Interdisciplinary Applied Mathematics

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9.2 Chaotic Advection

Homogenization of a tracer by a flow involves two processes: stirring and mixing. Stirring is the mechanical stretching of a material interface, while mixing is the diffusion of a substance across this interface. Mathematically, this process can be expressed by describing the passive advection of fluid particles given by

where v(x,y, z,t) = (u,v,w) is the velocity vector. When the flow is turbulent, the particle paths described by equation (9.1) are chaotic. Research in dynamical systems in the mid 1980s has shown that chaotic fluid particle motion can also be generated with simple velocity fields either two-dimensionally with time-dependent excitation or three-dimensionally with or without time dependence. This concept was introduced by (Aref, 1984), who coined the term chaotic advection. Streamlines and pathlines in steady three-dimensional flow coincide. However, they are not closed curves, and they are not confined to smooth surfaces unless the Lamb vector ш x v is nonzero everywhere in the field. In fact, in chaotic advection, particle paths diverge exponentially in time, which is simply translated in practice to very large residence time for fluid particles. This, in turn, offers the possibility for enhanced transport in the mixing of two coflowing liquid streams, a typical setup in applications. For unsteady flows, a necessary condition for chaos is the crossing of streamlines at two consecutive time instants. This is expressed mathematically by the concept of link twist maps (Wiggins and Ottino, 2004).

The stirring and mixing of a passive scalar described by the nondimensional concentration в is governed by the advection-diffusion equation

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