Stabilization of a Rigid Body in Two Fluids With Surface Tension

Abstract

This paper poses and solves a stabilization problem of a rigid body governed by nonlinear differential equations in two viscous incompressible fluids governed by Navier–Stokes equations (NSEs), where surface tension of the interface between the two fluids is considered, in a bounded domain in three dimensional space. Since only weak solutions of the NSEs exist globally while global existence of their strong/smooth solutions is a millennium problem, point-wise fluid forces and moments acting on the rigid body are not able to bound. This difficulty is overcome by designing an appropriate control law and performing stability analysis of the closed-loop system including the NSEs and surface tension, where “work and power of the two fluids” instead of forces and moments on the rigid body are used. A simulation is included to illustrate the results.