Stability of a viscous liquid contained between two rotating cylinders

Abstract

Part I .—The stability for symmetrical disturbances of a viscous fluid in steady motion between concentric rotating cylinders is investigated mathematically. It is shown that at slow speeds the motion is always stable, but that at high speeds the motion is only stable when the ratio of the speed of the outer cylinder to that of the inner one exceeds a certain value. When the ratio is less than this or when it is negative the motion becomes unstable at high speeds. The “criterion” for stability is found, and in cases suitable for experimental verification an approximate form for the “criterion” is developed which is useful for numerical computation. The type of instability which may be expected to appear when the speed of the cylinders is slowly increased is shown to consist of symmetrical ring-shaped vortices spaced at regular intervals along the length of the cylinders. These vortices rotate alternately in opposite directions. Their dimensions are calculated and it is shown that they are contained in partitions of rectangular cross-section. In the case when the instability arises while both cylinders are rotating in the same direction, these rectangles are squares, so that the vortices are spaced at distances apart equal to the thickness of the annular space between the two cylinders. In the case when the cylinders rotate in opposite directions the spacing, or distance between the centres of neighbouring vortices, is smaller than this; and at the same time two systems of vortices develop—an inner system which is similar to the system which appears when the two cylinders rotate in the same direction, and an outer system, which is much less vigorous and rotates in the opposite direction to the adjacent members of the inner system.