Lagrangian chaos and mixing process of fluids

1.对两个偏心圆柱体交替缓慢周期旋转引起的流体运动进行了数值模拟和实验研究.在Stokes近似下考察流体质点在速度场中的运动，发现：（1）流体区域分为规则区域和混沌区域（2）当圆柱体反向旋转且偏心距较小时，在每一个旋转周期之后，由流体粒子的位置定义的庞加莱映射的双曲不动点周围出现混沌。(3)混沌区域面积的增加与Poincare映射椭圆不动点的失稳有很强的相关性.在圆柱周期性旋转N个周期后进行时间反转的实验中，我们发现：（1）从规则区开始的染料几乎回到原来的位置，而从混沌区开始的染料最终偏离原来的位置很大，并且随着N的增加而迅速增加; (2)这种行为反映了混沌的轨道不稳定性.从动力系统理论出发，对流体的混合过程进行了研究，发现：（1）规则区域的面积越小，越有利于流体的均匀混合;因此，研究Poincare映射的椭圆不动点何时失稳是很重要的。(2)在规则区域和混沌区域的混合效率之间的差异是非常大的。(3)通过计算局部李雅普诺夫指数和两种流体边界线的运动和拉伸，将有限时间内的混合分为快混合区和慢混合区。重要的是要区分它们。

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.