Computational transport processes for micromixers* - Micromixers Fundamentals Design and Fabrication - page 78

B ¼ m m H (3.14)

J e ¼ s e E (3.15)

where 3 e , m m , and s e are permittivity (F/m), permeability (H/m), and conductivity (S/m), respectively.
Equations (3.8)-(3.11) are subject to the following boundary conditions:

n b ½½ E ¼ 0 (3.16)

½½ D $ b n ¼ r e ; s (3.17)

n b ½½ H ¼ J e ; s (3.18)

½½ B $ b n ¼ 0 (3.19)

where n , r e ; s ,and J e ; s are the unit normal vector, the surface charge, and the surface current density at the

boundary, respectively. The operator

b

represents the jump of a across the boundary. For a further
discussion of Maxwell's equations and the relevant boundary conditions, the reader is referred to [2-3] .
The system of equations governing electric and magnetic fields (3.8-3.15) are very complex and
demanding to solve. Fortunately, Eqns (3.8)-(3.15) can be greatly simplified for most cases encoun-
tered in the investigation of transport processes in micromixers. Two examples of these simplifications
are presented next.

½½ a

3.3.4.1 Electroosmotic flows
To describe the electric field in electroosmotic flows, only Eqns (3.8) and (3.9) are required. Since
magnetic field is not involved, vB

vt ¼

0. Equation (3.8) reduces to

V E ¼ 0 : (3.20)

This equation can be satisfied by introducing an electric potential 4 e of the form

(3.21)
It is assumed that the EDL formed is thin. In the bulk of the aqueous solution (i.e., except within the
EDL), the net charge density is zero:

E ¼V4 e :

r e ¼ :

0 (3.22)

Substituting Eqns (3.21) and (3.22) into Eqn (3.9) gives

(3.23)
Equation (3.23) governs the electric field in the domain of interest. It is assumed that the channel wall
vU e ; 2 is nonconducting. With this assumption, Eqn (3.23) is subject to the following boundary
conditions:

V $ ð3 e V4 e Þ¼ 0 :

4 e ¼ 4 e ; 1 ; c x ˛vU e ; 1 (3.24a)