Consider an elastic solid B fully submerged in a viscous liquid L that occupies
the whole space outside B. We suppose the motion of the liquid is governed by
the Navier-Stokes equations, and that the elastic solid is a St.Venant-Kirchhoff
We are interested in the mathematical analysis of certain motions of the coupled
system solid-liquid S. Consequently, the governing equations consist of a Navier-
Stokes system coupled with a nonlinear system of elasticity equations. Since no
constraints are imposed on the motion of B, the boundary values correspond to
those of a free traction
We shall assume that, with respect to an inertial frame I, B moves under the
action of a given constant body force b. No constraints are enforced on the motion
of B, namely, the solid moves freely.
Since b does not depend on time, it is natural to wonder whether there exists
a frame F with respect to which B may be in equilibrium, that is, the deformation
Φ evaluated in F from a stress-free reference configuration Ω is time independent.
However, the liquid is also producing a traction on the surface of B, which, therefore,
should be time-independent as well. This will be certainly the case if, with respect
to F, the motion of the liquid is steady. If such a frame F exists, the corresponding
motion of S in F will be referred to as steady motion.
Besides their intrinsic mathematical interest, steady motions may be also viewed
as “terminal states” of several important phenomena. A simple but significant ex-
ample is the free fall of a deformable body in a viscous liquid when the density
of the liquid is much smaller than that of the solid; see Remark 3.5. A more so-
phisticated example is furnished by the towing of airborne or underwater bodies,
such as antennas, banners, gliders, and targets, by powered aircraft or boats; see
for example [10, 49].
Objective of this paper is to prove the existence of steady motions. More
precisely, under suitable smoothness and geometric assumptions on the reference
configuration—that will be described subsequently—we show that there is always
a smooth steady motion, provided an appropriate dimensionless number, involving
the density and the Lam´ e coeﬃcients of the solid, the shear viscosity coeﬃcient of
the liquid, and the magnitude of b, see (3.30), is below a certain constant.
Before detailing the basic assumptions, the methods, and the significant fea-
tures of our approach, we would like first to comment on the physical character of
our results. Specifically, the most general steady motion of S that we obtain can be
described, with respect to the inertial frame I, as follows. The solid rotates, with
constant angular velocity directed along b, while its center of mass describes, with
called pure traction problem, or simply traction problem.