# Interdisciplinary Applied Mathematics

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Figure 7.17 (bottom) the velocity vectors at exit branch C show uniform plug profile of magnitude 2wHS, similar to that of inlet branch A, while the exit branch D reaches a uniform plug profile of magnitude wHS, similar to that of inlet branch B. Under the bias electric field, the cross-flow junction creates unique    opportunities    for    flow    control.    If    we    examine    the    stream-

FIGURE 7.16. Electric field lines (solid) and equipotential contours (dashed) in a cross-junction geometry, under various electric fields. Top: Ehor = Ever = EoBottom: Ehor = 2Ever = 2Eo.

lines in the figure, it is clear that 50% of the fluid leaving from channel C is coming    from    inlet    channel    A.    This is    required    by    the bias    electric field

strength and conservation of mass in the microfluidic system. Using this, we conclude that it is possible to control the amount of fluid in exit channel C that is coming from inlets A and B by controlling the ratio of the

FIGURE 7.17. The streamlines and velocity vectors for pure electroosmotic flow in a    cross-flow    junction    (only    25%    of    the    vectors    are    shown    for    clarity    of the

figure). Top: Ehor = Ever = Eo; Bottom: Ehor = 2Ever = 2Ea.

FIGURE 7.18. The y—PIV velocity measurements of electroosmotic cross-flow junction, where Ehor = Ever. (Courtesy of E. Cummings.)

flow in a

flowrates

following

(7.42)

horizontal to vertical electric fields (Ehor/Ever). The ratio of the from inlets A and B at the exit channel C can be written in the form:

QAC /А, I

Q BC    Ever

where Q shows the flowrate, and the subscripts AC and BC show the contributions of flow from inlets A and B to the total flowrate in channel C, respectively. The above formula is subject to the restriction

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