Mathematical and Numerical Analysis of Some FSI Problems - Fluid-Structure Interaction and Biomedical Applications - page 16

like to mention [ 99 ] where the continuity of the solution with respect to the data
was investigated a 2D/1D context and for >0. The study of the existence of weak
solution for unsteady fluid-structure interaction problem for shear-thickening flow
was investigated in [ 111 ]. This yields the existence of at least one weak solution of
the fully coupled unsteady fluid-structure interaction between the non-Newtonian
shear-dependent fluid and the elastic string. Note that in this case an additional
viscous term is added @ x @ t . For more details see Sect. 1.2.4 . Moreover the coupling
of a Newtonian (or resp. a generalized Newtonian flow) and a linearly elastic Koiter
shell has recently been studied in [ 118 ], (resp. [ 117 ]). In these studies the mid-
surface of the structure is not flat anymore and other types of arguments to prove
compactness than the ones we will develop here are used. To complete the references
see [ 49 , 68 , 90 , 91 , 139 , 141 ].
The aim of this section is to present some known results on existence of a solution
of fluid-beam (or rod) coupled problems and to show how these problems could be
approximated or decoupled, how compactness results could be derived. The section
is organized as follows: we will consider the unsteady problem and review some of
the existing results that can be found on the problem. Note that we will only consider
the case where D 0 which is the case that has been treated in the literature so far.
In a first part we will explain on a simplified linear problem the so-called added
mass effect and why it may lead to some mathematical (and numerical) difficulties.
Then we will review some results of existence of weak and strong solutions. In
particular, we will see how to prove existence of weak solutions and how one
can obtain compactness of a sequence of approximated solutions in the case of a
damped structure first and then in the undamped case. Next the general ideas of
the proof of existence of strong solution will be developed. We will explain the
decoupling strategy used. This strategy enables to exploit the properties of each
sub-problem but impose to treat carefully the so-called added mass effect. This
theoretical section is concluded by presenting the existence result for the interaction
of a non-Newtonian fluid with a viscoelastic structure. The proof inherits some
techniques from numerical simulations, such as the artificial compressibility and
the global iterations with respect to the moving domain (i.e., the Schauder fixed
point theorem). We will underline which kind of additional difficulties arise due to
the non-Newtonian behavior of the fluid.

1.2.1 A Linear Simplified Problem

Let us consider the following toy problem that has been introduced in [ 25 ]to
illustrate the role played by the added mass effect on the numerical stability of
explicit schemes for fluid-structure interaction problems. For the fluid we consider
a perfect, inviscid flow described by the following equations:

O f ;

f @ t u Crp D 0; in (1.30)