Active micromixers - Micromixers Fundamentals Design and Fabrication - page 283

Substituting X ( x ), Y ( y ) and T ( t ) back into (7.84) results in:

A sin mpx
L

C sin npy z mn ¼

W ð

E sin ut þ f cos ut Þ

(7.97)

sin mpx
L

sin npy

¼

W ðM sin ut þ N cos utÞ

with m .

The modal frequency f mn ¼

¼ 1, 2,

.

u /(2 p ) can be determined from:

ðu=cÞ

q

mp
L

2

k 2 ¼ (7.98)

or

mp
L

2 c 2 mp

L

2 c 2 np

W

2 c 2

u 2 k 2 c 2 ¼ þ ¼ þ : (7.99)

Thus,

s

T
s

m 2
L 2 þ

n 2
W 2

1
2 p

f mn ¼ (7.100)

with m . Fig. 7.34 shows the solution for the first few vibration modes of a rectangular

membrane. These vibration modes will affect the acoustically induced flow pattern inside the mixing
chamber. For a square membrane ( L

¼ 1, 2, .

¼ W ), the two modes mn and nm have the same frequency

( f mn ¼

f nm ). At this frequency, the membrane can vibrate with an infinite number of different shapes.
The different shapes results from the combination of a and b in the equation:
z ð x ; y ; t Þ¼ð az mn þ bz mn Þ cos u mn t (7.101)

FIGURE 7.34
Some vibration modes of a rectangular membrane (L:W¼ 1:2).