Computational transport processes for micromixers* - Micromixers Fundamentals Design and Fabrication - page 86

where

a E ¼ D e þ k F e ; 0 k (3.53b)

a W ¼ D w þk F w ; 0 k (3.53c)

a N ¼ D n þ k F n ; k

0 (3.53d)

a S ¼ D s þ k F s ; 0 k (3.53e)

a P a P ¼ a E þ a W þ a N þ a S þ S P D V (3.53f)

r P D

V

a P ¼ (3.53g)

D t

b ¼ S C DV þ a P F

P :

(3.53h)

3.4.2.1 Higher-order schemes
In the above discussion, the convective terms in Eqn (3.43) are modeled using a first-order upwind
scheme (Eqn 3.45). The first-order upwind scheme is numerically too diffusive. Higher-order scheme
is then introduced. The convection term is modeled using a second-order upwind scheme with flux
limiters. This is achieved easily within the present framework by employing first-order upwind scheme
with an additional source term to increase its accuracy to second order via deferred correction
approach. An additional source term S DC , included into S C (Eqn 3.53h), is given as [15]
S DC ¼

2 F e þð

r e Þ F E F P Þ

1

r e Þa e jð þ 1 a e Þjð

2 F w ð

r w Þ F P F W Þ

1

r w Þþa w jð þ 1 a w Þjð

(3.54a)

2 F n þð

r n Þ F N F P Þ

1

r n Þa n jð þ 1 a n Þjð

2 F s ð

Þ F P F S Þ

1

a s Þjðr s Þþa s jðr s þ 1

¼ ðF EE F E Þ=ð

x EE x E Þ

¼ ðF P F W Þ=ð

x P x W Þ

r e r e ; (3.54b)

ðF E F P Þ=ð x E x P Þ ðF E F P Þ=ð x E x P Þ

r w ¼ ðF E F P Þ=ð

x E x P Þ

r w ¼ ðF W F WW Þ=ð

x W x WW Þ

x W Þ ;

(3.54c)

ðF P F W Þ=ð x P ðF P F W Þ=ð x P x W Þ

r n ¼ ðF NN F N Þ=ð

x NN x N Þ

r n ¼ ðF P F S Þ=ð

x P x S Þ

; (3.54d)

ðF N F P Þ=ð x N x P Þ ðF N F P Þ=ð x N x P Þ

¼ ðF N F P Þ=ð

x P Þ
ðF P F S Þ=ðx P x S Þ ;

x N

¼ ðF S F SS Þ=ð

x SS Þ
ðF P F S Þ=ðx P x S Þ

x S

r s r s (3.54e)