# Interdisciplinary Applied Mathematics

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ffrw(x,y)dT

P

Cfl

(18.10)

where Cfl is the fluidic capacitance, w is the deflection, Г is the total surface area of the flexible membrane, and p is the pressure difference across the channel wall. For a rectangular membrane of dimensions axb, the fluidic capacitance from equation (18.10) is given by

Cfl

4 a

Tr5Dr

Ж

E

m=1,3,5

m — 1

(_!)— sin(*f)

m5

m7r

a

b

2 +

a

2mn

tanh(am )(3 + am tanh(am))]

where

a

m

mnb

and Dr is the rigidity of the membrane given by

Dr =

Emod hm

12(1 — v where hm is the thickness of the membrane, Emod is the elastic modulus of the membrane, and v is the Poisson ratio of the membrane.

The implementation of the electrical model and the fluidic model is carried out using the modified nodal analysis technique (Ogrodzki, 1994). Once the variations of ф and p are known, the flowrate in each channel can be computed using the constitutive relationship given in equation (18.7).

###### 18.1.3 Chemical Reactions: Device Models

Consider a scheme (shown in Figure 18.5) in which the chemical species A and B are    transported    to    the    reaction    chamber,    where    they    undergo    a

second-order reversible reaction process to produce species C. The governing equations for this reaction process are given by

= QaCa — ki(m>A)(m>B) + k2(mc), = QbCb — ki(m,A)(m,B) + k2(mc),

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