William McF Orr's Stability or Instability of Motions of a Liquid
Part I. A Perfect Liquid.
Introduction and Summary of Contents.
It is a well-known experimental fact that when a liquid of small viscosity, such as water, flows through a straight circular pipe under applied pressure or under the action of gravity, the steady motion - in which, of course, each particle describes a straight line - may be unstable. The subject has been investigated experimentally by Osborne Reynolds, who found that the motion is stable so long as the mean velocity does not exceed a certain limit depending on the radius of the pipe and on the nature of the liquid. This limit, beyond which instability sets in and the motion becomes turbulent, he found to vary directly as the kinematic viscosity, and inversely as the radius of the pipe - results to which he was led also by considerations of the theory of dimensions. The question has also been attacked theoretically, chiefly by Lord Rayleigh, Lord Kelvin, and Reynolds himself. Lord Rayleigh has ignored the effect of viscosity in the disturbed motion - a simplification which renders such problems much more amenable to mathematical treatment. ... Portions of Lord Rayleigh's argument have, however, been criticised adversely by Lord Kelvin and by Love. When viscosity is taken into account, the mathematical difficulties involved in a discussion of the question of stability are much greater. Lord Kelvin has attacked the question under such conditions. ... The investigation here presented deals exclusively with questions in which viscosity is altogether ignored. ... It is held that as far as this investigation goes no contradiction between theory and experiment is revealed. The apparent paradox that the motion of a liquid devoid of viscosity, if such existed, would be stable, while that of an actual liquid of small viscosity is found by experiment to be highly unstable, is disposed of by showing that though the perfect liquid may be said to be stable if the disturbance is small enough, yet the, limit of stability, or, to be accurate, the limit within which it is legitimate to rely on equations which take account of only the first powers of small quantities, depends on the nature of the disturbance, and may be diminished indefinitely by a suitable choice. ...
Part II. A Viscous Liquid.
Introduction and Summary of Contents.
In Part I reference was made to a well-known difficulty in reconciling theory and experiment in the case of the steady motion of liquids. The flow through pipes and between concentric cylinders, one of which is rotated, had been found experimentally to be unstable if the velocity is great enough; while, on the other hand, Lord Rayleigh had shown that, in these cases, if the effect of viscosity be neglected in the disturbed motion, the fundamental free disturbances are strictly periodic, the values of the "free periods" being real. An explanation of the difficulty was given by showing that it is necessary to push Lord Rayleigh's investigations a step farther by resolving a disturbance into its constituent fundamental ones by quasi-Fourier analysis, and that, when this is done for disturbances of initially simple type in some of the most important and simplest cases of flow, it is found that the disturbance will, for suitable values of the constants, increase very much, so that the motion is practically unstable. The present investigation attempts to discover how far this conclusion must be modified when viscosity is taken account of. It may be stated at once that I have not succeeded in throwing much additional light on this matter; but a good deal of the work had been done before I discovered that the slight extension of Lord Rayleigh's analysis which is contained in Part I. would explain the difficulty, at least qualitatively; and I therefore decided to carry the investigation as far as I could: I may moreover plead that I found some portions of the analysis interesting on their own account. ... As the fundamental modes of disturbance do, as is shown in Chapter II., possess stability of the simple exponential character, the "special" solution is, I believe, as a matter of fact, the solution for a given initial disturbance; if this be a simple trigonometrical function of the coordinates, the form of v is simple; but that of the "forced" disturbance in no case appears capable of being readily calculated. ...
JOC/EFR February 2016
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