Biomedical Engineering Reference
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FIGURE 3.21
Mixing in a serpentine microchannel.
FIGURE 3.22
Schematic of mixing within a droplet flowing in a straight microchannel.
ðr ¼ r þ ¼
For the Navier-Stokes equation,
the properties of oil and water are used
m 3
10 2 Pa s and m þ ¼
10 3
=
; m ¼
:
:
Þ
1000 kg
. For flow in microscale, the effect
of density is normally not important, and the density of oil is slightly lower than that of water. Thus,
identical density is used for both liquids. The surface tension between water and oil is set to
s ¼
3
0
1
0
Pa s
10 3 N/m. The inlet velocity is specified as u o ¼
0.003 m/s. Outflow boundary condition
is used at the outlet. At the walls, no-slip condition is enforced.
In the solution of the species conservation equation, a second-order upwind method with
Superbee flux limiter is employed for the convective terms. Even with a second-order scheme,
species X diffuses out of the droplet during the solution process because of the inherited numerical
diffusion of the scheme. This phenomenon is purely a numerical artifact and therefore is not
physical. An even higher-order scheme can be an alternative to alleviate this problem. However,
the authors choose to adopt a simpler approach, though less mathematically sophisticated. The
amount of species X diffused out of the droplet is redistributed back into the droplet uniformly by
making an appropriate correction to c within the droplet and setting c
3.65
¼
0 in the oil region at every
time step.
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