Biomedical Engineering Reference
In-Depth Information
Z
T
u
.k/
; //
h
.k/
..
..
O
u
; //
h
! 0:
O
(1.114)
0
.h
.k/
/ !
This concludes the limiting process in (
1.109
). We found out that
F
.h/,ash
.k/
D R
0
C ı
.k/
converges to h D R
0
C ı as k !1.
Finally, using the continuity of the mapping
F
F
, its relative compactness in Y and
the property
.B
Ǜ;K
/ B
Ǜ;K
we deduce from the Schauder
fixed point theorem,
that there exists at least one fixed point of the mapping
F
F
defined by the weak
formulation (
1.109
),
.R
0
C / D . Thus, we obtain the existence of at least one
weak solution (
1.90
) of the original unsteady fluid-structure interaction problem
(
1.73
)-(
1.89
). The proof of the Theorem
1.3
is now completed.
Let us point out that we have obtained the existence of weak solution until
some time T
. We remind that this time is obtained in order to achieve the fixed
point of the mapping
F
and to avoid the contact of the elastic boundary † with
the fixed boundary for given data p
@D
;R
0
and Ǜ; K. Similarly as in [
26
]andas
explained in the previous subsection for the Newtonian case, we can prolongate the
solution in time and even obtain the global existence until the contact with the solid
bottom.
F
Remark 1.13.
The result on the existence of weak solution for the coupled fluid-
structure interaction problem for shear-thickening power-law fluids is shown for
the generalized string equation with a regularizing term of type @
x
@
t
.The
same existence result can be obtained for other regularizing terms in the structure
equation. Instead of @
xx
C @
x
@
t
we can consider @
x
@
xx
@
t
. The regularity
of the domain deformation coming from the term @
x
is essential to obtain that
2 L
1
.0;T I W
1;
1
.0;L//. This is a sufficient condition for generalized Korn's
inequality for q ¤ 2. As seen in the Newtonian case (see also [
26
,
88
]) such a
condition for is not required for Korn's equality in the moving domain
.t/ to
hold. Consequently, in three-dimensional case a plate with no additional viscosity
may be used and in the two-dimensional case a string model with no additional
viscosity may be used.
In this section we have reviewed some existence results. In the unsteady state
case for the existence of weak solutions, we have seen that the nonlinear geometrical
term could be decoupled from the fluid-structure problem. In the case, when some
decoupling of the fluid and the structure is used (e.g., to take advantage of the
already known properties/regularity of solutions of the Stokes system or of the
elastic one [
119
,
120
] or because of the chosen strategy used to obtain the desired
result [
111
,
129
]), it is still necessary to decouple the fluid and the structure in an
appropriate way, because of the added mass effect. This is in particular true for
numerical schemes as we will detail it hereafter.
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