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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]

4.6 Taylor-Aris dispersion [flow patterning/microfluidic mixing top]

The flow in Figure 4.4 illustrates a one-dimensional diffusion process across gradients transverse to the flow direction. However, many microfluidic devices manipulate boluses of fluid, for example for chemical separation. In this case, we wish to explore the axial diffusion and dispersion of an isolated bolus of fluid as the flow moves through a long narrow microchannel.

We are concerned with the effects of cross-sectional variations in velocity and how these variations affect the measured cross-sectionally averaged concentration. This averaged concentration, for example, is measured by the detector in an electrophoretic separation apparatus. Taylor-Aris dispersiondescribes how axial convection, axial diffusion, and transverse diffusion combine to control analyte transport in pressure-driven flow through a microchannel. An illustration of how pressure-driven flow leads to dispersion is shown in Figure 4.9.


microfluidics textbook nanofluidics textbook Brian Kirby CornellFigure 4.9: Schematic of Taylor-Aris dispersion in a Poiseuille flow. a): initial bolus of fluid. b): fluid after elapsed time t in the absence of diffusion. c): fluid after elapsed time t with a finite axial diffusion. A slight broadening of the distribution is seen, but the effect of diffusion is minor as compared to the dispersive effect of the flow. d): fluid after elapsed time t with finite radial/transverse diffusion. The radial diffusion reduces the dispersive effects by causing the scalar to sample both slow- and fast-moving regions of the flow.


ConsiderPoiseuille flow through a circular microchannel of length L and radius R. The two governing parameters for this flow are thePeclet number Pe = uR∕D and the length ratio L∕R. u is the average velocity in the Poiseuille flow, for a circular microchannel given by
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(4.8)

The Peclet number characterizes the relative importance of species advection to species diffusion, while the length ratio characterizes the relative importance of radial diffusion to axial diffusion.

Pure Convection If Pe L∕R, diffusion can be ignored (Figure 4.9b). In this case, which uncommon in microfluidic devices, thewidth w of a thin injected sample bolus can be shown to grow linearly with time, and the averaged concentration within the bolus decreases inverselywith time:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(4.9)

and
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(4.10)

These relations are valid only for t w∕u. This fluid flow is inherently dispersive—the transverse variation of velocity leads to a spreading of the cross-section-average scalar distribution.

Convection-diffusion If Pe L∕R, we can solve the 2D convection-diffusion problem and show that an averaged 1-D equation can be written:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(4.11)

in which the effective diffusivityDeff is given by

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

and this leads to a diffusive/dispersive growth of the band width with a w t(12) dependence:

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

The 1 in Equation 4.12 comes from the diffusive component, and the other term is a dispersive component. Diffusion plays two roles in this equation. Axial diffusion leads to the unity term in this equation, and tends to increase the effective diffusion (albeit only significantly if Pe is small). Radial diffusion is what leads to the use of theTaylor-Aris dispersion relation and a w t(12) rather than a linear dependence. While axial diffusion causes band broadening, radial diffusion minimizes band broadening. Radial diffusion causes analyte molecules to sample both fast-moving and slow-moving parts of the Poiseuille flow, so each molecule sees an average velocity rather than the widely-varying radially-dependent x-velocities.

[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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