Mathematical and Numerical Analysis of Some FSI Problems - Fluid-Structure Interaction and Biomedical Applications - page 36

With this notation we have: D f M .@ tt /. Note that every intermediate

pressure is defined up to an additive constant that can be taken such that the average
of each pressure is zero. Consequently we have added a constant c in the right hand
side which is the Lagrange multiplier of the constraint R 0 @ t D 0.Nowweare
in a position to decouple the fluid from the structure equations and more precisely
we decouple ( 1.71 ) from ( 1.72 ). Then the fixed point procedure can be applied to
prove the existence of a regular solution of the linear nonhomogenous problem since
now the added mass is treated implicitly. Indeed the regularity in time of . w e ; e /
depends only on the regularity of @ t and not on the one of @ tt . Note that here one
needs to have LJ 2 >0in order to obtain enough regularity of the structure velocity.
Remark 1.10. To decouple the fluid from the structure in this way does not allow
to take advantage of the fluid dissipation that induces, on the full coupled problem,
dissipation of the elastic structure.

The second step is the fixed point procedure. It is based on the Picard fixed
point theorem. To a given .Q ; v; p/ we associated .;v;p/ solution of the linear
coupled problem, with right hand sides f;g;h depending on .Q ; v; p/. Typically
f contains terms of the form div..I A f /rv/ or .I B f /rq with A f and B f
defined by ( 1.65 ), ( 1.66 ), ( 1.67 ). The key argument of this step is that for a given
Q 2 L 2 .0;T I H # . O f // \ H 1 .0;T I L # . O f //, with Q .0/ D 0 then A f , B f will
stay, for small enough time, in a neighborhood of the identity matrix in spaces that
are multiplayer of L 2 .0;T I H # . O f //, which is the space to which rv belongs to
if v 2 L 2 .0;T I H # . O f // \ H 1 .0;T I L # . O f //. Note that, at this step, one has to
pay a particular attention on the dependency of the various constants with respect to
the time since one wants to prove existence for small time.
The open questions raised by this study are numerous. First could we prove
existence of strong solutions with no additional viscosity of the elastic part (i.e.,
LJ 2 D 0) and do these solutions exist till the elastic boundary touches the bottom of
the fluid cavity? Moreover could we consider other type of boundary conditions for
the fluid such as Neumann boundary conditions? One of the key points is then the
regularity of the fluid velocity and pressure that satisfy a Stokes like system with
mixed Neumann-Dirichlet boundary conditions. Note that considering ( 1.5 ), ( 1.6 )
or ( 1.8 ), ( 1.9 ) will not give the same type of regularities. Another question is: could
we include also longitudinal motion of the elastic part of the boundary as for the
steady state case (see [ 86 ])?

1.2.4 Non-Newtonian Shear-Dependent Fluid

The aim of this section is to present analysis of the fluid-structure interaction
problem for some non-Newtonian fluids. Moreover, we will also present a com-
plementary proof of the existence of the weak solution that is based on the so-called
global iterative method with respect to the domain deformation and special ";