Biomedical Engineering Reference
In-Depth Information
In order to consider these different factors, we have recently proposed a model
that focuses on three independent components: the far-field hydrodynamic interac-
tion of a RBC in a plasma solvent, the raise of viscosity of the suspension with the
hematocrit level, and the many-body collisional contributions to viscosity [ 26 ].
These three critical components conspire to produce large-scale hemorheology and
the local structuring of RBCs. The underlying idea is to represent the different
responses of the suspended bodies, emerging from the rigid body as much as the
vesicular nature of the globule, by distinct coupling mechanisms. These
mechanisms are entirely handled at kinetic level, that is, the dynamics of plasma
and RBC's is governed by appropriate collisional terms that avoid to compute
hydrodynamic forces and torques via the Green's function method, as employed in
Stokesian dynamics [ 7 ]. The fundamental advantage of hydrokinetic modeling is to
avoid such an expensive route and, at the same, enabling to handle finite Reynolds
conditions and complex boundaries or irregular vessels within the simple colli-
sional approach. At the macroscopic scale, the non-trivial rheological response
emerges spontaneously as a result of the underlying microdynamics.
The presence of suspended RBCs is included via the following forcing term
[see ( 10.1 )]:
c S G
w p G
c S þ ð
c p
G
c p Þð
u
c p Þ
u
F p ¼
;
(10.6)
c S
where G ( x , t ) is a local force-torque. This equation produces first-order accurate
body forces within the LB scheme. Higher-order methods, such as Guo's method
[ 16 ], could be adopted. However, given the non-trivial dependence of the forces
and torques on the fluid velocity and vorticity, Guo's method would require an
implicit numerical scheme, whereas it is preferable to employ an explicit, first-order
accurate numerical scheme.
The fluid-body hydrodynamic interaction is constructed according to the transfer
function
centered on the i -th particle position r i and having ellipsoidal
symmetry and compact support. The shape of the suspended body can be smaller
than the mesh spacing, allowing to simulate a ratio of order 1 : 1 between
suspended bodies and mesh nodes. In addition, the body is scale-adaptive, since it
is possible to reproduce from the near-field to the far-field hydrodynamic response
with desired accuracy [ 25 ]. The fluid-particle coupling requires the computation of
the following convolutions over the mesh points and for each configuration of the N
suspended bodies:
r i Þ
X x u
Þdð
~
u i ¼
ð
x
x
r i Þ;
X x
O i ¼
Þdð
x
x
r i Þ;
X x t
t i ¼
r i Þdð
ð
x
Þð
x
x
r i Þ;
(10.7)
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