Biomedical Engineering Reference
In-Depth Information
along the surface of
dV
. In (4.11), the notation
E
2
corresponds to
�
E
and
d
ik
is the
Kronecker delta function:
d
ik
= 0 if
i
¹
k
and
d
ii
; =
l
; and
i
,
k
=
x
,
y
, or
z
. On a general
standpoint, the net force acting on the liquid volume element is
=
ò
�
F
T n dA
(4.12)
i
ik ik
W
where we have used the Einstein summation convention (summation on the re-
peated indices). At the surface of a perfectly conducting liquid on the gas side, the
electric field is perpendicular to the surface (Figure 4.4), and related to the surface
density of electric charges by Gauss' law
�
�
σ ε
=
E n
.
(4.13)
s
0
where
�
n
is the outward unit normal vector. Moreover, the electric field vanishes in
the conducting liquid. If we consider the {
x,y,z
} axis system such as the x-axis is
aligned with
n
, the electric field is
E
= (
E
x
,0,0) in the gas domain, and
E
= (0,0,0) in
the liquid domain. In the gas domain (
e
= 1), the Maxwell tensor is
1
æ
ö
2
ε
E
0
0
0
x
ç
÷
2
ç
÷
1
[ ]
ç
2
÷
T
=
0
-
ε
E
0
(4.14)
0
x
ç
2
÷
ç
÷
1
2
0
0
-
ε
E
ç
÷
0
x
è
ø
2
and vanishes in the liquid domain
[ ]
= [0
T
(4.15)
We can now integrate (4.12); the cross terms
xy
,
yz
, and
zx
are all zero, the forces
in the
y
(respectively
z
) direction cancel out, and we find that the only nonvanishing
contribution is a force directed along the outward normal
�
n
�
�
F
�
ε
�
σ
0
s
2
=
P n
=
E n
=
E
(4.16)
e
δ
A
2
2
where
d
A
is an elementary surface area of the interface. In (4.16),
P
e
is the elec-
trostatic pressure defined by
P
e
=
e
0
/2
E
2
. The electrostatic pressure
P
e
acts on the
Figure 4.4
Electric force acting at the interface of a conducting liquid.
