Active micromixers - Micromixers Fundamentals Design and Fabrication - page 281

FIGURE 7.33
Vibration model for a rectangular membrane: (a) membrane dimension; (b) forces on a membrane element d x d y.

area density of s . Damping effect of the surrounding fluids is neglected. The forces acting at the two
edges at y and y

þ d y are - T sin a d x and T sin b d x. For a small displacement of d z or small a and b :

yþ d y

vz
vy

sin a z tan a ¼

y : (7.80)

vz
vy

sin b z tan b ¼

Thus, the restoring forces in x and y are:

T d x vz
vy

!

yþ d Y y

T d y v 2 z

vz
vy

f x ¼ ¼

vy 2 d x

(7.81)

T d y vz
vx

xþ d x y

T d x v 2 z

vz
vx

f y ¼ ¼

vx 2 d y

:

The total restoring force on the element d x d y is f ¼ f x þ f y . Newton's second law leads to the wave

equation:

T d x d y v 2 z

v 2 z
vy 2

s d x d y v 2 z
vt 2 vx 2 þ

¼ (7.82)

or

v 2 z
vx 2 þ

v 2 z
vt 2 ¼

v 2 z
vy 2

T
s

¼ c 2

2 x

V (7.83)

p is the propagation velocity. The wave equation can be solved by separation of
variables. Substituting

T where c ¼ =

z ð x ; y ; t Þ¼ X ð x Þ Y ð y Þ T ð t Þ

(7.84)