VIII. Stability of a viscous liquid contained between two rotating cylinders

TLDR

Fluid flow near solid boundaries has been extensively studied, yet a general mathematical description of instability remains elusive, especially for configurations such as two moving parallel planes. Experiments show steady motion is maintained only below a critical speed that depends on viscosity and boundary configuration, beyond which the flow becomes unstable and eddying, although small disturbances in the two‑plane case remain stable.

Abstract

In recent years much information has been accumulated about the flow of fluids past solid boundaries. All experiments so far carried out seem to indicate that in all cases steady motion is possible if the motion be sufficiently slow, but that if the velocity of the fluid exceeds a certain limit, depending on the viscosity of the fluid and the configuration of the boundaries, the steady motion breaks down and eddying flow sets in. A great many attempts have been made to discover some mathematical representation of fluid instability, but so far they have been unsuccessful in every case. The case, for instance, in which the fluid is contained between two infinite parallel planes which move with a uniform relative velocity has been discussed by Kelvin, Rayleigh, Sommerfeld, Orr, Mises, Hope, and others. Each of them cam e to the conclusion that the fundamental small disturbances of this system are stable. Though it is necessarily impossible to carry out experiments with infinite planes, it is generally believed that the motion in this case would be turbulent, provided the relative velocity of the two planes were sufficiently great.