# Interdisciplinary Applied Mathematics

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(ho. ^

p    _ _l 2 )_

J threshold —    m — i .    7

( 4a4    y-^00    (—1)    2 sinl^) _ gm tanh(am)+2 -|

i 7U Dr 2_^77г=1,3,5… m5    ЦТ 1    2cosh(a„,) bJ

where h is the height of the flow channel, a and b are the dimensions of the rectangular membrane acting as the valve, Dr and am are as defined in Section 18.1.2. Figure 18.14(a) shows the simulated flow distribution in the flow layer of the microfluidic circuit shown in Figure 18.13(b). A plus sign corresponding to a given flow channel indicates that the flow is “on;” otherwise, the flow is “off.” A cell associated with a given flow channel will receive fluid only if the flow is on. Figure 18.14(b) shows the nonlinear variation of the threshold pressure with the thickness of the membrane, and Figure 18.15(a) shows the nonlinear variation of the threshold pressure with the dimension of the square membrane. Thus, for a specified threshold pressure one can choose the thickness of the membrane from Figure 18.14(b) and the dimension of the membrane from Figure 18.15(a). The CPU time to simulate the flow distribution for the system shown in Figure 18.13(a) was 16 seconds. Figure 18.15(b) shows the simulation result for an array

Dimension of the square membrane acting as a valve (am)

(a)

An array of 60 by 126 chambers

(b)

FIGURE 18.15.    (a)    Variation    of    the    threshold    pressure    with    the    width    of    the

square membrane. (b) An example of large-scale integration, where the fluid is stored in a desired pattern in a microfluidic chip containing 60×126 chambers.

of 60×126 chambers. Fluid is stored in the chambers based on the filling mechanism described in Figure 18.13(a) (Thorsen et al., 2002). This result demonstrates that the fluid can be stored in any arbitrary pattern using large-scale integration of micro/nanochannels.

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