# Interdisciplinary Applied Mathematics

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Figure 7.19 shows an electrokinetic speed field in an array of circular posts at    a    45°    angle    with    respect to    the    applied    electric    field    of 2 V/mm.

The flow is from lower left toward upper right. Lines of constant gray scale are contours of constant speed. The magnitude of speed at any point can be estimated by counting and interpolating the fringes starting at a stagnation point. The interferogram fringe spacing in the figure corresponds to 24.5 p,m/s,    and the    stagnation    points    on    each    post    is    at    45°    and    225°

from the horizontal axis, aligning with the applied electric field.

Figure 7.20 shows an electrokinetic streamwise velocity field in an array of square posts. The electric field is applied from left to right at a value 1 V/mm, creating flow in the electric field direction. The interferogram fringe spacing in the figure corresponds to 9.8 p,m/s. The uniform flow between the top and bottom posts is pure electroosmotic flow. Two-dimensional flow is observed    in    the    region    between    the    two    posts,    where    the    flow expands

FIGURE 7.20. Electrokinetic streamwise velocity contours in an array of square posts.    The electric field    is    from    left    to    right    (1 V/mm). (Courtesy    of    E.    Cum

mings.)

and contracts; the velocity contours are symmetric in this zone.

In the flow examples given above, the EDL is infinitesimally small compared to the flow dimensions, and the total pressure is constant at the entry and exit ports. Hence, the flow conditions obey the ideal electroosmosis requirements (Cummings et al., 2000), with the following result:

• The flow field outside the EDL is proportional to the external electric field. Therefore, the bulk flow is potential flow.

The velocity and speed contours presented in Figures 7.19 and 7.20 closely follow the numerical solution of potential flow past circular and square array posts, as demonstrated in (Cummings, 2001).

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