Biomedical Engineering Reference
In-Depth Information
μ
∂
u
1
,
z
∂
−
p
1
(
z
|
ζ
a
)+
2
z
(
z
|
ζ
a
)=
−
n
d
1
|
Γ
a
|
p
2
(
x
,
y
,
z
|
Γ
a
)+
2
με
(
u
2
(
x
,
y
,
z
|
Γ
a
))
n
·
Γ
a
at
ζ
a
.
(9.52b)
Γ
a
Conversely, for this case we have the strong continuity (pointwise) of the velocity
field, in the sense of
H
1
/
2
(it is considered to be constant over the section) and
a weak continuity (mean value) of the dual field, in the sense of
(
Γ
a
)
.
When looking at (9.52a), the constant velocity profile may suggest that incon-
sistencies arise in the representation of no-slip boundary conditions over the lateral
boundary. For instance, this profile may lead us to think that no viscous effects should
be considered in the 3D model, while it is obvious that we should take into account
such viscous effects. In this respect, this model is not consistent in the sense of re-
producing the same profile. Nevertheless, this inconsistency can be removed if we
further refine the 1D model, for example, by modifying the velocity profile and intro-
ducing a parabolic, or Womersley-like, profile characterized by a function
R
. That
is, a more complex situation can be considered assuming that the velocity profile in
the 1D model is given by
ϕ
u
1
→
u
1
,
z
(
x
,
y
,
z
)
e
z
=
ϕ
(
x
,
y
)
u
1
,
z
(
z
)
e
z
,
(9.53)
Γ
a
ϕ
(
Γ
a
u
1
·
1
|
Γ
a
|
1
|
Γ
a
|
where
0(see
Fig. 9.3). The spaces are similar to those previously introduced. Eventually, the vis-
cous contribution to the dual variable in the 1D domain will change, according to
the specific form of the function
x
,
y
)
d
Γ
a
=
1 is such that
e
z
d
Γ
a
=
u
1
,
z
,and
ϕ
|
Γ
L
1
=
. Nevertheless, the way in which the problem is
formulated is completely equivalent.
ϕ
9.3.3 About fluid-structure interaction
In our present case, the fluid problem we consider both above and in the numerical
examples below is the blood flow problem in deforming domains. Such deformation
is induced by lateral wall displacements in the direction of the normal to the wall as
a function of the value of the pressure (in a pointwise sense). Therefore, we have an
independent ring model for the 3D wall (an algebraic equation -elasticity- or an ordi-
nary differential equation -viscoelasticity-) which does not introduce any difficulty
in the definition of the operator
R
2
for the 3D model, nor incompatibilities with the
structural model of the 1D representation (an independent ring model as well). This
is why we refer to our problem just as a “fluid problem”, although we clearly consider
deformations of the domains. However, and as explicitly mentioned before, we focus
our presentation in the role of the variational formulation of the fluid problem only.
In order to consider the fluid-structure interaction using our variational problem
for the structure, the operator
R
2
must be properly extended. In such case, not only
the fluid, but also the solid may undergo kinematical incompatibilities at coupling in-
terfaces. Hence, the same treatment used for the fluid formulation would be required
by the solid problem for the incompatibilities in the solid displacement between the
1D model and the 3D model. In this respect, the abstract formulation presented here
