Active micromixers - Micromixers Fundamentals Design and Fabrication - page 239

Adding a disturbance x 0 to the basic solution x 0 and substitute x 0 þ x 0 into the governing equation.

Subtracting terms that x 0 satisfies. The remaining equation is the disturbance equation.

For small disturbance analysis, ( x 0

x 0 ), the higher order of x 0 can be neglected. The disturbance

equation becomes linear.

Further simplifying the disturbance equation by assuming a prescribed form such as a travelling
wave.

The linearized disturbance equation should be homogenous and have homogenous boundary
conditions. Thus it can only be solved for specific values of the equation's parameters. The
analysis becomes an eigenvalue problem.

The eigenvalues found in the step above are examined for unstable, stable, or neutrally stable
behavior. The results are shown on a chart with neutral curves, which separate the stable and the
unstable regions.

7.2 PRESSURE-DRIVEN DISTURBANCE
7.2.1 Actuation concepts for pressure generation
In most cases, pressure-driven disturbance is created by an external actuator. This section reviews the
most common microactuator concepts, which can be integrated in a micromixer. These concepts can
be categorized according to their physical effects as:

Pneumatic;

Thermopneumatic;

Thermomechanical;

Piezoelectric;

Electrokinetic;

Electromagnetic;

Electrochemical and chemical;

Capillary force; and

Centrifugal and Coriolis forces.

As explained later in Section 7.2.2 , the two key parameters for hydrodynamic instability needed in
active micromixers are the magnitude and the frequency of the disturbance. Thus, the pressure
generated by an actuator and its dynamic response are of interest for designers of active micromixers.
Figure 7.1 shows the typical ranges of actuation pressure and actuation frequency. Because actuation
magnitude is associated with the storage capacity of the microactuator, thermal concepts, such as
thermo-pneumatic and thermomechanical, are the most powerful in term of actuation pressure.
However, thermal processes are usually slow and the actuation frequencies are on the order of a few
hertz. In contrast, surface-based concepts, such as electrostatic and piezoelectric microactuator, can
work at a very high frequency.
Because the total energy delivered by a microactuator is proportional to its volume, a small actuator
also means a small actuation energy. The energy density of actuators usually does not scale with
miniaturization and can be estimated as follows:
The energy density of a thermomechanical actuator, which is based on the thermal expansion to
convert electrical energy to mechanical energy through thermal energy, can be estimated as: