Biomedical Engineering Reference
In-Depth Information
where
u
and
are the velocity and the stress fields, the blood density and
f
the
volume force per unit mass of fluid. The components of the stress tensor are defined
by the Stokes' law:
D
p
I
C
2
©
.
u
/
(2)
where
p
is the pressure,
I
the unit tensor, the dynamical viscosity and
.
u
/ the
strain rate tensor. Neglecting body forces, conservation of mass and momentum (
1
)
become:
"
@
u
u
@t
C
u
:
r
Dr
p
C
r
u
r
:
u
D
0
(3)
This equation system (
3
) can be solved for the velocity and the pressure given
appropriate boundary and initial conditions. In this study the biochemical and
mechanical interactions between blood and vascular tissue are neglected. The
innermost lining of the arterial wall in contact with the blood is a layer of firmly
attached endothelial cells and it appears to be reasonable to assume no slip at
the interface with the rigid vessel wall; at the flow entrance Dirichelet boundary
conditions for all points are considered prescribing the time dependent value
u
D
for
the velocity on the portion
D
of the boundary:
u
.
x
;t/
D
u
D
.
x
;t/;
x
2
D
(4)
At an outflow boundary
N
the condition describing surface traction force
h
is
assumed. This can be described mathematically by the condition:
@
u
i
n
j
@
u
j
@x
j
pı
ij
C
@x
j
C
D
D
h
i
i; j
1; 2; 3
on
N
(5)
where n
j
are the components of the outward pointing unit vector at the outflow
boundary.
3
Finite Element Formulation
The finite element method is a mathematical technique for obtaining approximate
numerical solution of the physical phenomena subject to initial and boundary
conditions. Two different finite element models of the Navier-Stokes equations are
considered in this chapter, the mixed model and the penalty finite element model.
