p(x, y)= p+ 6Cx -i (Q'()- Q'(;))
(4-30)
where pc is a reference pressure, an arbitrary constant, and Q'(4) is the derivative of Q(;)
with respect to From the periodic boundary condition in Equation 4-14, the constant C is
determined as
C
where
(4-31)
K P (L/2;z)2
pvro (4-32)
is a dimensionless parameter representing the ratio of pressure force to electrokinetic body force
in the fluid flow. Therefore, the stream function (Equation 4-29), pressure (Equation 4-30), and
velocities are expressed as,
K
/(x, y) = By2 Ky3 + 29 [R(;) + yQ(\)]
12 (4-33)
p(x,y)
K
Pc -+23[Q'()]
2Jt
K
u(x, y)= 2By-- y2 +
4;r
a a
2 3[R()] + 2[Q()] + 2y 3[Q()]
cx Ox
v(x, y)= -2--[R()]+2y -- [Q(i)]
Using the Cauchy integral formula, R(,;) and Q(e() can be written as
R(;) = (x) dx) + R '(x)dx
Ti .(x)- 2;i P(x)-
1
Q( )= 2-
2zz
Q(-(x) 1 I+ QP(x)
S '(x)dx+- '(x)dx
S((x)-; 2 8(x)- "(4-38)
(4-34)
(4-35)
(4-36)
(4-37)