Chapter 4 Passive scalar transport: dispersion, patterning, and mixing

Many microfluidic systems are used to manipulate the distribution of chemical species. Chemical separations, for example, physically separate components of a multispecies mixture so that the quantities of each component can be analyzed or so that useful species can be concentrated or purified from a mixture. Many biochemical assays, for example DNA microarrays, require that a reagent be brought into contact with the entirety of a functionalized surface, i.e., that the reagents in the system be well mixed. Studies of homogeneous kinetics in solution require that a system become well mixed on a time scale faster than the kinetics of the reaction. In contrast to these, extracting functionality from a spatial variation of surface chemistry often depends on the ability to pattern surface chemistry with flow techniques, which requires that components of the solution remain unmixed.

These topics all motivate discussion of the passive scalar transport equation. This convection-diffusion equation governs the transport of any conserved property that is carried along with a fluid flow, moves with the fluid, and does not affect that fluid flow. Chemical species and temperature are two examples of properties that can be handled in this way, as long as (1) the chemical concentration or temperature variations are low enough that transport properties such as density or viscosity can safely be assumed uniform, and (2) we neglect electric fields, which can cause migration of chemical species relative to the fluid.

We start by introducing the scalar convection-diffusion equation, which describes species transport, and discuss the physics of mixing. We then note that, owing to the nature of microfabrication techniques and the species of interest in biochemical analysis systems, many microfluidic species transport systems reside in the limit of low Reynoldsnumber (laminar) but high Pecletnumber (minimal diffusion). This limit makes it straightforward to isolate chemical species in microdevices, enablinglaminar flow patterning techniques, which can be analyzed with simple 1D arguments that are a minor extension of the hydraulic circuit analyses presented in Chapter 3. Unfortunately, this same situation leads to challenges when species must be mixed, leading to the so-calledmicrofluidic mixing problem. The challenges of mixing in these systems has given rise to interest in chaotic advection, which employs flow fields with special properties that can exponentially increase the scalar mixing by using a deterministic flow field to amplify the random effects of species diffusion. In chemical separation systems, we are interested in the mixing of species in the axial direction, because mixing in this direction decreases the resolution of a chemical separation. This motivates our study ofTaylor-Aris dispersion, in which the dispersive nature of the flow leads to a significant increase in the effective diffusion of the scalar in the axial direction.

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