Steady Motion of a Self-Propelled Body in a Viscous Fluid: Dirichlet Boundary Conditions with Nonzero Flux

Abstract

We study the steady self-propelled motion of a rigid body immersed in an incompressible viscous fluid occupying an exterior domain in $\mathbb{R}^3$. In a body-fixed reference frame, the problem is described by a coupled fluid-rigid body system posed in a fixed exterior domain, where the Navier--Stokes equations are coupled with the unknown translational and angular velocities of the rigid body. The self-propulsion is generated by prescribing a boundary velocity on the body surface. Our main result asserts the existence of weak solutions under the sole assumption that the prescribed boundary flux is sufficiently small. The crucial step is the construction of a divergence-free extension of the boundary data that does not rely on either the zero-flux condition or the smallness of the boundary data. As a result, we generalize the result of Galdi [Theorem 5.1 in On the steady self-propelled motion of a body in a viscous incompressible fluid. Arch. Rational Mech. Anal., 148 (1999), 53-88], where both assumptions were required.