Interdisciplinary Applied Mathematics

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The first generation of mixers sought to increase the mixing surface area between two streams of fluids by splitting them into n substreams and rejoining them again in a single stream at a certain location downstream. This mixing method is called parallel lamination, and it can potentially reduce the mixing time by a factor proportional to n2. Similarly, sequential lamination has been used based on vertical and horizontal splittings in many stages, with the potential of reducing the mixing time by a factor of 4n-1, where n is the number of splitting stages. Passive mixers involve appropriate geometric modifications, while active mixers rely on the unsteady action of a stirring force using a mechanical, acoustic, magnetic, or electroosmotic actuation    (Bau    et    al.,    2001; Oddy    et    al.,    2001; Moroney    et    al.,

1991). A magnetohydrodynamic (MHD) mixer was designed and tested in (Bau et al., 2001); it uses Lorentz forcing to produce cellular advection patterns (with induced speed up to 1.5 mm/s), and this can enhance mixing in microchannels. Such an MHD micromixer can be useful in biological fluids that are partially conductive. However, the mixing rate achieved in (Bau et al., 2001), for a steady magnetic field was only slightly faster than linear,    suggesting    that    no    complete    mixing    was    possible    with    this    excita

tion. A time-varying magnetic field could potentially increase the mixing rate significantly. Good mixing with exponentially fast rate is achieved if chaotic advection is implemented properly using either time-dependent forcing or appropriate geometric modifications. We present some details of this concept and examples in the following section, focusing in particular on techniques that have the potential of achieving complete mixing. Representative designs of both passive and active mixers are studied in some detail in the special volume on mixing at microscales edited by (Ottino and Wiggins, 2004).

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