Computational transport processes for micromixers* - Micromixers Fundamentals Design and Fabrication - page 82

To mitigate the mass loss problem, a particle correction procedure is adopted. Basically, two sets of
particles offering subcell resolution are included to keep track of the interface. The particle correction
procedure will not be discussed here. Interested readers are referred to [10] . It should be mentioned
here again that the level-set method is the authors' personal choice. Other method of treating an
evolving interface, e.g., the VOF method [11] and front-tracking method [12-13] , can also be used.

3.4 SOLUTION PROCEDURE
3.4.1 General transient convection-diffusion equation
The species conservation (Eqn 3.1 ), the Navier-Stokes (Eqns 3.3 and 3.4 ), the energy (Eqn 3.6 ),
electric potential (Eqn 3.23 ), and magnetic potential (Eqn 3.30 ) equations can be recast into a general
transient convection-diffusion equation of the form
v ð r FÞ
vt
|{z}
Transient

þV $ ðruFÞ
|{z}
convection

¼ V $ ðGVFÞ
|{z}
Diffusion

þ S

|{z}
Source

(3.39)

where r , are the appropriate “density”, “diffusion coefficient”, and source term, respectively.

The source term contains all other terms that cannot be fitted neatly into the convection or diffusion
terms. The solution procedure for this equation is presented next.

G , and S

F

3.4.2 Finite volume formulation
The finite volume method is employed to solve the general transient convection-diffusion equation
numerically. This solution procedure follows the idea described in [14-15] . Integration of Eqn (3.39)
over an arbitrary control volume (CV) gives
Z Z Z Z

v ð r FÞ
vt

d V þ

DV V $ ðru

FÞ d V ¼

DV V $ ðGVFÞ

d V þ S d V : (3.40)

DV DV

Employing Gauss' divergence theorem, the volume integration is converted into a surface inte-
gration as

Z I I Z

v ð r FÞ
vt

d V þ

S ðru

FÞ $ d S ¼

S ðGVFÞ $

d S þ S d V : (3.41)

D V D V

Equation (3.41) states the conservation principle for the quantity within the CV. This equation is

applied to every CV to derive discretized governing equations relating the dependent variable of that
CV to those of the neighboring CVs. Then, the discretized governing equations express the conser-
vation principle for the CV in a discrete sense.
Application of Eqn (3.41) for a two-dimensional domain will be demonstrated. Equation (3.39) can
be expressed in a two-dimensional Cartesian coordinate system as
v ð r FÞ
vt þ

F

v
vx ðru

v
vy ðrv

v
vx

v vx v

vy

v vy

FÞþ FÞ¼ G þ G þ S : (3.42)