Active micromixers - Micromixers Fundamentals Design and Fabrication - page 275

where s el is the conductivity of the fluid. Assuming that the channel height is much smaller then the
channel width and the electrode distance h w , d , the problem can be reduced to a two-dimensional

problem for the x-y plane. With a v el ocity sc ale u sE el Bh 2 / m , the length scale, the time scale, and the

pressure can be normalized by d , d / u and m ud / h 2 , respectively. Using the stream function j ( x *, y *), the
steady-state Navier-Stokes equation has the form:

¼

cos ð

2 N
n

px *

2 n þ 1 Þ

2 j

V ¼

(7.62)

2

¼ 0

with the boundary conditions j ( x *, w */2) ¼ 0 and j ( x *, < 1, y *) ¼ 0. With the stream function

from (7.62) the velocity components can be determined as:

n ln ( cosh

p þ

2 p N
n

y * w * nw * px *

þ = 2 þ Þ cos ð Þ

1

u *

¼

0 ð

Þ

cosh p

1

y * w * nw * px *

þ = 2 þ Þ cos ð Þ

¼

)

cosh y * p w * nw * px *

þ = 2 þ Þ cos ð Þ

cosh y * Þp þ

(7.63)

w * nw * ðpx *

þ = 2 þ cos Þ

n " arctan (

)

p N

ðpx *

1
2 sgn

1 sin Þ

y *

¼ ð x Þ

n¼ 0 ð

1 Þ

sinh p y * w * nw *

þ = 2 þ Þ

arctan (

)#

px *

sin ð Þ

sinh y * p

(7.64)

w * nw *

þ = 2 þ Þ

where the sign function is defined as

( 1

x * ð > 0 Þ

N

1 sin 2 n

px * :

4
p

1

x * sign ð Þ¼

x *

þ 1 Þ (7.65)

0 ð ¼ 0 Þ z

2 n þ

n ¼ 0

x * 1 ð < 0 Þ

The solution of the velocity field ( u *, y *) is depicted in Fig. 7.28 . The kinetic equations of a fluid
particle can be formulated by superposition of the previous steady-state solution with the time-
dependent switching function of the electrodes. This time-dependent flow field can lead to chaotic
advection and improves mixing.

7.6.2 Curved-channel configuration [44]
Figure 7.29 shows an annulus channel configuration. The electrodes are placed on the inner and outer
channel walls. Assuming a two-dimensional model ( h W ), the Navier-Stokes equation with the

MHD body force is:

d 2 y
d r 2 þ

1
r

d y
d r

y
r 2 ¼

a
r

(7.66)

where a hm represents the EHD force. I , B , and m are the current, the magnetic field strength

and the dynamic viscosity of the fluid, respectively. The ratio between the channel's half width and the

¼ - BI /4 p