From the previous section, we see that mixing can be efficient if Pe is low, in which case diffusion occurs quickly,
or if the kinematics of the flowfield leads to a repetitive process that shortens the length scales over which
diffusion must act. However, if Pe is high and the kinematic structure of the flowfield does little to shorten
diffusion length scales, then mixing is negligible. In low-Re systems with simple, steady boundary
conditions, the laminar flow kinematically does not shorten diffusion length scales. Thus flows in most
microfluidic devices do not lead to a shortening of diffusion length scales unless specifically designed to do
so.

VIDEO: Low-Re, high-Pe limit discussed in terms of momentum and scalar diffusivities.

4.4.1 The high-Pe limit

Recall from Section E.2.2 that the mass transfer Peclet number for a dilute species i is given by Pe= Uℓ∕D_{i}, where
D_{i} is the binary diffusivity of the species i in the solvent, ℓ is a characteristic length, and U is a characteristic
velocity. Here, ℓ should characterize the length over which species must diffuse, while U should characterize the
velocity transverse to this diffusion. Table 4.1 shows diffusivities for some example ions, molecules, and
particles—from this table, we can see that the diffusivities of particles and macromolecules can be quite small, and
since many of the species of interest are large and slow-diffusing, the Peclet number is often large for these
flows.

Analyte

D

Na^{+}

1×10^{-9}

bovine serum albumin, 66 kDa

1×10^{-10}

10 nm particle

1×10^{-14}

1 μm particle

1×10^{-16}

10 μm particle

1×10^{-17}

Table 4.1: Diffusivities for dilute analytes in water at 25^{∘} C. Particle diffusivities calculated from the
Stokes-Einstein relation D = k_{B}T ∕6πηa, where a is the particle radius.

4.4.2 The low-Re limit

The results of the previous section all are relevant if the flow has a characteristic direction and diffusion acts
transverse to that to distribute chemical species. This is typically true if flow moves in an orderly, laminar fashion
through long, narrow channels (as are typical of many microfluidic chips). In this case, flow is unidirectional and the
Peclet numbergoverns diffusion. However, if the flow has no characteristic direction, but rather turns
about, especially if the flow varies with time, the transport of chemical species is a function of both (1)
diffusion, as specified by the mass transfer Peclet number, and (2) the flow itself (as specified by the
Reynoldsnumber). Note from Appendix E that the Reynolds number is given by Re= ρUl∕η, where U
is the velocity, ℓ is a characteristic length, and η and ρ are the dynamic viscosity and density of the
fluid. The flow is almost always in the low-Re limit in microfluidic devices, and thus in the laminar
regime—this means the flow is stable and any shortening of diffusion length scales must come from complex
geometric boundary conditions, and does not occur naturally from flow instability as is the case in large,
high-Re systems.