Advection, chaotic

To facilitate mixing, mixers based on chaotic advection in special microstructures were constructed. For instance, in a PDMS fluidic mixer, small chevron-shaped indentations, which were not centered, were constructed in the channel. Such an arrangement forced the fluid to recirculate in order to achieve a mixing effect [184], 

FIGURE 3.39 Photomicrograph of the microfabricated mixer based on eddy diffusion. This mixer is about 100 by 200 pm wide and 10 pm in depth. Effect of mixing is evaluated by bringing in two fluids from channels A and B, and flowing to D. Channel C is not used in this work. The inset show the SEM image of the mixer [493]. Reprinted with permission from the American Chemical Society. 

FIGURE 3.40 (a) Configuration of the experimental setup and white light microscopy image of an imprinted T-channel with a series of ablated wells, (b) Fluorescence images of electroosmotic flow past the mixer at flow rates of 0.06 cm/s [193]. Reprinted with permission from the American Chemical Society. 

FIGURE 3.41 Optical micrograph taken from above of a stream of black dye flowing in a microchannel containing square grooves in the bottom of the channel. A narrow stream of black dye in water is injected alongside a broad stream of clear water (flow rate of the clear stream is 20 times that of the dyed one). The average flow speed in the channel is 1 cm/s (Reynolds number Re 1) [471], Reprinted with permission from the American Chemical Society. 

A mixer was constructed by porous hydrogel formed at a Y-junction. It was found that the mixing efficiency was higher for a more porous structure [496]. 

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Aref, H., Stirring by chaotic advection. J. Fluid Mech. 143, 1-21 (1984). 

Kumar, S., and Homsy, G. M., Chaotic advection in creeping flow of viscoelastic fluids between slowly modulated eccentric cylinders. Phys. Fluids 8,1774-1787 (19%). 

Taylor s dispersion is one of the most well-known examples of the role of transport in dispersing a flow carrying a dissolved solute. The simplest setting for observing it is the injection of a solute into a slit channel. The solute is transported by Poiseuille s flow. In fact this problem could be studied in three distinct regimes (a) diffusion-dominated mixing, (b) Taylor dispersion-mediated mixing and (c) chaotic advection. 

In the following, the use of periodical potentials will be described the periodicity will be denoted T[28]. Switching between two ore more flow patterns is performed inducing chaotic advection. One flow field is maintained in one time interval and another flow field in a second interval. This is repeated with the period T. The switching of the flow fields is accomplished by controlling the distribution of the C, potential created by the electrodes. By flow field alternation, particles virtually expose a zig-zag path, thereby distributing material all over the channel s cross-section. Such transport is similar to efficient stirring. 

P 7] The topic has only been treated theoretically so far [28], A mathematical model was set up slip boundary conditions were used and the Navier-Stokes equation was solved to obtain two-dimensional electroosmotic flows for various distributions of the C, potential. The flow field was determined analytically using a Fourier series to allow one tracking of passive tracer particles for flow visualization. It was chosen to study the asymptotic behavior of the series components to overcome the limits of Fourier series with regard to slow convergence. In this way, with only a few terms highly accurate solutions are yielded. Then, alternation between two flow fields is used to induce chaotic advection. This is achieved by periodic alteration of the electrodes potentials. 

Time-dependent electroosmotic flows - chaotic advection... 

On further increasing T to 2, 4 and 6, the complexity of the flow becomes more pronounced (see Figure 1.21) [28]. First, the particles wander around the superimposed image. Then, particles stray further away from the regular path and sample most of the cell s area. Chaotic advection is now present. 

Deformation of a blob by chaotic advection - simulation of a stirring process... 

A quantitative analysis shows that the interface increases slightly faster than a linear function of time [96], This is better than for having diffusion only, but is behind the performance of chaotic advection. 

Mixing in a PMMA T-junction was achieved by chaotic advection in the channel with slanted grooves created by laser photoablation (see Figure 3.40) [193,257,470]. Mixing could also be achieved in a PDMS microchannel using square grooves in the channel bottom (see Figure 3.41) [471]. 

Passive mixing by chaotic advection is demonstrated in a PDMS chip consisting of a winding channel (see Figure 3.43). An essential component is a water-... 

A 3D serpentine microchannel was fabricated on a Si-glass chip to enhance mixing by chaotic advection (see Figure 3.44). It was found that mixing in the 3-D channel was faster and was more uniform than in either a square wave channel or straight channel [477]. 

Other micromixers based on various principles have also been constructed. These principles include vortex [492], eddy diffusion [493-501,654,955], rotary stirring [502], turbulence [495,503], EK instability [504—506], chaotic advection [248,507-513], magnetic stirring [514], bubble-induced acoustic mixing [515], and piezoelectric actuation [516,517]. 

Potentiometric measurement using a Pt electrode has been employed in a titration (between Cr,072 and Fe(CN)63 ) which was carried out on a PDMS-glass chip. An 11-channel serial dilutor was used to produce the different titrant concentrations, and a chaotic advective mixer was incorporated to facilitate solution mixing [769]. 

Results demonstrate that when agitators are switched the slope of the pathline becomes discontinuous. We will see later in this chapter how this mechanism may produce an essentially stochastic response in the Lagrangian sense. Aref termed this chaotic advection, which he suggested to be a new intermediate regime between turbulent and laminar advection. The chaos has a kinematic origin, it is temporal—that is, along trajectories associated with the motion of individual fluid particles. Chaos is used in the sense of sensitivity of the motion to the initial position of the particle, and exponential divergence of adjacent trajectories. 

Fig. 7.7 Sample particle trajectories in a twin vortex flow. The agitator location is set by the amplitude, which is 0.5 (i.e., it is at midpoint between the center and the perimeter) and marked by the crosses. The dimensionless period for each vortex is 0.5. The mixing protocol is to activate one agitator for a period of time and then switch to the other agitator. [Reprinted by permission from H. Aref, Stirring Chaotic Advection, J. Fluid Meek, 143, 1-21 (1984).)...

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