Biomedical Engineering Reference
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where h WD
dist.
B
1
.t/;B..2;0;0/;1// (D
dist.
B
1
.t/;B..2;0;0/;1// by sym-
F.h/diverges like 1=h when h ! 0 and the vectors e
C
;e
metry), the drag
are
defined by:
e
C
D .0;0;a/ .2;0;0/; e
D .0;0;a/ .2;0;0/:
In (
4.103
) the first contribution on the right-hand side is an approximation of the
drag force exerted by the fluid on
B
1
due to the presence of the hole B..2;0;0/;1/
whereas the second term stands for the drag force due to the presence of the hole
B..2;0;0/;1/. Consequently, a good approximation of (
4.102
) reads:
m
1
a D2 h F.h/.e
C
e
3
/e
3
C f:
Integrating once, we obtain:
Z
t
2 h F.h/.e
C
e
3
/ds D ft:
m
1
Œa.t/ a.0/ C
0
where, introducing that the total energy of the system decreases with time, we obtain
that jŒa.t/ a.0/jC
0
is bounded w.r.t. initial data only before contact. Finally,
computing e
C
e
3
w.r.t. h, we obtain that:
C
p
h
0 .h/ D F.h/.e
C
e
3
/
for small h.
This entails that before contact:
2
Z
h.t/
h.0/
.h/dh C
0
C ft:
and we prove that a contact must occur by a contradiction argument. Indeed, 2
L
1
.0;h.0// so that the left-hand side of this last inequality is bounded from below by
a fixed constant depending on initial data, whereas the right-hand side goes to 1
when t !1. Hence, contact must occur in finite time. More details and rigorous
estimates for remainder terms in this construction (for the full system (FRBI)) can
be found in [
40
].
From the Stokes System to (FRBI).
As shown in the previous construction, lack
of contact in solutions to the Stokes problem comes from overestimating the drag
force that rigid bodies undergo. This property is transferred to solutions to (FRBI)
thanks to the Lorentz reciprocal theorem:
be an open bounded set of
R
d
having a smooth boundary.
Proposition 4.11.
Let
O
Let
.
u
i
;p
i
/ 2 H
2
.
/ H
1
.
O
O
/.iD a;b/
such that:
r
u
a
Dr
u
b
D 0;
on
O
:
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