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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]

8.4 Micro-PIV [Stokes flow top]

Particle-image velocimetryisoften used in microscale systems to visualize fluid flow. Since it typically involves flow of a suspension of spheres at low Re, it is an important example of Stokes flow and the applicability of Stokes flow analysis. Particle-image velocimetry operates as follows: first, a fluid flow is seeded with particulate fluid tracers. In microsystems, the most common particles are fluorescent polystyrene latex beads.Then, two images of the fluid tracers are recorded in rapid succession. For measurements with high temporal resolution, this usually involves mating a dual-pulselaser to amicroscope and recording the fluorescence from the beads with aCCD (charge-coupled device) camera. Figure 8.5 shows an example of a micro-PIV setup. Finally, the two images are correlated. If the two images are separate, this is called across-correlation. If the two images are recorded on the same camera image, this is called anautocorrelation. If image one is treated as a q×p array of brightness values f(i,j) and the second image as g(i,j), then the cross-correlation Φ is given by:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(8.43)

and the autocorrelation of a single image f is given by
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(8.44)

Here, m and n indicate pixel offsets in the i and j directions. The cross-correlation thus gives—as a function of the offsets—a measure of how well two images match each other. Figure 8.6 shows this cross-correlation algorithm. The cross-correlation Φ is large when the offsets m and n cause the two images to overlap as perfectly as possible, and small when the offsets cause the two images to misalign. The most likely distance traveled by the fluid between images corresponds to the pixel offset with the maximum value of Φ. The process of finding the maximum of the cross-correlation function is analogous to printing one image on a sheet of paper and printing the second image on a transparent sheet. By moving these two sheets with respect to each other, one can find an offset the best matches the two images to each other. To get a complete flow field, small interrogation regions throughoutthe image are each analyzed to give the velocity in that region, and the flow field is the product of systematically evaluating velocities in interrogation regions throughout the image. Such spatial correlations are straightforward using fast Fourier transforms.


microfluidics textbook nanofluidics textbook Brian Kirby CornellFigure 8.5: Schematic of a micro-PIV system. Courtesy LaVision (http://www.lavision.de). A pulsed laser creates two laser pulses that are focused by an epifluorescent microscope with a high numerical aperture onto a microfluidic device. A microscope lens collects the fluorescent signal from fluorescent particles and focuses the image onto a dual-frame CCD camera. The z-axis resolution comes from the narrowfocal depth of the collection objective.



microfluidics textbook nanofluidics textbook Brian Kirby CornellFigure 8.6: A cross-correlation algorithm for PIV.


8.4.1 Deterministicparticle lag

PIV measures the particle velocity field. If this is to be used to infer the fluid velocity field, then the relation of the particle velocity field to the fluid velocity field must be established using Stokes flow arguments, and two key error sources must be avoided. First, body forces on the particles, e.g., gravity or electric fields, will induce a particle velocity relative to the flow. Second, if the particles are too large or too dense, or the velocity gradients are too high, the particles lag the flow because of their finite inertia. The former issue means that micro-PIV works best in the absence of electric fields and furthermore benefits if neutrally buoyant particles are used. The second issue can be addressed by evaluating the Stokes number of the particle. If the Stokes number is small, then the particle follows the flow and its velocity can be used to infer the fluid velocity—these particles are called Lagrangian flow tracers.

8.4.2 Brownianmotion

Brownian motion of microparticles is the random motion due to the statistical nature of fluid forces on particles. Brownian motion becomes more important as particles become small and the importance of individual particle-fluid collisions becomes larger. The root-mean-square PIV measurement error because of Brownian motion can be related to the particle diffusivity D:

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

where Δt is the time between exposures, and the diffusion coefficient for small particles in Stokes flow is given by the Stokes-Einstein relation:

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

where a is the particle radius, kB isBoltzmann’s constant, and T is temperature. Equation 8.45 assumes that exposures are infinitely fast and determines the errors owing to the random velocity fluctuations between exposures. If the exposure time is long, the image can be blurred and there can be errors in the inferred velocity field due to theBrownian motion during each exposure.

Some examples of our research where Stokes flow analysis is relevant include our circulating tumor cell capture microchips and our dielectrophoretic manipulation of microparticles.

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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]

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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