A case study of chaotic mixing in deterministic flows: The partitioned-pipe mixer

We exploit the connection between the kinematics of mixing and the theory of dynamical systems. The presentation takes the form of a case study of a novel continuous flow mixer-the partitioned-pipe mixer-to exemplify the application of theoretical concepts relating fluid mixing in deterministic chaotic systems. Two general points are stressed: firstly, the complexities that are invariably encountered during the course of analysis limit the detail to which it may be carried out, and secondly, naive analysis based on direct use of the theory may result in misleading conclusions. Starting with an approximate Stokes flow velocity field in the partitioned-pipe mixer, we study the mixing in terms of the flow patterns in the cross-section (Poincaré sections) and their relation to the conventionally used continuous mixing diagnostic, the residence time distribution, as well as to the local specific rate of stretching of material lines and the mixing efficiency. Some of the limitations of each of these methods of characterizing the mixing are exposed; however, together they provide a broad description of the mixing in the partitioned-pipe mixer, and indicate the utility of the theory. The applications of the ideas, within and outside chemical engineering, are many. The most obvious are the mixing of viscous liquids (such as molten polymers), the design of mixing devices for shear sensitive molecules and cells under non-turbulent conditions, prototype models of porous media, enhanced mass transfer devices, etc. Other applications can be expected in geophysics, environmental fluid mechanics, and condensed matter and plasma physics.