Lagrangian chaos and mixing process of fluids

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1. The motion of a fluid due to the alternate slow periodic rotations of two eccentric cylinders was exmined numerically and experimentally.2. Examining the motion of fluid particles in the velocity field under the Stokes approximation, we found that (1) The fluid region is divided into a regular region and a chaotic region (celled Lagrangian chaos) (2) When the cylinders rotate in the opposite directions and the eccentricity is small, chaos appears around the hyperbolic fixed point of the Poincare map defined by the position of a fluid particle after every periods of the rotation. (3) There is strong correlation between the increase in the area of the chaotic region and the destabilization of the elliptic fixed point of the Poincare map.3. In the experiments on the periodic rotation of the cylinders for N periods followed by its time-reversal, we found that (1) The dye starting from the regular region almost returns to its original position, whereas the final deviation of the dye starting from the chaotic region from its original position is quite large, and increases rapidly with N.(2) This behavior reflects the orbital instability of chaos.4. From the consideration of the mixing process of fluids based on the theory of dynamical systems, we found that (1) The smaller area of the regular region is better for the uniform mixing. Therefore, it is important to examine when the destabilization of the elliptic fixed point of the Poincare map occurs. (2) The difference between the mixing efficiencies in the regular and chaotic regions is very large. (3) According to the computations of the local Lyapunov exponent and the motion and the stretching of several boundary lines of two fluids, the chaotic region is divided into the regions of fast and slow mixing when we consider the mixing in a finite time. It is important to distinguish them.