Biomedical Engineering Reference
In-Depth Information
FIGURE 2.14
Example 2.9: Velocity field and streamlines at t
¼
2(a
¼
1, b
¼
2, and c
¼
p/2).
particle position ( x , y , z ) at a given time t . The solution of ( x , y , z ) is then used for describing the
motion of fluid particles in a region R , such as a channel cross section, and thus can be called
a mapping function S [26] . From an initial condition ( t
0), R can be transformed into a new region
using S . This transformation or mapping can mathematically be described as S ( R ). Each trans-
formation is called an advection cycle , which corresponds to a mixing element in different micro-
mixer designs such as sequential lamination, discussed later in this topic. Repeating these mixing
elements n times refers to repeated application of S to R ,or S n ( R ). With the discrete number of
advection cycles, the transformation S is understood as a discrete, not continuous, operation as in the
case of the time function.
If the fluid is assumed to be incompressible, then the volume (in a three-dimensional case) or
the area (in a two-dimensional case) are preserved after each transformation. Thus, the above
mapping function S is called a volume-preserving transformation or a area-preserving trans-
formation.
A trajectory of a point after applying many discrete transformations is called an orbit . If a point p
returns to the same place after N transformations, the orbit is periodic with a period of N . The two
typical periodic orbits are the stable elliptic orbit and the unstable hyperbolic orbit. Elliptic orbits
lead to a region that does not mix with the surrounding fluid and thus are bad for mixing. Hyperbolic
orbits lead to squeezing and stretching of fluid regions and thus are good for mixing. In an unstable
orbit, if the fluid particle changes its path at the intersection of the same orbit, the orbit is called
homoclinic . If the fluid particle changes its path at the intersection with another orbit, the orbit is
called heteroclinic .
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