Fluid-plate interaction under periodic forcing
The motion of a thin elastic plate interacting with a viscous fluid is investigated. A periodic force acting on the plate is considered, which in a setting without damping could lead to a resonant response. The interaction with the viscous fluid provides a damping mechanism due to the energy dissipation in the fluid. Moreover, an internal damping mechanism in the plate is introduced. In this setting, we show that the periodic forcing leads to a time-periodic (non-resonant) solution. We employ the Navier-Stokes and the Kirchhoff-Love plate equation in a periodic cell structure to model the motion of the viscous fluid and the elastic plate, respectively. Maximal Lp regularity for the linearized system is established in a framework of time-periodic function spaces. Existence of a solution to the fully nonlinear system is subsequently shown with a fixed-point argument.