STOCHASTIC PARTICLE MOTION IN LAMINAR FLOWS

The practical actions referred to as stirring, mixing, blending, and so on rely on our ability to produce stochastic trajectories of particles within given flow fields. There are time-honored ways of achieving stochastic particle motion within a fluid, in particular, by exploiting molecular diffusion and through advection by a turbulent flow. There are, however, also other methods, particularly useful for laminar flows, that rely on the notion of chaos in a low-dimensional dynamical system for their explanation and elucidation. Indeed, the theory of dynamical systems is one of the few places where the term "mixing" has a precise technical meaning. The paper reviews several of these concepts, many of them taken from ergodic theory, and attempts to explain their application to the fluid mechanical problems. All these developments rely on a correspondence between the "phase space" of the general dynamical system, and the real-space motion of the fluid in the stirring/mixing problem. Other aspects of theoretical physics that appear useful are the kinetic theory of Brownian motion and equilibrium statistical mechanics. Again, pertinent concepts, definitions, and results are stated. The theoretical ideas presented offer some measure of general understanding for simple problems of stirring and mixing.