On the stability of a viscous liquid between rotating coaxial cylinders

Abstract

1. From the fullness of G. I. Taylor’s theoretical discussion (1923) of the stability of a viscous liquid contained between rotating coaxial cylinders and the remarkable experimental confirmation of his results by himself and by Lewis (1927), it might be thought that the problem has been dealt with completely. However, it must be borne in mind that Taylor’s results depend on the solution of an infinite determinantal equation, which presents a problem insoluble from a practical standpoint except when simplifications arise in a limiting case. Taylor’s deductions deal only with the case when the distance between the cylinders is small in comparison with the radius of either, and even then the calculations are formidable. General results concerning stability, obtained without heavy calculations and not limited by any approximation, appear worthy of attention. The present paper contains one such result, namely, the proof of stability if ω 2 r 2 2 >ω 1 r 2 1 , (1.1) where ω 1 , r 1 = angular velocity and radius of the inner cylinder, ω 2 , r 2 = angular velocity and radius of the outer cylinder, ω 1 being (without loss of generality) assumed to be positive. Stability under the condition (1·1) was established by Taylor for the limiting case he considered, and has been confirmed by the experiments mentioned. With regard to the disturbance, Taylor assumed (i) independence of the azimuthal angle ϕ, (ii) spatial periodicity in the direction of the generators of the cylinders: these assumptions are also made in the present paper.