Biomedical Engineering Reference
In-Depth Information
for given f;g;h and with the additional constraint that @
t
has a zero mean.
This system has to be completed by initial conditions and we assume that periodic
boundary conditions in x are satisfied. In the fixed point procedure the right hand
side terms will typically contain nonlinear terms that write div ...I A
f
/r/v/
for f or the second component of ..I A
f
/r/v n for h, with v a given velocity
in L
2
.0;T I H
#
.
O
f
// \ H
1
.0;T
I L
#
.
O
f
//. Under the assumption that
0
D 0
these terms shall be small in suitable spaces for small time since at initial time
A
f
D B
f
D I.
Note that, since @
t
has a zero average, g has to satisfy the compatibility
condition
R
†
[
0
g n D 0. This property enables to consider the following lifting of
the nonhomogeneous divergence:
O
f
;
z
Cr D 0;
in
O
f
;
div
z
D div g; in
† [
0
;
z
D 0;
on
x
periodic:
Let us underline that this lifting does not modify the kinematic coupling condition
and consequently the new velocity v
z
is still equal to .0;@
t
/
T
on the fluid-
structure interface. It is strongly linked to the only transverse motion. Note moreover
that, in the fixed point procedure, g will write .I B
f
/
T
v, with v satisfying
v
1
D 0 on † and v D 0 on
0
. Consequently g D 0 on †. It implies
that if @
t
g 2 L
2
.0;T I L
#
.
O
f
// then div .@
t
g/ belongs to L
2
.0;T I .H
#
.
O
f
//
0
/.
Furthermore, div g D .I B
f
/ Wrv since B
f
is the cofactor matrix of the
deformation gradient r
f
and thanks to the Piola's identity [
31
]. Consequently
div g will have the regularity of .I B
f
/ Wrv.
With this lifting we have to study the following coupled problem, by changing
the right hand side still denoted here f and h:
f
@
t
w
w
Cr D f; in
O
f
:
div
w
D 0; in
O
f
:
w
D .0;@
t
/
T
; on †:
w
D 0; on
0
:
s
e@
tt
Ǜ
2
@
xx
LJ
2
@
xx
@
t
D ..
w
;/n/
2
C h; on .0;L/;
x
(1.70)
periodic:
As we have seen in Sect.
1.2.1
, to naively decouple the fluid from the structure to
prove existence of a solution of (
1.70
) leads to impose a condition on the density
of structure (that has to be large enough with respect to the fluid one). The key
idea is then to rewrite this coupled system thanks to an added mass operator and to
spilt the fluid problem into two sub-problems. We decompose
w
D
w
e
C
w
s
, with
w
s
D .I P/
w
,
w
e
D P
w
,whereP denotes the Leray operator on the subspace of
L
2
.
O
f
/ of divergence free vectors with zero normal trace on the boundary. In the
same way D
e
C
s
.Then
w
s
Dr,where is the solution (up to an additive
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