Biomedical Engineering Reference
In-Depth Information
moving in an unlimited container is provided with existence of classical solutions
[
8
,
9
,
24
,
58
]. Contrary to the assumptions herein, the densities of the rigid bodies
are assumed to be constant in most of the references above.
4.3.1
Many Bodies in a Container: Weak Solutions
In the first part of this section, we detail the theory of weak solutions for (FRBI):
we provide a definition that is adapted to the case of non-smooth bodies, we recall
the main difficulties to be handled by a construction and we conclude by describing
the answers given in the references above. The first part of this section relies on
[
4
,
20
,
43
,
51
].
Let ..
i
;!
i
;
i
/
i
D
1;:::;n
;
u
f
/ be an initial condition with a smooth
u
f
B
and
consider that the collection ..
i
;!
i
;
B
i
/
i
D
1;:::;n
;
u
f
;p
f
/ represents an associated
classical solution to (FRBI) on .0;T/ satisfying (
4.18
). To derive a weak formu-
lation, we assume that rigid bodies remain far from contact i.e.:
0
i
;
0
j
/>0;
0
i
;@/>0;
dist.
B
B
dist.
B
8 i ¤ j;
(4.19)
dist.
B
i
.t/;
B
j
.t// > 0; dist.
B
i
.t/;@/ > 0; 8 i ¤ j;
8 t 2 .0;T/:
(4.20)
We also eliminate gravity for simplicity: g D 0.
We introduce the extended velocity-fields (see (
4.5
) for a definition of the body
velocity-fields
u
i
):
n
n
X
X
u
0
0
u
f
C
i
u
i
:
u
WD
1
F
.t/
u
f
C
1
B
i
.t/
u
i
;
WD
1
F
1
(4.21)
0
B
i
D
1
i
D
1
Because of the no-slip boundary conditions (
4.10
)-(
4.11
), we obtain divergence-free
vector-fields which are defined on and continuous through fluid-body interfaces.
We first define Sobolev-like function spaces adapted to such velocity-fields. Namely,
given an open domain
O
, we introduce:
•
the classical function spaces of incompressible hydrodynamics:
/ WD f
u
2 C
c
.
D
.
O
O
/ s.t. r
u
D 0g;
(4.22)
/ in L
2
.
/ (resp. H
0
.
and H.
O
/ (resp. V.
O
/) the closure of
D
.
O
O
O
/),
•
the set of rigid velocity-fields:
R
WD
n
C ! x;
R
d
o
:
8 x 2
R
d
.;!/ 2
R
d
I
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