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Research Papers: Micro/Nanoscale Heat Transfer

# Numerical Modeling of Chaotic Mixing in Electroosmotically Stirred Continuous Flow Mixers

[+] Author and Article Information
Ho Jun Kim

Department of Aerospace Engineering, Old Dominion University, Norfolk, VA 23529-0247

Ali Beskok1

Department of Aerospace Engineering, Old Dominion University, Norfolk, VA 23529-0247abeskok@odu.edu

1

Corresponding author.

J. Heat Transfer 131(9), 092403 (Jun 24, 2009) (11 pages) doi:10.1115/1.3139109 History: Received July 07, 2008; Revised October 22, 2008; Published June 24, 2009

## Abstract

We present numerical studies of particle dispersion and species mixing in a $ζ$ potential patterned straight microchannel. A continuous flow is generated by superposition of a steady pressure-driven flow and time-periodic electroosmotic flow induced by a streamwise ac electric field. $ζ$ potential patterns are placed critically in the channel to achieve spatially asymmetric time-dependent flow fields that lead to chaotic stirring. Parametric studies are performed as a function of the Strouhal number (normalized ac frequency), while the mixer geometry, ratio of the Poiseuille flow and electroosmotic velocities, and the flow kinematics (Reynolds number) are kept constant. Lagrangian particle tracking is employed for observations of particle dispersion. Poincaré sections are constructed to identify the chaotic and regular zones in the mixer. Filament stretching and the probability density function of the stretching field are utilized to quantify the “locally optimum” stirring conditions and to demonstrate the statistical behavior of fully and partially chaotic flows. Numerical solutions of the species transport equation are performed as a function of the Peclet number (Pe) at fixed kinematic conditions. Mixing efficiency is quantified using the mixing index, based on standard deviation of the scalar species distribution. The mixing length $(lm)$ is characterized as a function of the Peclet number and $lm∝ln(Pe)$ scaling is observed for the fully chaotic flow case. Objectives of this study include the presentation and characterization of the new continuous flow mixer concept and the demonstration of the Lagrangian-based particle tracking tools for quantification of chaotic strength and stirring efficiency in continuous flow systems.

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## Figures

Figure 1

The mixer consists of spatially repeating mixing blocks with ζ potential patterned surfaces as shown in (a) for pattern A and (b) for pattern B. Electroosmotic flow streamlines induced by an axial electric field are shown by solid lines in (c) for pattern A and in (d) for pattern B, while a unidirectional pressure-driven flow enables continuous flow in the system (not shown in the figure). Arrows on the top and bottom of the mixing blocks in (c) and (d) indicate the directions of the Helmholtz–Smoluchowski velocity UHS. Combining the pressure-driven flow with electroosmotic flow under time-periodic external electric field (in the form of a cosine wave with frequency ω) results in two-dimensional time-periodic flow suitable to induce chaotic stirring in the mixer.

Figure 2

Snapshots of passive particle dispersion at time t=20π: (a) case A at St=1/2π, (b) case A at St=1/π, (c) case A at St=3/2π, (d) case B at St=1/4π, (e) case B at St=1/2π, and (f) case B at St=3/2π

Figure 3

Poincaré sections for (a) case A at St=1/2π, 1/π, 3/2π, and (b) case B at St=1/4π, 1/2π, 3/2π, on the α-y plane. Passive tracer particles were initially placed at (0.02, 0.0), which is near the entrance of the channel and on the interface between the two fluids.

Figure 4

Stretching contours, h, at t=100 for case A at (a) St=1/2π, (b) 1/π, (c) 3/2π, and case B at (d) St=1/4π, (e) 1/2π, (f) 3/2π. Red and blue parts are regions of high and low stretching values, respectively.

Figure 5

Time evolution of the PDF of stretching for case A at (a) St=1/2π, (b) 1/π, (c) 3/2π, and case B at (d) St=1/4π, (e) 1/2π, (f) 3/2π

Figure 6

Species concentration contours at Re=0.01 and Pe=1000 for case A at (a) St=1/2π, (b) 1/π, (c) 3/2π, and case B at (d) St=1/4π, (e) 1/2π, (f) 3/2π. The mixing domain is within the range of 0≤x≤24 (six blocks). Each computation has reached a time-periodic state, while results are shown at time t=20π.

Figure 7

M−1 variation for case A at St=1/π and case B at St=1/2π as a function of the number of mixing blocks at Pe=1000 and Re=0.01

Figure 8

Temporal evolution of M−1 for case B at St=1/2π within eight mixing blocks. (a) A series of snapshots obtained during a period with a time interval of 0.25T. (b) Time history of M−1 in the eighth mixing block exhibits consistent and stable mixing quality.

Figure 9

Number of mixing blocks as a function of the Peclet number required to obtain 90% mixing. The results for case B at St=1/2π are shown at Re=0.01 and various Pe.

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