Stability and chaotic advection in the eccentric Taylor problem

Abstract

Theoretically, Lagrangian chaos or chaotic advection can occur by forcing a steady, two-dimensional velocity field with a time-periodic perturbation. This idea has recently been confirmed experimentally by Chaiken et al.1 and Chien et al.2 In principle, chaotic advection should also occur in a three-dimensional steady flow. To investigate this problem we constructed an eccentric Taylor–Couette apparatus with a rotating inner cylinder and a stationary outer cylinder. We obtain a 3-D steady flow in the following way. We first create a 2-D velocity field at small rotation rates of the inner cylinder by imposing a sufficiently large ecccentricity to obtain separated flow. (That is, an eddy in the region of largest gap is created by the separation of the fluid from the outer boundary and its reattachment downstream.) The inner cylinder rotation rate is then increased until Taylor vortices appear. The Taylor vortices modify the separated flow but do not destroy it. It is in this mixed Taylor vortex-separated flow regime that we carry out our studies of chaotic advection. On account of mathematical difficulties there exist few theoretical or experimental studies of stability of the flows in the eccentric geometry. We have therefore conducted numerical and laboratory experiments to identify the regions of parameter space where 3-D steady flow exists and where a transition of time dependence (i.e., 4-D flows) occurs. In our numerical work we used a commercial computational fluid dynamics program and we will report on our assessment of its accuracy in predicting the three-dimensional flow features observed in the experiments and its potential for investigating chaotic advection. In particular, we use it to find the ‘‘skeleton’’ of the flow and the fluxes between the associated regions of different flow type that it defines, and compare these with experimental observations.