Biomedical Engineering Reference
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V4 e $ b
n
¼
0
; c
x
˛vU e ; 2
(3.24b)
where
vU ¼ vU e ; 1 W vU e ; 2 :
The boundary conditions for the Navier-Stokes equations have to be modified to account for the
electric-osmotic effect. For a thin EDL, the Helmholtz-Smoluchowski slip velocity is imposed at the
wall of the channel vU e ; 2 as
I
n
3 e z
m
u
¼
b
n
b
$ V4 e
(3.25)
where z is the zeta potential of the charged walls.
3.3.4.2 Ferrofluid flows
For ferrofluid flows, electric field is not involved. Therefore, only Eqns (3.10) and (3.11) are required
to describe the magnetic field. As ferrofluid is assumed to be nonconductive, J e ¼
0 and D
¼
0 , Eqn
(3.10) reduces to
V
H
¼
0
:
(3.26)
This equation can be satisfied by introducing a magnetic potential 4 m in the form of
H
¼V4 m :
(3.27)
For a ferrofluid, the magnetic flux in Eqn (3.14) can be written as
B
¼ m m ; o ð
H
þ
M
Þ
(3.28)
10 7 H
where the free space permeability m m ; o ¼
m and M is the magnetization. With the
assumptions of a linear ferrofluid with instant and collinear magnetization, M is given by
4
p
=
M
¼ cH
(3.29)
where c is the magnetic susceptibility.
Upon substitution of Eqns (3.27) , (3.28) , and (3.29) into Eqn (3.11) , we have
V $ ½ð
1
þ cÞV4 m ¼
0
:
(3.30)
The boundary equation for Eqn (3.30) governing the magnetic field is
v4 m
B
$ n
¼ð
1
þ cÞ
vn ; c
x
˛vU:
(3.31)
The magnetic field imparts an additional force on the ferrofluid. This force is to be included in the
Navier-Stokes equations [4-5] (Eqn 3.4 )as
Z H
vyM
vy
f u ¼V
d H
þ m m ; o ð
M
$
H
(3.32)
T $
0
H
;
where y is the specific volume.
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