Biomedical Engineering Reference
In-Depth Information
∇
c
, is the attraction to nutrients (based on
chemotaxis) and the second term,
K
v
p
, is the drag force exerted by the porous
media.
M
The first term on the right side,
M
Jm
3
are two constants that will be determined exper-
imentally. The shear stress for laminar flow of a Newtonian fluid is linearly related
to the shear rate
[
/
mol
]
and
K
[
Ns
/
m
]
(
d
V
/
d
r
)
in terms of cylindrical coordinates [
25
,
26
]:
d
V
d
r
˄
=−
μ
(11.19)
m
2
] is the dynamic
viscosity of the fluid. Wall shear stress for turbulent flow is large compared to laminar
flow. For either case, laminar or turbulent, the wall shear stress can be determined
from:
where
V
is the velocity [m
/
s] at radial position
r
[m] and
μ
[N s
/
d
4
p
L
˄
=−
(11.20)
Wall shear stress of Newtonian fluids in tubular vessels can be calculated as a
function of volumetric flow rate:
4
μ
Q
˄
=
(11.21)
ˀ
r
3
based on Hagen-Poseuille equation:
128
μ
LQ
ˀ
p
=
(11.22)
d
4
The diffusion constant in the fluid for nutrients was estimated by the Stokes-
Einstein diffusivity equation for diffusion of spherical particles through liquid with
a low Reynolds number [
20
,
27
]:
k
B
T
D
=
(11.23)
ˀμ
6
r
where
k
B
is the Boltzmann's Constant,
T
is the absolute temperature and
r
is the
radius of the spherical particles.
r
was estimated from the assumed molecular weight
and the following equation for a sphere:
3
3
·
M
w
·
V
p
r
=
(11.24)
4
ˀ
N
A
where
V
p
is the molecule's specific volume,
M
w
is the molecular weight, and
N
A
is
Avogadro's number. The characteristic Knudsen Number (Kn) is defined as [
19
]:
L
Kn
=
(11.25)
Search WWH ::
Custom Search