Biomedical Engineering Reference
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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
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