Biomedical Engineering Reference
In-Depth Information
Z
t
m
1
h.t/ C
Z
u
.s/
u
h.s/
t
0
h.s/F.h.s//ds D
0
F
.s/
Z
t
Z
u
@
t
u
h.s/
C
u
r
u
h.s/
: (4.109)
C mg e
d
t
0
F
.s/
where F is computed by Proposition
4.9
and has been shown to diverge like h
3=2
when h ! 0. Proving that the
no-collision paradox
extends to the nonlinear system
(FRBI) then reduces to bound the remainder term on the right-hand side locally
in time, having in mind that
u
satisfies (
4.34
). The main difficulty here is thus to
extract fine properties of the solution to the Stokes problem in the limit h ! 0.
Hesla applies this method in [
37
] to the case of one cylinder inside a cylindrical
domain and proves no contact occurs between the cylinder and the domain boundary
in finite time.
An alternative method, proposed in [
38
], is to set
h
D
u
h
the approximate Stokes
solution constructed in the previous section. One then note that, in the gap between
the cylinder and the wall, there holds
u
h
Dr
?
(see (
4.101
) for a definition of )
so that:
u
h
D
@
112
@
222
@
111
C @
221
Setting
p
h
.x
1
;x
2
/ D@
1
@
222
.s;x
2
/ds @
12
yields:
0
@
111
C 2@
221
:
u
h
r p
h
D
We emphasize that a remarkable feature of here is that it is polynomial of degree
3 in x
2
so that the @
1
@
222
does not depend on x
2
. The pressure is extended to the
whole
R
d
C
n B
h
by truncation. This yields (see [
26
, Proposition 9] for instance):
Proposition 4.12.
There exists a constant
C<1
s.t. for all
h<1
and
2
KŒB
h
;
R
d
there holds:
LJ
LJ
LJ
LJ
LJ
C
.
u
h
r p
h
/
LJ
LJ
LJ
LJ
LJ
Z
CkI H
1
.
R
d
C
/k:
R
d
C
n
B
h
With these test-function
u
h
and associated pressure
p
h
we rewrite the right-hand
side of (
4.108
)as
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