Fluid motion in a curved channel

Abstract

Experimental work due to Prof. J. Eustice has shown that there is no marked critical velocity for a fluid flowing through a curved pipe. If the pipe is straight there is a sudden increase in the loss of head as soon as the velocity exceeds its critical value; below the critical the loss of head varies as the first power of the velocity, but above it approximately as the second power. But if the pipe is curved there does not appear to be such a sudden change at any velocity of flow. One possibility is that flow through a curved pipe is stream-line at velocities much greater than the critical for a straight pipe, but experiment seems to show that the critical velocity is smaller in a curved pipe than in a straight one. If then the motion in a curved pipe becomes unstable at a velocity somewhat less than the critical for a straight pipe, the absence of a sudden increase in the loss of head in this region suggests that the stream-line motion in a curved pipe (unlike that in a straight pipe) is unstable for small disturbances. A similar problem shows that it is not unlikely that curvature may have such an effect: it is believed that uniform shearing motion between flat plates is stable for small disturbances, but Prof. G. I. Taylor has shown that shearing motion between concentric cylinders can in certain conditions become unstable for small disturbances. A theoretical investigation of the stability of flow in a curved pipe is certain to be a matter of great difficulty, and therefore a simplified form of the problem, the stability of flow under pressure through a curved channel ( i. e. , between concentric cylinders), is here considered. It is shown that the motion can become unstable for a small disturbance of exactly the type found by Taylor to be possible in motion between rotating cylinders.