Biomedical Engineering Reference
In-Depth Information
Chapter 4
Topics in the Mathematical Theory
of Interactions of Incompressible Viscous Fluid
with Rigid Bodies
Matthieu Hillairet
Abstract
In this paper, we review recent results devoted to the interactions between
a collection of rigid bodies .
B
i
/
i
D
1;:::;n
and a surrounding viscous fluid
L
, the whole
system filling a container . We assume that the motion of
(resp. the rigid bodies
B
i
) is governed by the incompressible Navier Stokes equations (resp. Newton laws),
and that velocities and stress tensors are continuous at the fluid/body interfaces. Our
concern is the well-posedness of the associated Cauchy problem, with a specific
eye towards the handling of contact between bodies or between one body and the
container boundary.
L
Keywords
Cauchy theory • Contact issue • Fluid-solid interactions
MSC2010:
35Q35, 35B44, 35Q74, 74F10, 76D03, 76D05
4.1
Introduction
Studying the motion of rigid bodies inside a viscous fluid is crucial to many natural
and engineering problems such as sedimentation, filtration or slurry erosion, to men-
tion a few. In biological flows also, a disperse phase containing rigid bodies appears
in many contexts: in the modeling of rigid tracers [
11
]orsprays[
17
], in rheological
studies on active suspensions [
14
,
36
]. In all these cases, bodies/swimmers might
exhibit a complex behavior because of their elastic properties or their elaborate
retroaction on the fluid. They also can be numerous so that their collective behavior
is efficiently described by an equation of Vlasov type (see [
31
,
32
]and[
63
],
for instance). Nevertheless, a toy-model to tackle such complex problems is to
assume that the bodies/swimmers behave as a finite number of undeformable bodies
submitted to Newton laws. As for the surrounding fluid, assumptions on its behavior
might contain more-or-less complexity. In studies on micro swimmers, for instance,
Reynolds numbers are so small that a stationary Stokes system is relevant. To keep
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