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Copyright Brian J. Kirby. With questions, contact Prof. Kirby here.
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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.
[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]
[InducedCharge Effects]
[DEP]
[Solution Chemistry]
The flow in Figure 4.4 illustrates a onedimensional 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 crosssectional variations in velocity and how these variations affect the
measured crosssectionally averaged concentration. This averaged concentration, for example, is measured by the
detector in an electrophoretic separation apparatus. TaylorAris dispersiondescribes how axial convection,
axial diffusion, and transverse diffusion combine to control analyte transport in pressuredriven flow
through a microchannel. An illustration of how pressuredriven flow leads to dispersion is shown in
Figure 4.9.
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
 (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:
 (4.9) 
and
 (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 crosssectionaverage scalar distribution.
Convectiondiffusion
If Pe ≪ L∕R, we can solve the 2D convectiondiffusion problem and show that an averaged 1D equation can be
written:
 (4.11) 
in which the effective diffusivityD_{eff} is given by
and this leads to a diffusive/dispersive growth of the band width with a w ∝ t^{(1∕2)} dependence:
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 theTaylorAris dispersion relation and a w ∝ t^{(1∕2)} rather than a linear dependence.
While axial diffusion causes band broadening, radial diffusion minimizes band broadening. Radial
diffusion causes analyte molecules to sample both fastmoving and slowmoving parts of the Poiseuille
flow, so each molecule sees an average velocity rather than the widelyvarying radiallydependent
xvelocities.
[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]
[InducedCharge 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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