Biomedical Engineering Reference
In-Depth Information
for a suitable choice of the positive constant K. Therefore, the liquid adds a further
damping term into the energy equation, which, however, being restricted to the
interface , is not able to spread the dissipation also in the interior of the structure,
in a way to ensure the
uniform decay
of the energy.
However, in the case of a “thin” structure that can be modeled as a two-
dimensional manifold,
16
like a plate or a membrane, the interface liquid/solid
coincides
with the elastic structure, and one may expect that the viscosity effects
of the liquid are “strong” enough as to prevent the occurrence of resonance.
In the present and following sections we shall show that this is indeed the case
for two prototypical models mostly used in blood flow, and investigated in [
12
]and
[
37
], respectively. This circumstance is indicative of the fact that
the “thickness”
of the elastic wall may act in favor of the event of resonance
. In that regard, we
refer the reader to Sect.
3.6
, where numerical tests are presented that confirm the
theoretical prediction.
In this section we shall consider the case when the “thin” structure is flat
(a smooth portion of a plane). This model, introduced in [
12
], can be roughly
regarded as a “drum completely filled with a viscous liquid,” and will be specified
next. Consider a sufficiently regular domain
R
3
with a connected boundary
constituted by two open components,
1
and such that
1
\ D;. Moreover,
we assume that (the elastic structure) is flat, namely,
fx D .x
1
;x
2
;0/W x
0
WD .x
1
;x
2
/ 2
R
2
g ;
with a smooth boundary @, while
1
is a surface lying in the half-space fx
3
0g.
The domain is completely filled with a viscous liquid that moves in the vanishing
Reynolds number approximation, so that its motion is governed by the Stokes
equations
in .0; 1/:
v
t
divT.v;p/D 0
divv D 0
(3.97)
As for the motion of the “plate” , we assume that it can only undergo transversal
displacements
u
D
u
.x;t/, therefore directed along the x
3
axis. In such a case, the
governing equations become (e.g., [
29
])
u
tt
C
2
u
De
3
T.v;p/ e
3
C f; in .0; 1/.
(3.98)
To (
3.97
)-(
3.98
) we have to append boundary conditions. As for the liquid, it
adheres at the “rigid” as well as the elastic walls:
v.x;t/ D 0;.x;t/2
1
.0; 1/I v.x
0
;t/D
u
t
.x
0
;t/e
3
;.x
0
;t/2 .0; 1/;
(3.99)
16
In the case of three-dimensional flow, or else as a “string,” in the two-dimensional case.
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