Stability of viscous flow between rotating cylinders. II. Cylinders rotating in opposite directions

Abstract

In a previous paper (Meksyn 1946; it will be referred to as Part I) the problem of stability of viscous fluid between coaxial cylinders, rotating in the same direction, was solved by expanding the integrals in inverse powers of a large parameter. The question was first considered theoretically and experimentally by Taylor (1923), the problem being solved by expanding the integrals in orthogonal Bessel functions. The aim of the present paper is to extend the solution to the case when the cylinders rotate.in opposite directions. The important difference between the two cases in its mathematical aspect consists of the following. It is necessary to find the asymptotic integrals of a certain linear differential equation. In the case when the cylinders rotate in opposite directions, these integrals become infinite within the range under consideration; namely, approximately at the point where the mean velocity of rotation is equal to zero, and the asymptotic expansions change their form in passing through this point. It is, therefore, necessary to find the law of transformation of these integrals; that requires a rather extensive mathematical investigation. For the sake of convenience the work is divided into two separate parts, hydro - dynamical (Part II) and mathematical (Part III). In the present paper (Part II) the transformations are only quoted, whereas in the mathematical part the detailed solution of the equations is developed. The main results obtained are as follows.