Fundamentals of mass transport in the microscale - Micromixers Fundamentals Design and Fabrication - page 38

FIGURE 2.13
Example 2.9: Pathlines of three particles with initial position at (x 0 ¼ 1, y 0 ¼ 1), (x 0 ¼ 1, y 0 ¼ 2), and (x 0 ¼ 1,
y 0 ¼

3) (a ¼ 1, b ¼ 2, and c ¼ p/2).

The stream function j can be solved for the explicit form using the equation system (2.87) . Taking
the time t as constant, u ( x , y , t ) ¼ x sin( at ), and v(x , y , t ) ¼ y sin( bt þ c ), we have:

Z u Z y ZZ d u

ð x ; y ; t Þ

j ð x ; y ; t Þ¼ ð x ; y ; t Þ d y ð x ; y ; t Þ d x d y d x

d x

:
It is apparent that streamlines are time dependent. Figure 2.14 shows the streamlines as the level
sets of stream function j .
Equation (2.85) represents a system of coupled ordinary differential equations (ODEs). Similar
dynamical systems in engineering and physics have shown a strong chaotic behavior. Poiseuille flow in
a straight microchannel is considered as a one-dimensional incompressible flow at low Reynolds
number. The dynamics of this flow is simple and nonchaotic. In the case of a two-dimensional flow, the
dynamic behavior of the flow is more interesting. The two-dimensional continuity equation:
vu
vx þ

j ð x ; y ; t Þ¼ xy sin bt þ c

vy
vy ¼

0 ; (2.88)

is fulfilled by the stream function j (2.87) . Equation system (2.87) has the same form of the Hamilton
equation of motion, where the stream function j plays the role of the Hamiltonian. Thus, steady
two-dimensional incompressible flow and time-independent Hamiltonians with one degree of freedom
are integrable and deterministic dynamics. Adding one more dimension to the systems, such as
unsteady two-dimensional incompressible flow and independent Hamiltonians, makes the equations
nonintegrable and causes chaotic dynamics.
The terminologies for chaotic advection can be borrowed from the more established field of
dynamical systems theory. The advection equations (2.85) can be solved explicitly for the fluid