Biomedical Engineering Reference
In-Depth Information
Figure 4.5
Schematic of the electric charge distribution in the vicinity of the triple contact line.
Electric charges are located at the solid-liquid interface and at the liquid-gas interface, approximately
within a distance
d
from the solid surface. In reality, this is a very simplified view since the liquid
interface is distorted by the electric forces very close to the triple contact line.
liquid surface and brings a negative contribution to the total pressure within the
liquid. The liquid interface is distorted by the electric forces, according to (4.16).
Electric charges at the liquid-gas interface are located close to the triple contact
line, as sketched in Figure 4.5, within a distance roughly equal to the dielectric
thickness
d
.
This problem has been investigated by Kang [8] and Vallet et al. [10] using the
Schwarz-Christoffel conformal mapping [11]. In our case, the electric potential
f
satisfies the Laplace equation Ñ
2
f
= 0 and consequently is a harmonic function. The
theory of analytic and harmonic function shows that a conformal mapping exists
(the Schwarz-Christoffel mapping, or SC mapping) that transforms the functions
E
and
f
for a half plane into the same functions for a wedge (Figure 4.6). This map-
ping shows that the charges concentrate at the tip of the wedge, and, after some
complicated algebra, produce the resultant of the electric forces
Figure 4.6
Principle of the Schwarz-Christoffel conformal mapping. The two orthogonal fields
E
and
f
are transformed from a simple half plane geometry (with an evident solution) to the geometry
of a wedge. Note the concentration of the electric field around the tip of the wedge.
