Fundamentals of mass transport in the microscale - Micromixers Fundamentals Design and Fabrication - page 15

where V is the nabla operator and D = D t is the total derivative operator, which is defined as:

D
D t ¼

v
vt þ

u v y v w v

v
vt þ

vx þ vy þ vz ¼

v $V (2.12)

where v ¼ ( u , y , w ) is the velocity vector.

2.1.2.2 Conservation of momentum
Conservation of momentum is described by Newton's second law:
r D v

D t ¼

f ¼ f body þ f surface : (2.13)

The left-hand side of the above equation represents the acceleration force, while the right-hand side
consists of forces per unit volume acting on the fluid particle. The force f may consist of body force and
surface forces. In microscale, surface forces such as viscous force, electrostatic force, or surface stress
are dominant over body forces such as gravity. If the only body force is gravity and surface forces are
caused by a pressure gradient and viscous force and both the density and viscosity are constant, the
Navier-Stokes equation can be derived from the general conservation Eqn (2.13) :
r D v

2 v

D t ¼

rg V p þ m V (2.14)

where r and m are the density and the dynamic viscosity of the fluid, respectively. In micromixers, we
often encounter a pressure-driven flow in a straight microchannel. If the flow in the axial x -direction is
fully developed, the continuity equation is automatically satisfied with v

¼ w ¼ 0 and d u /d x ¼ 0. The

Navier-Stokes Eqn (2.14) can then be further simplified to the two-dimensional form:

v 2 u
vy 2 þ

v 2 u
vz 2 ¼

1
m

d p
d x

(2.15)

where the right-hand side is a constant. Applying the no-slip boundary condition at the channel wall,
an analytical solution can be obtained for channels with simple cross-section geometry, such as circle,
ellipse, concentric annulus, rectangle, and equilateral triangle ( Fig. 2.2 ). Table 2.2 and Fig. 2.3 show

FIGURE 2.2

Cross section of channels considered in Table 2.2 : (a) circle; (b) ellipse; (c) concentric annulus; and
(d) rectangle.