Chaotic mixing and mixing efficiency in a short time

Several studies of the chaotic motion of fluid particles by two-dimensional time-periodic flows or three-dimensional steady flows, called Lagrangian chaos, are first introduced. Secondly, some of the studies on efficient mixing caused by Lagrangian chaos, called chaotic mixing, are reviewed with discussion of several indices for the estimation of mixing efficiency. Finally, several indices to estimate the efficiency of mixing in a short time, such as those related to transport matrices, stable and unstable manifolds of hyperbolic periodic points of Poincaré maps, and lines of separation, are explained by showing examples of mixing by two-dimensional time-periodic flows between eccentric rotating cylinders and mixing by three-dimensional steady flows in a model of static mixers.