Topics in the Mathematical Theory of Interactions of Incompressible Viscous Fluid with Rigid Bodies - Fluid-Structure Interaction and Biomedical Applications - page 299

n B

0 and initial data .. B

By construction, the new container D R d

i ; O i / i D 1;:::;n 0 ; u /
satisfy the assumptions of Theorem 4.1 . This yields T 0 and a weak solution to
(FRBI) .. ' i ; O i / i D 1;:::;n 0 ; u /. Applying the associated isometries to the

B

0

i , we might

then split

n

B i .t/ D [
j 2 M i

X
i D 1 ' i

O i D X

B j .t/;

' i D

j 2 M i Q i :

where:

' j .t; / D 1 B j .t/ ; Q j DO i ' j 8 i 2f1;:::;ng8j 2 M i :

Finally straightforward computations show that we obtain a weak solution on
.0;T 0 C T 0 / when concatenating ..' i ; i / i D 1;:::;n ; u / and .. ' i ; Q i / i D 1;:::;n ; u /.This
ends the proof.

Properties of Weak Solutions with Contact. We extended the definition of weak
solutions for (FRBI) after contact. However, even if the initial body shapes had
smooth boundaries and the fluid velocity-field u f of the weak solution were smooth,
there would remain several discrepancies between the weak solution and what we
expect a classical solution of (FRBI) to be.
The first difference comes from the set of test-functions that we have chosen.
Indeed, we require that a test-function w coincides with a rigid velocity-field on
a neighborhood of

.t/ for all t. This restriction does not enable to dis tingu ish
between rigid bodies which belong to the same connected component of

S

.t/,or
equivalen tly, b etw een r igid bodies in contact. For instance, let two indices i and
j satisfy

S

B i .t/ \ B j .t/ ¤;for t 2 .0;T/. Then, we have that any test-function

w 2 K . Q S / satisfy:

8 x 2 B i .t/ [ B j .t/:

w .t;x/ D C ! x;

Consequently, setting ! D 0, letting take arbitrary values and reproducing
formally the computations of Sect. 4.3.1 , we do not get ( 4.12 )fori and j separately
but merely the equation:

Z

dt m i i C m j j .t/ D

d

@ B i .t/ [ @ B j .t/ T

nd:

Similarly, only one equation holds for a combination of ! i and ! j . Consequently,
the system is algebraically underconstrained unless a compatibility condition
between . i ;! i / and . j ;! j / is implied by the property that contact holds between
B i and

B j .