Topics in the Mathematical Theory of Interactions of Incompressible Viscous Fluid with Rigid Bodies - Fluid-Structure Interaction and Biomedical Applications - page 296

We note here that .5d/=4is always smaller than 1 so that this last inequality does
not rule out contact. Nevertheless, if contact occurs in T we might integrate the
above inequality between t and T for arbitrary t<T , this yields that:

D o..T t/ 2 /; as Z T
0

d 1

kr u .t; /I L 2 ./k

2 dt<1:

jh.t/j

4

Adding that dŒf B i .t/g i D 0;:::;n h.t/, we finally obtain:
• dŒf B i .t/g i D 0;:::;n D o..T t/ 2 / when d D 2, yielding that a contact is glueing
in the sense that both relative velocity and acceleration of colliding bodies vanish
at contact-time;
• dŒf B i .t/g i D 0;:::;n D o.T t/when d D 3.
In the two-dimensional case, the content of this remark is due to [ 51 ].
The second criterion ( 4.69 ) appears in [ 40 ]. We generalize it here to the case of
a container having a smooth but arbitrary boundary. Prior to contact, a solution
.. i ;! i / i D 1;:::;n ; u f ;p f / with smooth initial datum satisfies:
Z T

kr

2 dt<1: 2 u f I L 2 . 2

Ckrp f I L 2 .

F F

.t//k .t//k

0

The second criterion shows that this estimate does not hold any longer after
a contact. We believe it is possible to generalize the computation to arbitrary
geometries. The difficulties here are that, on the one hand, the second order
derivatives of the extended velocity-field are not a priori L 2 -functions on , because
of the discontinuities in the stress tensor on solid boundaries. So, we might not
reduce our computations to the case of rigid disks as in the first criterion. On
the other hand, we need the symmetries of the ball B 1 in order that the lateral

flux cancels on @ B 1 and @. To generalize the computations above to arbitrary

configurations, it should be possible to define domains . l / l>0 adapted to the shape
of the rigid bodies in order that this cancellation property is preserved.

4.4.2 On Weak Solutions with Contact

In Sect. 4.3.1 , we presented a proof of existence of local-in-time weak solutions to
(FRBI). Prior to studying the meaning of weak solutions in case of contact, we show
in this section that it is possible to construct global-in-time weak solutions, whether
contact occurs or not.

Global Existence of Weak Solutions. Following the arguments in [ 20 ], we obtain:

Theorem 4.4. Let the following assumptions hold true:

• is a container having a smooth boundary,