Lagrangian chaos and mixing process of fluids

1.从数值和实验两方面分析了两个偏心圆柱体交替慢周期旋转引起的流体运动。在Stokes近似下考察流体颗粒在速度场中的运动，我们发现：(1)流体区域分为规则区域和混沌区域(胞格拉格朗日混沌)。(2)当圆柱体反向旋转且偏心率较小时，在每个旋转周期后，流体粒子位置所定义的庞加莱映射的双曲不动点附近出现混沌。(3)混沌区域面积的增大与Poincare MAP椭圆不动点的失稳有很强的相关性。在圆柱体周期旋转N周后时间反转的实验中，我们发现：(1)从规则区域开始的染料几乎返回到其原始位置，而从混沌区域开始的染料最终偏离其原始位置相当大，并且随着N的增加而迅速增加。(2)这一行为反映了混沌轨道的不稳定性。从动力系统理论考虑流体的混合过程，我们发现：(1)规则区域越小，混合越均匀。因此，重要的是要检查Poincare映射的椭圆不动点何时发生失稳。(2)规则区和混乱区的混合效率相差很大。(3)根据局部Lyapunov指数的计算和两种流体的几条边界线的运动和伸展，在考虑有限时间内混合时，将混沌区域划分为快混合区和慢混合区。区分它们是很重要的。

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.