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An analytical model of mixing in the staggered herringbone mixer (SHM) was derived to estimate mixing parameters and provide practical expressions to guide mixer design and operation for a wide range of possible solutes and flow conditions. Mixing in microfluidic systems has historically been characterized by the mixing of a specific solute system or by the redistribution of flow streams; this approach does not give any insight into the ideal operational parameters of the mixer with an arbitrary real system. For Stokes-flow mixers, mixing can be computed from a relationship between solute diffusivity, flow rate, and mixer length. Confocal microscopy and computational fluid dynamics (CFD) modeling were used to directly determine the extent of mixing for several solutes in the staggered herringbone mixer over a range of Reynolds numbers (Re) and Péclet numbers (Pe); the results were used to develop and evaluate an analytical model of its behavior. Mixing was found to be a function of only Pe and downstream position in the mixer. Required mixer length was proportional to log(Pe); this analytical model matched well with the confocal data and CFD model for Pe<5 x 10(4), at which point the experiments reached the limit of resolution. For particular solutes, required length and mixing time depend upon Re and diffusivity. This analytical model is applicable to other solute systems, and possibly to other embodiments of the mixer, to enable optimal design, operation, and estimation of performance.
Williams et al. (Tue,) studied this question.