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Copyright Brian J. Kirby. With questions, contact Prof. Kirby here. This material may not be distributed without the author's consent. When linking to these pages, please use the URL http://www.kirbyresearch.com/textbook.

This web posting is a draft, abridged version of the Cambridge University Press text. Follow the links to buy at Cambridge or Amazon or Powell's or Barnes and Noble. Contact Prof. Kirby here. Click here for the most recent version of the errata for the print version.

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Jump To: [Kinematics] [Couette/Poiseuille Flow] [Fluid Circuits] [Mixing] [Electrodynamics] [Electroosmosis] [Potential Flow] [Stokes Flow] [Debye Layer] [Zeta Potential] [Species Transport] [Separations] [Particle Electrophoresis] [DNA] [Nanofluidics] [Induced-Charge Effects] [DEP] [Solution Chemistry]

4.4 The low-Re, high-Pe limit [flow patterning/microfluidic mixing top]

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∕Di, where Di is the binary diffusivity of the species i in the solvent, is a characteristic length, and U is a characteristic velocity. Here, 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
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 = kBT ∕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, 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.

[Return to Table of Contents]



Jump To: [Kinematics] [Couette/Poiseuille Flow] [Fluid Circuits] [Mixing] [Electrodynamics] [Electroosmosis] [Potential Flow] [Stokes Flow] [Debye Layer] [Zeta Potential] [Species Transport] [Separations] [Particle Electrophoresis] [DNA] [Nanofluidics] [Induced-Charge Effects] [DEP] [Solution Chemistry]

Copyright Brian J. Kirby. Please contact Prof. Kirby here with questions or corrections. This material may not be distributed without the author's consent. When linking to these pages, please use the URL http://www.kirbyresearch.com/textbook.

This web posting is a draft, abridged version of the Cambridge University Press text. Follow the links to buy at Cambridge or Amazon or Powell's or Barnes and Noble. Contact Prof. Kirby here. Click here for the most recent version of the errata for the print version.


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