Quasiperiodic drift flow in the Couette-Taylor problem

Abstract

The Couette-Taylor problem deals with the flow of an incompressible, viscous fluid between two coaxial rotating cylinders. Depending on the angular velocities of the cylinders, different flow patterns are observed in experiments. Mathematically, transitions between different flow patterns can be described by instabilities and bifurcations of certain solutions of the Navier-Stokes equations for this problem. In this paper we consider the counterrotating case, i.e. the cylinders rotate in opposite directions. We describe a sequence of three successive instabilities and corresponding bifurcations which occur in a certain parameter regime when the Reynolds number is increased. The primary bifurcation is the classical bifurcation from Couette flow to Görtler-Taylor vortex flow (cf. Taylor [1923]), the secondary one leads to wavy vortex flow, and the tertiary one leads to what we call quasiperiodic drift flow. We also indicate how this result is linked to related work on the Couette-Taylor problem and give an outline of the method by which we have obtained it. A key step of our method is the reduction of the Navier-Stokes equations to a system of ordinary differential equations. This is achieved by invariant manifold theory and ideas from the theory of dynamical systems with symmetry.