investigated by them. It is, therefore, interesting to examine the equations arising in investigations of small perturbations of stationary flows by the method of conformal mapping, to determine its limits of applicability and, also, to solve Cauchy's problem in terms of perturbation equations.