Biomedical Engineering Reference
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where
H
is the extra body force that will be used to incorporate further details of the
ESL and particulate effects, such as the RBC interface force density (pressure step)
defined below. To solve governing Equations (
10.8
) and (
10.9
) we combine the LB
methods of Guo and Zhao [
15
] with the model of Halliday et al. [
18
], that allows for
the introduction of two immiscible fluid components and the formation of interfaces
that embed correct kinematic and surface tension laws. To complete the algorithm,
we must mention that, for multiple fluid LB, the propagation step is augmented by a
fluid segregation process that ensures the correct kinematics and dynamics and
good integrity for an interface between completely immiscible fluid components,
representing RBC and plasma, as discussed above [
18
]. The propagation step is
expressed as:
R
r
RB
r
f
p
þ
R
p
ð
x
þ
c
p
D
t
;
t
þ D
t
Þ¼
w
p
b
c
p
n
;
(10.10)
B
r
RB
r
f
p
B
p
ð
x
þ
c
p
D
t
;
t
þ D
t
Þ¼
w
p
b
c
p
n
;
where the density of each fluid component is given by
R
¼ ∑
p
R
p
(
x
,
t
)and
B
¼ ∑
p
B
p
(
x
,
t
) , the combined particle distribution function is
f
p
¼
R
p
þ
B
p
,
and
f
p
accounts for the propagated combined distribution. In (
10.10
)
represents
an interfacial segregation parameter and
n
the interfacial unit normal vector. We
also note that, if only one fluid component exists, (
10.10
) reduce to the standard
LB propagation step (
10.1
). Returning to the definition of the extra body force
term,
H
in (
10.9
), this incorporates both particulate and glycocalyx forces and is
defined as
b
2
r
prr
N
þ
H
¼
E
:
(10.11)
s
The left-hand side term imposes an interfacial tension
on multicomponent
particles. Here,
is a
phase field indicator. The right-hand term
E
is a glycocalyx force that acts upon the
particles as defined in the next section.
p ¼ ∇
n
is the local curvature, and
r
N
¼ð
R
B
Þ=ð
R
þ
B
Þ
10.5 The RBC: Glycocalyx Interplay
In the proposed model of the ESL as a porous layer, the porosity is reduced by a
compressive encounter with an erythrocyte. As a consequence, the ESL is squashed
locally transporting the same mass into a smaller volume and consequently decreas-
ing the porosity in that region. Even in the simplest situation, the ESL-lumen
boundary should not be regarded as sharp, and there is an
uncertainty region
between bulk, lumen, and glycocalyx material [
33
]. Let us define a variable porosity
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