Steady Self-Propelled Motion of a Rigid Body in a Viscous Fluid with Navier-Slip Boundary Conditions

TL;DR

This paper proves the existence of steady self-propelled motion of a rigid body in a viscous fluid with Navier-slip boundary conditions, highlighting boundary effects' role in propulsion.

Contribution

It introduces a new analytical framework for steady motion with Navier-slip conditions, including a Korn-type inequality and a characterization of boundary-driven propulsion.

Findings

Existence of weak steady solutions under small boundary flux conditions.

A necessary and sufficient condition for boundary slip to induce motion.

Development of a finite-dimensional thrust space for boundary-driven propulsion.

Abstract

We investigate the steady self-propelled motion of a rigid body immersed in a three-dimensional incompressible viscous fluid governed by the Navier-Stokes equations. The analysis is performed in a body-fixed reference frame, so that the fluid occupies an exterior domain and the propulsion mechanism is modeled through nonhomogeneous Navier-slip boundary conditions at the fluid-body interface. Such conditions provide a realistic description of propulsion in microfluidic and rough-surface regimes, where partial slip effects are significant. Under suitable smallness assumptions on the boundary flux and on the normal component of the prescribed surface velocity, we establish the existence of weak steady solutions to the coupled fluid-structure system. A key analytical ingredient is the derivation of a Korn-type inequality adapted to exterior domains with rigid-body motion and Navier-slip interfaces, which yields uniform control of both the fluid velocity and the translational and rotational velocities of the body. Beyond existence, we provide a necessary and sufficient condition under which a prescribed slip velocity on the body surface induces nontrivial translational or rotational motion of the rigid body. This is achieved through the introduction of a finite-dimensional thrust space, defined via auxiliary exterior Stokes problems with Navier boundary conditions, which captures the effective contribution of boundary-driven flows to the rigid-body motion. Our results clarify how boundary effects generate propulsion and extend the classical Dirichlet-based theory to the Navier-slip setting.