Engineers and scientists generally think in terms of three means to mix fluids, which can be easily explained in terms of mixing cream in coffee. Diffusion disperses the cream in a cup of coffee via molecular collisions; chaotic advection, or stirring, generates thin layers of coffee and cream that merge; turbulence mixes the coffee and cream by strong agitation or vigorous stirring. However, the simple technique of cutting and shuffling, much like what takes place in shuffling a deck of cards or mixing the colors of a Rubik's cube, is also a means of mixing that is commonplace. However, it is not well understood in the context of engineering systems such as mixing powders and granules for pharmaceuticals, plastic resins, and chemicals. This award supports the development of a formal mathematical framework to predict mixing by cutting and shuffling that is validated by computational and experimental results. The theory is inspired by abstract mathematical concepts, but is motivated by systems for blending and mixing bulk particles or granules, a common process in the chemical and pharmaceutical industries. The research links fundamental mathematics and practical applications while training the next generation of scientists and engineers.
While molecular diffusion, chaotic advection, and turbulence as mixing mechanisms have long been studied, mixing by cutting and shuffling is not well explored or understood. This award centers on a relatively new mathematical approach that describes cutting and shuffling known as Piecewise Isometries (PWI). The idea is to apply PWI concepts to mixing as well as to connect PWI with traditional dynamical systems approaches. The objective is to develop and utilize PWI within a dynamical systems framework to predict mixing by cutting and shuffling in 2D and 3D geometries with eventual applications to practical mixing systems. The development of the 3D dynamical systems framework will be informed by the mixing of granular materials, leading to an integrated theory inspired by abstract mathematical concepts but connected to practical applications of cutting and shuffling. The mathematics of PWI will be integrated into the dynamical systems toolkit, while complementary experiments and simulations will be used to confirm the applicability of the theoretical approaches to a model granular mixing system. The merging of new mathematics, dynamical systems approaches, and physical applications could result in an entirely new paradigm for understanding and predicting mixing.
