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The stability of Couette flow and flow due to an azimuthal pressure gradient between arbitrarily spaced concentric cylindrical surfaces is investigated. The stability problems are solved by using the Galerkin method in conjunction with a simple set of polynomial expansion functions. Results are given for a wide range of spacings. For Couette flow, in the case that the cylinders rotate in the same direction, a simple formula for predicting the critical speed is derived. The effect of a radial temperature gradient on the stability of Couette flow is also considered. It is found that positive and negative temperature gradients are destabilizing and stabilizing, respectively.

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A unified treatment is given of the Eckhaus mechanism of stability or instability of two-dimensional flows, which are periodic in one spatial dimension, and the Benjamin—Feir instability mechanism of the two-dimensional Stokes water wave. The method of the amplitude equation is used, following the lead of Newell in a related context. This method easily allows the analysis of the so-called side-band perturbations, which are a crucial feature of the Eckhaus and Benjamin—Feir resonance mechanisms. In particular, it is shown that Eckhaus’s result, that a periodic flow is stable only within a particular band of wavenumbers narrower than the span of the neutral curve of linearized theory, is only valid when the eigenvalues and other parameters are real. A corrected and extended form of the result is given for the general case of complex eigenvalues and coefficients. It is noted, however, that Eckhaus’s result is valid for the important examples of Taylor vortices and Bénard cells.

This paper deals with a system of equations which includes as special cases the equations governing such hydrodynamic stability problems as the Taylor problem, the Bénard problem, and the stability of plane parallel flow. A non-linear analysis is made of disturbances to a basic flow. The basic flow depends on a single co-ordinate η. The disturbances that are considered are represented as a superposition of many functions each of which is periodic in a co-ordinate ξ normal to η and is independent of the third co-ordinate direction. The paper considers problems in which the disturbance energy is initially concentrated in a denumerable set of ‘most dangerous’ modes whose wave-numbers are close to the critical wave-number selected by linear stability theory. It is a major result of the analysis that this concentration persists as time passes. Because of this the problem can be reduced to the study of a single non-linear partial differential equation for a special Fourier transform of the modal amplitudes. It is a striking feature of the present work that the study of a wide class of problems reduces to the study of this single fundamental equation which does not essentially depend on the specific forms ofthe operators in the original system of governing equations. Certain general conclusions are drawn from this equation, for example for some problems there exist multi-modal steady solutions which are a combination of a number of modes with different spatial periods. (Whether any such solutions are stable remains an open question.) It is also shown in other circumstances that there are solutions (at least for some interval of time) which are non-linear travelling waves whose kinematic behaviour can be clarified by the concept of group speed.