Topics in the Mathematical Theory of Interactions of Incompressible Viscous Fluid with Rigid Bodies - Fluid-Structure Interaction and Biomedical Applications - page 303

As a consequence, it is unlikely to obtain an equivalent bound to ( 4.78 )priortoa
detailed analysis of the contact issue.
First studies tackling the question on contact occurrence, in the multi-dimen-
sional case, go back to the 1960s. In a series of papers, M.E. O'Neill and his
collaborators consider a rigid sphere moving close to a plane wall and compute
the forces exerted by a solution to the stationary Stokes system on the sphere
[ 6 , 7 , 13 , 50 ]. They apply methods previously developed by Brenner [ 3 ] to the case
where the ratio distance between the ramp and the body vs radius of the body is
small. They show that the drag, which the sphere undergoes, diverges rapidly when
the ratio goes to 0 and prevents the sphere from touching the wall in finite time.
This is called afterwards the no-collision paradox . Similar computations in the
lubrication approximation are gathered in [ 10 ]. In the more recent papers [ 26 , 38 -
40 ], the authors show that the no-collision paradox for the Stokes system extends
to solutions to (FRBI) in many cases. In [ 38 , 39 ], the two-dimensional and three-
dimensional cases of a sphere, or a cylinder, moving close to a plane wall, are
considered. As in the studies of M.E. O'Neill and his collaborators, it is proven
that no-contact between the rigid body and the wall occurs in finite time. In [ 39 ], an
example of a three-dimensional configuration in which contact occurs is exhibited.
However, this construction is limited to a very peculiar configuration so that the
no-collision paradox seems to hold generically.
All the results concerning the full system (FRBI) are obtained by applying a
multiplier method. This multiplier is constructed thanks to a detailed analysis of
solutions to the stationary Stokes problem and the associated drag. In this section,
we first present an efficient method to compute the drag in the frame of the Stokes
problem. We then recall the Lorentz formula associated with the Stokes problem
and discuss its application in the extension of the no-collision paradox to solutions
to (FRBI) yielding the results in [ 26 , 38 - 40 ].
To conclude this introductory part on the contact issue, we note that all the
mentioned results underline that the system (FRBI) is not relevant to describe the
close-contact interactions between rigid bodies. In real life, one expects contact
between rigid bodies to occur in much more general contexts than the one exhibited
in [ 40 ], see [ 46 ] for experiments. In order to derive (FRBI) we assumed implicitly
that the relative velocity of two bodies is slow w.r.t. their distance so that we might
neglect several phenomena in the fluid and bodies behaviors. In particular, when
the distance between rigid bodies becomes very small with the bodies having fast
relative velocities:

• the fluid pressure diverges so that non-newtonian properties [ 2 ] and compress-

ibility [ 48 ] might become critical in the fluid behavior;

• the fluid strain-tensor becomes large so that slip at the fluid/bodies interface [ 42 ]

and also elasticity in the bodies equations [ 12 ] should be considered;

• the fluid-layer is thin so that asperities in the description of the bodies surfaces

and container boundaries should be included [ 62 ].