As shown in Figure 1-2, a local Cartesian coordinate system is set up with the origin on the
surface (stern plane actually, in order to simplify the exponential index in electric potential in
Equation 1-6), the first axis x tangential to the wall surface, and the second axis y normal to the
wall surface.
The set of governing equations for a steady, low Reynolds number EOF is given by
1 Dp
p Dt (1-8)
Vp -+ PV V qp =
P P (1-9)
and
V2 = Pe
8 (1-10)
where p is fluid density, v is kinematic viscosity, and q is electric potential of the fluid.
The overall electric potential inside the channel is a superposition of 0r, the electric
potential associated with the surface charge, and 0ex that caused by the external electric field
= 0f + Oext (1-11)
Since the external electric field will cause no local accumulation of electric charge, the
Laplacian of ext drops to zero,
V2 =ext Z 0
S. (1-12)
is assumed invariant along the glass plates, it is independent ofx coordinate, hence
= 0. Thus, the Laplacian of s,,uf reduces to
8x
2 a2 a2 020 s
V2.. = "+ f
S 9x2 +y 2 f 2y 2 (1-13)
Plug Equation 1-11 into Equation 1-10, and make use of Equations 1-12 andl-13. The
Poisson's equation is reduced to