Computational transport processes for micromixers* - Micromixers Fundamentals Design and Fabrication - page 76

and temporally, due to, for example, a nonuniform temperature field T , as discussed in Section 2.2.3.
In the case of more than one species, additional equations of the form similar to that of Eqn (3.1)
governing the transport of each species are needed [1] . Equation (3.1) is subject to the initial
condition:

c ¼ c o ; c x ˛U (3.2a)

and boundary conditions of

c ¼ c 1 ; c x˛vU c; 1 (3.2b)

q c ¼ D V c

$ b

n ; c x ˛vU c; 2 (3.2c)

where

vU ¼ vU c; 1 W vU c; 2 :

3.3.2 Fluid transport
For the sake of simplicity, yet without sacrificing the essence of the computational framework, only an
incompressible Newtonian fluid is considered. Non-Newtonian effect, e.g. of a generalized Newtonian
fluid, can be accommodated within the framework accordingly. The motion of an incompressible
Newtonian fluid is governed by the Navier-Stokes equations (see Section 2.1.2.2):

V $ u ¼ 0 (3.3)

þ V $ m V u T þ

vðru Þ

vt þ V $ ðruu

Þ¼V p u þ V f u

(3.4)

where r , u , p , and m are density, velocity, pressure, and viscosity, respectively. The viscosity can be a
constant or a function of T and/or c .InEqn (3.4) , f u is the additional external force per unit volume,
e.g., electrical and magnetic forces.
The initial conditions for Eqn (3.4) are

(3.5a)
The relevant boundary conditions can be a combination of inlet velocity, outflow, and no slip, which
are mathematically expressed as

u ¼ u o ; p ¼ p o ; c x ˛U:

u

$ n

¼ u inlet ; c x ˛vU u; 1 (3.5b)

vu
vn ¼

0 ; c x ˛vU u; 2 (3.5c)

u ¼ 0; c x˛vU u; 3 (3.5d)

where the boundary vU is given by

vU ¼ vU u; 1 W vU u; 2 W vU u; 3 :