Small Forced Oscillation of a Rigid Body in a Viscous Liquid

The chapter delves into the critical problem of viscous flow around oscillating bodies, relevant across scales from microfluidics to marine vehicles. It introduces a simplified model of a two-dimensional rectangular structure subject to elastic restoring forces in a Navier-Stokes liquid, driven by a periodic velocity field. The primary objective is to investigate the existence of T-periodic solutions and the potential for resonance. The authors employ the Stokes approximation for creeping flow and demonstrate the unique solvability of the problem for arbitrary T > 0, ruling out resonance in this approximation. The methodology combines ideas from previous works, establishing uniform bounds on the Fourier modes of the displacement. The chapter concludes with a detailed proof and discussions on potential extensions to nonlinear problems and vortex-induced oscillations, setting the stage for future research.