Biomedical Engineering Reference
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liquid is slow enough as to apply the Stokes approximation. We thus have
u
tt
u
. C /rdiv
u
D f in
S
.0; 1/;
v
t
divT.v;p/D 0
divv D 0
in
F
.0; 1/
(3.88)
where v;pare velocity and pressure fields of the liquid,
T.v;p/Dp
1
C
rv C .rv/
>
(3.89)
is the Cauchy stress tensor, and >0the shear viscosity coefficient. To (
3.88
)we
have to append the conditions at the interface WD @
S
, of continuity of stress
vector and velocity:
.
u
/ n DT.v;p/ n; v D
u
t
on ;
(3.90)
where
WD .r
u
C .r
u
/
>
/ C .div
u
/
1
;
is the (linearized) Cauchy stress tensor, and n is the unit outer normal to
S
, along
with the adherence condition for v
v D 0 on @
F
:
(3.91)
In [
4
] it is shown, among other things, that the problem (
3.88
)-(
3.91
)definesa
strongly continuous semigroup of contractions,
U
.t/, on the space of “finite energy”
X WD
ǚ
.
u
;
u
t
;v/ 2 ŒH
1
./
3
H./
where, we recall, H./is defined
in (
3.83
). Moreover, in [
4
, Theorem 4.2(ii)] it is also proved that the infinitesimal
generator,
ŒL
2
./
3
, of the semigroup has 0 as an eigenvalue, with corresponding one-
dimensional eigenspace. As we mentioned earlier, this is due to the circumstance
that the pressure field associated with steady-state solutions to (
3.88
)-(
3.95
)is
determined only up to a constant, due to the (restrictive) hypothesis that the interface
is
fixed
. To see this, if in the above equations with f 0 we assume
u
;v and p
independent of t,(
3.88
)
1
and (
3.88
)
2;3
decouple
. In particular, from (
3.88
), (
3.90
)
2
,
and (
3.91
)wegetv 0, p D , arbitrary 2
R
A
,and
u
D
u
0
.x/ satisfying the
following pure traction problem
div.
u
0
/ D 0 in
S
;
.
u
0
/ n Dn on :
A way of avoiding this “unrealistic” family of solutions is to restrict the study of
the evolution to the space orthogonal to
N
.
/. This is exactly what is done in [
3
],
where it is shown that, in fact, the restriction
A
U
0
.t/ of
U
.t/ to the space X
0
WD
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