Computational transport processes for micromixers* - Micromixers Fundamentals Design and Fabrication - page 81

where

8
<

0 f þ 3

2 3 þ

if f < 3
if

sin pf
3

1
2

H ðfÞ¼ ; jfj 3

if f > þ3

(3.35)

:

p

1 ;

With this, the Navier-Stokes equations presented in Section 3.3.2 apply. No additional modification
on the Navier-Stokes equations is required to account for the second fluid. The continuum surface
force [7] model is used for the interfacial force between the two fluids. An additional volumetric
force localized around the interface is added to the Navier-Stokes equation (Eqn 3.4 )as

f u ¼ks NdðfÞþðN VsÞ NdðfÞ

(3.36a)

N , and the curvature k are defined

where the Dirac delta function d ( f ), the unit normal to the interface

respectively as

8
<

þ ðpf=3Þ

2 3

1 cos

if jfj < 3

;

dðfÞ¼ (3.36b)

:

0 ;

otherwise

¼ V f
jVfj

N

(3.36c)

k ¼ V $ N

(3.36d)

The first and second terms in Eqn (3.36a) represent the capillarity and Marangoni effects, respectively.
The interface is advected by the underlying velocity field u . Its evolution is governed by

(3.37)
Generally, u is nonuniform. Upon advection, f ceases to be a distance function. This is further
exacerbated by the unavoidable numerical errors incurred in advecting f .
To maintain f as a distance function after the advection of f via Eqn (3.37) , f is set to the steady-
state solution of Eqns (3.38) [8] :

f t þ u $ Vf ¼ 0 :

vf 0
vt þ

ðfÞðjVf 0 j

sign 1 Þ¼ 0 (3.38a)

where t is a pseudo-time for f 0 and sign

ðfÞ is given by [9]

f
f 2

sign ðfÞ¼ q (3.38b)

2

þjVfj ðDx 2 Þ

and is subject to the following initial condition:
f 0 ðx; 0 Þ¼fðxÞ (3.38c)