Fluid mixing and dynamical systems

Abstract

We have discussed several aspects of mixing in fluid flows, first by concentrating on a specific mixing problem - that of a simple diffusion flame with fast chemical kinetics in an arbitrary flow. The dependence of the local reactant consumption rate on the history of the local stretching of the flame sheet was determined up to the point where the flame sheet folds over on itself and the product zones collide. The analysis was applied to a simple, steady vortex flow producing a spiral flame sheet. In this case, the specific mixing problem is completely solved. Next we considered the flow due to a pair of translating point vortices subjected to a time-periodic potential flow. In this flow fluid is continually entrained into and detrained from a “turbulent” zone that moves with the translating pair. The original mixing problem was not solved but a considerable amount of information about entrainment rates, size of the mixing zone, etc., was obtained by using concepts and techniques from dynamical systems theory. In addition good insights into the mixing process were obtained. Tin particular, it was shown how the simple lobe structures of the stable and stable maniforlds, with their ordered behavior under the Poincare' map, are responsible for chaotic mixing and mass transport.