Biomedical Engineering Reference
In-Depth Information
3.3.3
Energy transport
In cases where temperature field is important, i.e., heat transfer occurs, the energy equation is required.
The energy equation can be expressed as (see Section 2.1.2.3):
vðrc
p
T
Þ
þ V
$
ðrc
p
uT
Þ¼V
$
ð
k
V
T
Þþ
S
T
(3.6)
vt
where
c
p
and
T
are the specific heat and temperature, respectively.
S
T
is the additional volumetric heat
source/sink. The boundary condition for temperature can be a combination of a given temperature and
a given heat flux. Mathematically, these can be expressed as
T
¼
T
1
;
c
x
˛vU
T;
1
(3.7a)
q
T
¼
k
V
T
$
b
n
;
c
x
˛vU
T;
2
(3.7b)
where
vU ¼ vU
T;
1
W
vU
T;
2
:
3.3.4
Electric and magnetic fields
Depending on the fluid type under consideration, its motion can be manipulated by an electric or
a magnetic field. For example, in electroosmotic flows, the motion of an aqueous solution is actuated
by an applied electric field. Ferrofluids are colloidal liquids with stabilized magnetic particles sus-
pended in a carrier fluid. Under the influence of an applied magnetic field, additional magnetic force
acting on the ferrofluid is induced. The motion of the magnetized ferrofluids can then be controlled via
the applied magnetic field. In the context of micromixers, these electric and magnetic forces can be
used to enhance mixing. To account for the electric and magnetic force in the current computational
framework, Maxwell's equations along with the continuity equation for charges are required:
vB
vt
V
E
¼
(3.8)
V
$
D
¼ r
e
(3.9)
vD
vt
V H ¼ J
e
þ
(3.10)
V
$
B
¼
0
(3.11)
vr
vt
V
$
J
e
¼
(3.12)
where
E
,
D
,
H
,
B
,
J
e
, and
r
e
are the electric field intensity (V/m), electric flux density (C/m
2
),
magnetic field intensity (A/m), magnetic flux density (Wb/m
2
), electric current density (A/m
2
), and
electric charge density (C/m
3
), respectively. Here, Eqns
(3.8)
,
(3.9)
,
(3.10)
,
(3.11)
, and
(3.12)
are
called Faraday's law, Gauss' law, Ampere's law, the magnetic Gauss' law, and the continuity equation
for charges, respectively. For a given medium,
D
,
B
, and
J
e
can be expressed as
D ¼ 3
e
E
(3.13)
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