Biomedical Engineering Reference
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and its inertia
J
i
, which reads
Z
i
.Œ
t
i
1
.x//.jx G
i
.t/j
2
J
i
WD
M
I
3
.x G
i
.t//ǝ.x G
i
.t///dx; (4.15)
B
i
.t/
in the three-dimensional case, and
Z
i
.Œ
t
i
1
.x//jx G
i
.t/j
2
dx;
J
i
WD
M
(4.16)
B
i
.t/
in the two-dimensional case. The mass m
i
and the two-dimensional inertia
J
i
are
constant scalars whereas the three-dimensional inertia is a 3 3 time-dependent
positive-definite symmetric matrix. In the three-dimensional case, we recall that
stands for the classical vector product, whereas in the two-dimensional case we need
to define a new operator:
a b D a
?
b;
8 .a;b/ 2
R
2
:
We underline that is defined in two different ways in the two-dimensional case
depending on whether the first operand is a vector or a scalar.
We denote (FRBI) (for fluid rigid-body interaction system) the full system (
4.2
)-
(
4.6
)-(
4.7
)-(
4.9
)-(
4.10
)-(
4.11
)-(
4.12
)-(
4.13
). The unknowns of this system are
..
B
i
.t/;
i
;!
i
/
i
D
1;:::;n
;
u
f
;p
f
/. It is completed with initial conditions:
i
;
u
f
.0; / D
u
f
;
i
.0/ D
i
; !
i
.0/ D !
i
;
B
i
.0/ D
B
8 i D 1;:::;n:
(4.17)
Multiplying formally (
4.9
) with
u
f
and combining with (
4.12
)-(
4.13
), we obtain
that reasonable solutions to this system should satisfy:
"
Z
i
j/g G
i
#
m
i
j
i
j
X
n
1
2
d
dt
2
2
0
C
J
i
!
i
!
i
C 2.m
i
j
B
.t/
j
u
f
j
C
F
i
D
1
C 2
Z
2
F
.t/
jD.
u
f
/j
D 0: (4.18)
This formal estimate states the decay of the total energy of the system. It is then
natural to consider initial conditions with bounded kinetic energy:
"
Z
i
!
i
!
i
#
;
m
i
j
i
j
X
n
1
2
0
c
WD
0
j
u
f
j
2
2
0
E
C
C
J
F
i
D
1
which amounts to require that
u
f
2 L
2
.
0
/. We shall restrict to this case throughout
F
the paper.
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