On Random Mixing

Abstract

We study the relaxation of initially segregated scalar mixtures in randomly stirred media, aiming at describing the overall concentration distribution of the mixture, its shape, and rate of deformation as it evolves towards uniformity. Two distinct experiments, one involving an ever dispersing mixture, the other a mixture confined in a channel, both in high Reynolds, three dimensional flows, behave very di erently. We show how these differences are reminiscent of two concomitant aspects of the process of mixing, namely the distribution of individual histories on one hand examined in the present review, and the interaction between the fluid particles on the other, examined separately in [2008]. The particles are stretched sheets whose rates of di usive smoothing and coalescence build up the overall mixture concentration distribution. The randomness of the particle’s net elongation at a given instant of time induces a distribution of the mixing time from which molecular di usion becomes e ective in erasing the concentration di erences. This ingredient is shown to rule the composition of an ever dispersing mixture, providing a detailed analytic description of the overall concentration distribution. It compares favorably with experiments using three di erent passive scalars suggesting that the mixture composition results from a one step lengthening process distributed among the sheets. Consequences of these processes on the spectral, and some geometrical facets of random mixtures are also examined.