# Interdisciplinary Applied Mathematics

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many biological applications, e.g., mixing a stream of proteins in an aqueous buffer, the diffusion coefficient is very small, of order 10~10 m2/s, and thus mixing by laminar diffusion is a very slow process. In order to enhance mixing and thus reduce the corresponding time, various subcritical excitation techniques have been developed, some of them resembling heat transfer enhancement methods used in microelectronic cooling applications. A more systematic procedure based on rigorous theory is to exploit the concept of chaotic advection or Lagrangian chaos that can be achieved for low Reynolds number flows or even for Stokes flows. In this chapter, we present the basic ideas behind chaotic advection and give analytical solutions for prototypical problems. We then discuss examples of passive and active mixers that have been used in microfluidic applications. Finally, we provide some quantitative measures of characterizing mixing based on the concept of Lyapunov exponent from chaos theory as well as some convenient ways to compute them.

##### 9.1 The Need for Mixing at Microscales

Mixing of fluids in microchannels is important in many applications, including homogenization of solutions of reagents in chemical and biological reactions, sequencing of nucleic acids, and drug solution dilution. Mixing reduces longitudinal dispersion, which is important in determining performance in pressure-driven chromatography, which is the transfer of fractions from a separation column to a point detector, where it leads to peak broadening (Stroock et al., 2002). G.I. Taylor has studied longitudinal dispersion of a scalar in a pipe, demonstrating that the effective diffusion coefficient is inversely proportional to the transverse diffusivity (see Section 7.5.3). In microchannels the flow is laminar, with the typical Reynolds number at least two orders of magnitude lower than the critical value for laminar-to-turbulence transition. Mixing of tracers is then based on molecular diffusion, with mixing times of order h2/D, where h is the channel height and D is the molecular diffusivity. Even though h is less than 100 pm in most applications, tracers with large molecules have very small D, leading to intolerably large mixing times. In Table 9.1 we present the molecular diffusivity of some relevant substances in water. Diffusivities of solutions containing large molecules (e.g., hemoglobin, myosin, or viruses) are typically two orders of magnitude lower than for most liquids. For example, at room temperature myosin’s coefficient of diffusion in water is 10-11 m2/s, which for h = 100 pm requires a time of about 1000 s, which is unacceptable in practice (Bau et al., 2001).

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