# Interdisciplinary Applied Mathematics

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are shown as electrical switches. The resistances (or conductances) in the fluidic circuit of Figure 18.13(b) are the fluidic resistances of the channels in the flow layer. The “on-off” position of the valves depends on the gauge pressure in the control channel compared to the pressure in the flow channel. Thus, the control layer is represented in the fluidic circuit through its gauge pressure.    In    Figure    18.13(b)    the    pressure    difference    of    the    ith    con

trol channel is represented by “Vi”. The notation “V” is used because of the analogy between electrical voltage and pressure. The “on” position of a switch (in Figure 18.13(b)) is represented by a vertical dash connecting two consecutive resistances (e.g., “A0”), and the “off” position of a switch

Top View of the Chambers (flow rate in each chamber is 250 pl)

О

О 475

+ + + +

2 75    3 75    4 75    5 75    6 75

Flow Channels —>

Thickness of the PDMS membrane used as valves (am)

(b)

(a)

FIGURE 18.14. (a) Simulation of fluid flow through the microfluidic system shown in    Figure    18.13(a).    The plus    signs    indicate    presence    of flow.    (b)    Varia

tion of the threshold pressure with the thickness of the membrane.

is represented by a slanted dash causing a break between two consecutive resistances (e.g., “B4”). The hydraulic conductances (or hydraulic resistances) can be modeled using the approach explained in Section 18.1.2. The pressure-actuated control valves can be modeled as switches, which are considered “off” if the pressure in the control channel is above the “threshold pressure” and are considered “on” if the pressure in the control channel is below the threshold pressure. The threshold pressure can be computed from the expression (Timoshenko and Woionowsky-Krieger, 1959)

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