Mathematical Methods for Chaotic Advection in Three-dimensional Fluid Flows
PI: Igor Mezic
The goal of the proposed research is to extend the existing theory of chaotic advection and mixing in two-dimensional, time-dependent flows. In particular, we wish to study three-dimensional incompressible steady and unsteady and two- and three-dimensional compressible flows using the methods of geometric theory of dynamical systems. We will also study effects of reaction and diffusion on the motion of particles in these flows, properties of chaotic advection in general, and particular models of great relevance in engineering applications: flows between concentric and eccentric rotating cylinders. These problems will be addressed starting from our recent developments in geometric theory of three-dimensional divergence-free vector fields. The issues of cantori, resonances and lobe dynamics in volume-preserving maps and flows will be addressed through theoretical analysis and computer simulation. Connection with experiments on mixing in three-dimensional flows will be made. Effects of inertia and viscosity on chaotic mixing will be studied via asymptotic, large Reynolds number analysis in conjunction with the transport theory of dynamical systems. This will allow for discussion of the change of mixing properties of laminar, incompressible, viscous flows with the change of the Reynolds number. Further, we will study the effects of molecular diffusion on chaotic advection based on the perturbative multiple-scales method combined with ergodic theory, and through numerical simulations. We will investigate a general relationship between the pure advection problem (possibly with chaotic mixing) and the full advection-diffusion problem. We will study reaction-diffusion-advection equations through the combination of tools mentioned before. We shall pursue nonlinear stability analysis in search of effects of chaotic advection on stability. The particular examples we will be studying are flows between concentric rotating cylinders. Compressible flows received virtually no attention in chaotic advection studies. We propose to remedy this situation by pursuing a basic study of this problem. Simple model flows will be identified starting with a compressible vortex flow in a cylindrical container. Comparison with the behavior of incompressible flows will be pursued.
The above study is useful in a variety of technological contexts. There is a recent surge of interest in micro- and nano-scale technology that poses a number of mathematical challenges. For example, flows in devices such as large mixers and combustion chambers are designed to mix well by moving the flow in a turbulent regime and thus causing enhanced mixing by rapid random movements of fluid in the flow. This is not possible in microscopic devices. The process of mixing needs to be understood much better in order to design microscopic mixers, combustion chambers etc. necessary as the building block of microscopic processing devices and microengines. The mixing process in such devices is typically three-dimensional. Thus, the design of microscopic mixing devices will benefit from the fundamental study of three-dimensional mixing processes outlined above. In addition, the performance of macroscopic devices is challenged by new requirements on the levels of environmental chemical pollution (NO_x) and sound pollution. The improvement in the design of these devices will be based on a better understanding of the underlying mixing processes. The above study will provide some of the crucial concepts for such design by unraveling the fundamentals of mixing in three-dimensional flows and using these concepts to study the effect of mixing on combustion and noise production.
