Biomedical Engineering Reference
In-Depth Information
0.7
Carreau model
Cross model
Oldroyd
Experimental data
0.6
0.5
0.4
0.3
0.2
0.1
0
10
-3
10
-2
10
-1
10
0
10
1
10
2
10
3
Shear rate(S
-1
)
40 %, T = 37
◦
C,
Fig. 3
Apparent viscosity as a function of shear rate for whole blood at
H
t
=
taken from [
9
]
Plots of some non-Newtonian models and the experimental data are shown in
Fig.
3
. Experimental data for low shear rates is difficult to obtain, resulting in very
different behavior as the shear rate approaches zero.
3.3
Outflow Boundary Conditions
Equations (
4
)or(
6
) have to be provided with initial and boundary conditions, in
order to be mathematically well defined and prepared to be solved by numerical
methods. The prescription of proper initial and boundary conditions is a crucial step
in the numerical procedure to obtain accurate and meaningful computed solutions.
After defining the initial condition,
u
=
u
0
,for
t
=
0in
Ω
, an appropriate set
of conditions must be imposed on the boundary of the domain
. In particular, for
the problem of blood flow in arteries, the computational domain is bounded by a
physical boundary that is the arterial wall, and by artificial boundaries on the fluid
domain due to truncation of the artery, detailed in Fig.
2
.
On the physical boundary corresponding to the vascular wall a no-slip condition
is imposed, describing the complete adherence of the fluid to the wall. In this study,
the compliance of the artery wall will be neglected, that is, a fixed geometry is
considered, so that the velocity at the wall is zero. Thus, an homogeneous Dirichlet
boundary condition,
u
Ω
0, is imposed at the physical wall of the fluid.
The boundary conditions at the artificial sections cannot be obtained from
physical arguments and can be a significant source of numerical inaccuracies in
resolving the problem [
3
]. At these interfaces the remaining parts of the arterial
system need to be accounted for and modeled. Typically, it is very difficult to obtain
appropriate patient data for the flow boundary conditions.
=
0
, ∀
t
>
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