The Thurston-Nielsen theory provides a powerful collection of tools to analyze and understand two-dimensional dynamical systems. The theory has potential for significant advances in the study of two-dimensional systems of physical interest. Among the most interesting and important come from fluid mechanics. Dr. Boyland will use Thurston-Nielsen theory in conjunction with other dynamical tools to obtain a deeper understanding of two-dimensional fluid mixing. Various mathematical extensions of the Thurston-Nielsen theory will be obtained. These extensions would enhance the applicability of the theory as well as being of intrinsic mathematical interest. In addition, numerical and laboratory experiments are proposed with the goal of designing effective mixers and understanding many vortices as a model for two-dimensional turbulence.
The mixing of different species of materials is extremely common in natural systems and industrial applications. The main scientific question is to understand how the mixing takes place, while in industrial applications the focus is usually on the development of schemes for effective mixing. In many applications high viscosity, sensitivity of materials, the need to avoid foaming, and other constraints make turbulence mixing infeasible and undesirable. Thus in most instances the design of useful mixing procedures requires close attention to efficiency. The added mathematical understanding made possible by this project should help in this design process, providing novel ideas for efficient mixing procedures and helping to eliminate inefficient ones prior to expensive experimentation. In addition, this project will make a powerful body of pure mathematics more accessible for application to problems of physical and industrial importance.
