4.3 Measuring and quantifying mixing and related parameters [flow patterning/microfluidic mixing top]

Quantifying mixing typically involves some characterization of the spatial inhomogeneity of a scalar distribution.

The effectiveness of a mixing scheme is typically measured by monitoring the mixing time or the mixing distance, depending on the application, and evaluating how this mixing time or distance depends on the Pecletnumber.

Diffusive mixing regime. As an example, consider a flow of two miscible streams of fluid through a channel of width L at a characteristic velocity U. The Peclet number for this flow is UL∕D. The characteristic time for diffusive mixing (as illustrated in the solution of the diffusion equation above) is L²∕D or Pe and the characteristic length is UL²∕D or Pe L. Thus, for a purely diffusive mixing process, the characteristic length or time for mixing is proportional to the Peclet number. For example, for a solution of bovine serum albumin (D = 1×10^-10) flowing at u = 100 μm∕s through a channel of width 100 μm, the Peclet number Pe = 100, which means that the fluid remains largely unmixed until the flow has traveled a distance 100 times the width of the channel, i.e., 1 cm.

Chaotic mixing regime. Chaotic mixing is a term commonly used in the low-Re mixing literature, and indicates mixing processes with flows that lead to an exponential decay of the characteristic length over which diffusion must act. The term chaotic mixing implies that (1) trajectories in the flow become separated by a distance that grows exponentially with time, or, alternately, that (2) that the interfacial area between two fluids grows exponentially with time. This, in turn, implies that the net effect of diffusion, which is inherently random on a macroscopic scale, is deterministically amplified by the fluid flow. Thus a deterministic fluid flow can lead to a chaotic mixing result if the fluid flow amplifies the random aspect of the molecular diffusion. For chaotic mixing processes, the characteristic mixing time or length is proportional to lnPe. This regime is possible only far away from walls, and thus this scaling is only observable if the majority of the observed mixing is happening far from walls.

Chaotic Batchelor regime. The chaotic Batchelor regime implies the situation in which the flow is partially chaotic but the mixing is eventually limited by the non-chaotic flow near the wall. In this limit, the characteristic mixing time or length is proportional to Pe ^{1 4} . While the difference between the characteristic times of pure diffusion and chaotic mixing (with or without boundaries) is enormous, the effect of boundaries on the Peclet number dependence of chaotic mixing is only important from an engineering standpoint if the Peclet number range under investigation is on the order of 1×10⁴ or more.

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