Mathematical and Numerical Analysis of Some FSI Problems - Fluid-Structure Interaction and Biomedical Applications - page 14

In [ 102 , 104 ] a no-collision result is proven when there is only one body in a
bounded two-dimensional cavity. Later on, the result was extended to the three-
dimensional situation in [ 105 ]. The case of grazing collisions was studied in [ 106 ].
In [ 52 ] (see p. 287) the case of a rigid sphere surrounded by an incompressible
viscous fluid inside a cavity was considered and a “paradoxical” solution to the
subsequent problem in which the sphere sticks to the ceiling of the cavity without
falling down was constructed. Moreover in [ 151 ], collisions, if any, are proved to
occur with zero relative translational velocity as soon as the boundaries of the
rigid objects are smooth and the gradient of the underlying velocity field is square
integrable—a hypothesis satisfied by any Newtonian fluid flow of finite energy. The
possibility or impossibility of collisions in a viscous fluid is related to the properties
of the velocity gradient. A simple argument reveals that the velocity gradient must
become singular (unbounded) at the contact point since otherwise the streamlines
would be well defined, in particular, they could never meet each other.
In [ 45 ], inspired by Feireisl et al. [ 54 ], the motion of several rigid bodies in a non-
Newtonian fluid of power-law type (see Chap. 1 of [ 123 ] for details) is considered. It
is shown that not only that a weak solution exists but also that collisions cannot occur
in such viscous fluids. The question of the influence of the smoothness of boundary
on the existence of collisions was recently investigated in [ 78 ]and[ 77 ]. Moreover
slip boundary conditions at the fluid-structure interface have been considered in [ 79 ]
and [ 80 ] where, respectively, existence of weak solutions is proven up to collision
and it is showed that collision may occur. Recently, the existence result was extended
up to the contact [ 28 ] but for a slightly different problem.
Concerning the motion of rigid bodies in a viscous compressible fluid let us
mention [ 40 ], then [ 52 ], where the existence of global-in-time weak solutions was
proved. This case was extended to case with self-gravitation force in [ 46 ]. See also
regularity results [ 17 ].
Comparing with results in the incompressible case, there is no restriction on the
existence time, regardless of possible collisions of two or more bodies or contact of
a body with the boundary.
In [ 53 ] the problem of the long time behavior of global-in-time solutions is
addressed. The authors restrict themselves to the simplest situation of a rigid ball
in a viscous fluid occupying a two-dimensional bounded domain. Assuming there
is a body force (gravity) acting in the vertical direction, they show that the rigid
body approaches, as time tends to infinity, a static state when the body touches the
boundary. Thus the contact albeit possibly absent in any finite instant must occur in
the asymptotic regime in the long run.
Concerning investigation of motion of bodies in inviscid case one can refer to
[ 83 , 84 ].
Concerning an elastic structure evolving in incompressible flow, we can refer to
[ 41 ]and[ 19 ] where the structure is described by a finite number of eigenmodes
or to [ 10 , 16 , 119 , 120 ] for an artificially damped elastic structure. In [ 15 ]the
case of a compressible fluid was also considered. Note that, in these cases, the
displacement of the structure remains regular enough and that we have either a