Introduction

Abstract

About one hundred years ago, Maurice Couette, a French physicist, designed an apparatus consisting of two coaxial cylinders, the space between the cylinders being filled with a viscous fluid and the outer cylinder being rotated at angular velocity Ω₂. The purpose of this experiment was, following an idea of the Austrian physicist Max Margules, to deduce the viscosity of the fluid from measurements of the torque exerted by the fluid on the inner cylinder (the fluid is assumed to adhere to the walls of the cylinders). At least when Ω₂ is not too large, the fluid flow is nearly laminar and the method of Couette is valuable because the torque is then proportional to νΩ₂ where ν is the kinematic viscosity of the fluid. If, however, Ω₂ is increased to a very large value, the flow becomes eventually turbulent. A few years later, Arnulph Mallock designed a similar apparatus but allowed the inner cylinder to rotate with angular velocity Ω₁, while Ω₂ = 0. The surprise was that the laminar flow, now known as the Couette flow, was not observable when Ω₁ exceeded a certain “low” critical value Ω_1c, even though, as we shall see in Chapter II, it is a solution of the model equations for any values of Ω₁ and Ω₂. In fact, Mallock did not find a value of Ω₁ at which the Couette flow would be stable. However, as shown by Geoffrey I. Taylor in a celebrated paper of 1923 [Tay], an instability occurs for the Couette flow when Ω₁exceeds the value Ω_1c, which is lower than any of the values of Ω₁ used be Mallock.