Active micromixers - Micromixers Fundamentals Design and Fabrication - page 284

FIGURE 7.35

Some degenerate vibration modes of a square membrane at m , n ¼ 1, 2 (the dashed lines are the nodal lines):

(a) a ¼ sin 0, b ¼ cos 0; (b) a ¼ sin p /6, b ¼ cos p /6; (c) a ¼ sin p /3, b ¼ cos p /3; (d) a ¼ sin p /2, b ¼ cos p /2.

where a 2 þ b 2 ¼ 1. These vibration modes are called degenerate modes. Fig. 7.35 shows the few

possible combinations of z 12 and z 21 . This mode can be used for creating chaotic advection in a mixing
chamber without hoping from one resonant mode to another.

7.7.2 Vibration of a circular membrane [49]
The wave equation (7.83) can be formulated for the cylindrical coordinate as:
v 2 z
vt 2 ¼

c 2 v 2 z

v 2 z
vq 2

1
r

vz
vr þ

1
r 2

vr 2 þ

: (7.102)

The separated solution for the deflection z has the form:
z

ð r ; q ; t Þ¼ R ð r ÞQð q Þ $ exp ð iut Þ: (7.103)

The separated equations have the forms:
d 2 R
d r 2 þ

u 2
c 2

m 2
r 2

1
r

d R
d r þ

¼ 0 (7.104)

and

d 2 Q
d q þ

m 2 Q ¼ 0 : (7.105)