Axisymmetric and non-axisymmetric modes of instability for flow between rotating cylinders with a transverse pressure gradient

Abstract

The stability of the flow of a Newtonian fluid between concentric cylinders is considered where the flow is also subjected to a transverse pressure gradient. The problem considered is thus a combination of the problem first considered by Taylor (1923) and Dean (1938) for pure rotation and pure pressure flow, respectively. The investigation is restricted to the case where the annulus between the cylinders is small and the numerical results are confined to the case where the outer cylinder is stationary. In considering the stability of such a system one seeks to determine the reaction of the system when subjected to small disturbances. If the disturbances decay with time so that the system approaches the original steady state condition as time →∞, the flow is considered stable. On the other hand, if the disturbances increase in magnitude with time so that the flow progressively departs from the steady state condition and never reverts to it, the flow is considered unstable. States of marginal stability (states separating the stable from unstable flows) can be one of two kinds, depending on whether the amplitude of a small disturbance grow (or are damped) aperiodically or grow (or are damped) by oscillations of increasing amplitude. In the former case, the transition from stability to instability takes place via a marginal state exhibiting a stationary pattern of flow. In the latter case, the transition takes place via a marginal state exhibiting oscillatory motion with a definite characteristic frequency. Previous theoretical investigations of this problem have assumed axisymmetric disturbances and that the critical mode of instability was of a stationary cellular motion. This investigation has not been restricted to this assumption but considered three conditions: (1) Assuming axisymmetric disturbances imposed on the original flow and a stationary marginal state, (2) Assuming axisymmetric disturbances and an oscillatory marginal state, and (3) Assuming non-axisymmetric disturbances and an oscillatory marginal state. The governing equations for the problem must be solved subject to certain necessary boundary conditions. In general the equations will not admit non-trivial solutions for an arbitrary set of system parameters but allow non-trivial solutions only for certain characteristic values. The resulting characteristic value problem has. been solved by a direct numerical process which is particularly useful in obtaining the eigenfunctions associated with the various modes of instability. The critical Taylor number (non-dimensional parameter characterizing the onset of instability for the problem) has been determined for a wide range of values of λ. The parameter λ is defined as the ratio of the average velocity of pumping to average velocity of rotation and thus defines the initial steady state velocity distribution. This investigation considered a larger range of values for λ (both positive and negative) than had been considered in the past for the case of axisymmetric disturbances and a stationary marginal state. The existence of oscillatory marginal states arising from axisymmetric disturbances has been demonstrated for certain negative values of λ. The existence of oscillatory modes arising from non-axisymmetric disturbances has also been demonstrated and these have been shown to be the critical mode of instability for the region between approximately λ=-1.5 and λ=-4.0. Outside this range the critical mode of instability results from axisymmetric disturbances and the marginal state is stationary. The existence of oscillatory modes for this problem had not been previously reported. Numerous curves are shown relating the various system parameters at the onset of instability and the FORTRAN computer programs used in the numerical calculations have also been included.