Biomedical Engineering Reference
In-Depth Information
Introducing this information into (
4.98
), we obtain that there exists a constant C
0
depending only on initial data for which:
h.t/ C
0
exp
Œmg e
d
t
;
8 t 2 .0;T
/:
This yields the expected results.
t
This proposition states in particular that no-contact between the sphere, or the
cylinder, and the ramp occurs in finite time. In the three-dimensional case, it
corresponds to the
no-collision paradox
that was pointed out in [
7
].
We note that, in the last proof, applying Proposition
4.5
furnishes a bound from
below that may only enable to prove a no-contact result. In order to obtain contact,
or simply to ensure that the bound from below which we obtain is asymptotically
equivalent to F.h/, we compute a bound from above for F.h/. To this end, we apply
again the characterization (
4.96
). This yields that any
u
2
Y
h
satisfies:
Z
2
C
n
B
h
jr
u
j
F.h/;
R
d
so that extracting a sharp bound reduces to construct a good candidate
u
(of course,
the solution to the Stokes problem (
4.92
)-(
4.95
) would be the best choice) We detail
here the construction provided in [
27
,
41
] in the two-dimensional case.
We recall notations from the proof of Proposition
4.5
.Wehave@
R
2
C
WD
f.x
1
;0/; x
1
2
R
g
and, close to the origin, @B
h
WD f.x
1
;
h
.x
1
//; jx
1
j <1=2g
with
p
1 s
2
;
h
.s/ D h C 1
8 s 2 .1;1/:
(4.99)
We also denote:
1=2
WD f.x
1
;x
2
/ 2
R
2
s.t. jx
1
j <1=2 0<x
2
<
h
.x
1
/g:
C
We focus on the construction in
1=2
.AsB
h
remains away from @
R
2
C
outside this
domain, one may extend the constructed vector-field with something independent
of h and yielding O.1/ terms when computing the H
1
-norm to be minimized. In
1=2
,any
u
2
Y
h
reads
u
Dr
?
, where, normalizing to vanish in the origin,
boundary conditions satisfied by
u
imply that:
.x
1
;0/D 0; .x
1
;
h
.x
1
// D x
1
C C:
(4.100)
Extracting the dominating part of the minimizing problem characterizing F.h/,we
look for solution to the approximate minimizing problem:
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