Biomedical Engineering Reference
In-Depth Information
pative interaction term of the kind
F
ij
=
−
γ
C
e
ij
e
ij
·
T
1
T
v
ij
−
γ
C
+
γ
v
ij
=
−
γ
(
v
ij
·
e
ij
)
e
ij
,
(10.22)
where the second term is analogous to the dissipative force in DPD. From the
fluctuation-dissipation theorem, random interactions are given by
2
k
B
T
2
1
3
T
tr
[
d
W
ij
]
3
F
ij
dt
T
d
W
ij
+
=
γ
γ
C
−
γ
·
e
ij
,
(10.23)
where
tr
is the trace of a random matrix of independent Wiener increments
d
W
ij
,and
d
W
ij
=
[
d
W
ij
]
d
W
ij
−
d
W
ij
]
3 is the traceless symmetric part, while
d
W
ij
=
tr
[
1
/
d
W
ij
]
/
[
+
d
W
ij
2 is the symmetric part. Note, that the last equation imposes the
T
. The defined dissipative and random forces in combination with
an elastic spring constitute a viscoelastic spring whose equilibrium temperature
k
B
T
is constant. To relate the membrane shear viscosity
C
condition 3
γ
>
γ
η
m
and the dissipative parameters
C
we employ the idea used for the derivation of membrane elastic properties
(see [37, 51] for details) and obtain the following relation
T
,
γ
γ
√
3
√
3
C
γ
T
η
m
=
γ
+
.
(10.24)
4
T
C
is
Clearly,
γ
accounts for a large portion of viscous contribution, and therefore
γ
T
set to
γ
/
3 in all simulations.
10.2.3.3 RBC-solvent boundary conditions
The RBC membrane encloses a volume of fluid and is itself suspended in a solvent.
In particle methods, such as DPD, fluids are represented as a collection of interacting
particles. Thus, in order to impose appropriate boundary conditions (BCs) between
the membrane and the external/internal fluids two matters need to be addressed:
i) enforcement of membrane impenetrability to prevent mixing of the inner and the
outer fluids;
ii) no-slip BCs imposed through pairwise point interactions between the fluid parti-
cles and the membrane vertices.
Membrane impenetrability is enforced by imposing bounce-back reflection of
fluid particles at the moving membrane triangular plaquettes. The bounce-back re-
flection enhances the no-slip boundary conditions at the membrane surface as com-
pared to specular reflection; however, it does not guarantee no-slip. Additional dis-
sipation enhancement between the fluid and the membrane is required to achieve
no-slip at the membrane boundary. For this purpose, the DPD dissipative force be-
tween fluid particles and membrane vertices needs to be properly set based on the
idealized case of linear shear flow over a flat plate. In continuum, the total shear force
exerted by the fluid on the area
A
is equal to
A
η
γ
˙
,where
η
is the fluid's viscosity
and ˙
is the local wall shear-rate. In DPD, we distribute a number of particles on the
wall to mimic the membrane vertices. The force on a single wall particle exerted by
γ