Biomedical Engineering Reference
In-Depth Information
In [
26
-
29
], the influence of various roughness models for fluid/bodies interactions
(singular body boundaries and slip at the fluid/body interface) is discussed. With
any of these models, the no-collision paradox is ruled out.
Analysis of the Stokes System.
Let us first consider the same system as in [
7
].
Namely, we assume that D
R
d
C
B
1
.t/ D B.G.t/;1/ has
constant density
1
>0. Without restricting the generality, we set the radius of
the moving body to be 1. The fluid/disk system evolves according to the following
simplified system:
,thatn D 1 and that
u
f
rp
f
D 0;
in
Q
F
;
(4.79)
r
u
f
D 0;
in
Q
F
;
(4.80)
u
f
D C ! .x G/; on @
B
1
.t/;
(4.81)
u
f
D 0;
on @ and at infinity;
(4.82)
G D ;
(4.83)
Z
m
1
P
D
B
1
.t/
T
.
u
f
;p
f
/nd C mg
(4.84)
@
Z
J ! D
.x G.t//
T
.
u
f
;p
f
/nd:
(4.85)
@
B
1
.t/
This system is obtained from (FRBI) by deleting the convective terms in the Navier
Stokes system (yielding a stationary Stokes system). We normalized the pressure
in order that the gravity does not appear in the fluid equation. This yields the term
mg corresponding to Archimedes' force, we denoted m D m
1
4=3
f
where
m
1
is the mass of the rigid body and
f
D 1. We also applied that the body is a
homogeneous sphere so that its inertia reduces to a scalar matrix
J
1
D J
I
d
.
B
1
.t/;;!/;
u
f
;p
f
/ a priori. However, we
remark that the body domain is completely fixed by G.t/ so that we might reduce the
unknown
In (
4.79
)-(
4.85
), the unknowns are ..
B
1
.t/ to G.t/. Furthermore, we note that (
4.79
)-(
4.82
) is a Stokes system
with unknowns .
u
f
;p
f
/ and data .G.t/;;!/. As this system is well-posed (see
[
23
, Chap. V]), .
u
f
;p
f
/ might be seen as a function of these data. Finally (
4.79
)-
(
4.85
) reduces to an autonomous differential system:
G D
(4.86)
m
P
D F.G;;!/C m
1
g
(4.87)
J ! D T.G;;!/
(4.88)
in the unknown .G;;!/and where F.G;;!/(resp. T.G;;!/) is the force (resp.
torque) exerted on
B
1
by the solution .
u
f
;p
f
/ to the Stokes system (
4.79
)-(
4.82
).
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