How Fluids Mix: Stretching Fields and Nonlinear Dynamics

The mixing of an impurity into a flowing fluid is an important process in many areas of science, including geophysical fluid dynamics and chemical systems. This problem also played an important role in the foundations of statistical mechanics. Mixing can be produced by either chaotic advection or turbulence, which stretch fluid elements, in conjunction with diffusion. However, a clear understanding of the mixing process is still lacking. In some cases, for example periodic flows, the concepts of nonlinear dynamics provide a deep theoretical basis for understanding mixing. Unfortunately, the building blocks of this theory, i.e. the fixed points and invariant manifolds of the associated Poincare map of the flow, have remained inaccessible to direct experimental study, thus limiting the insight that could be obtained.

In this Lecture, I will describe precision measurements of tracer particle trajectories that allow these dynamical objects to be determined for the first time. A particular advance is the measurement of the time-dependent stretching field, which controls the evolution of mixing. The dynamic range of the stretching field extends over 12 orders of magnitude, and it outlines outline the invariant manifolds of the flow map. The talk will include a number of striking digital video clips illustrating mixing in fluids and its explanation using basic nonlinear dynamics.