Three-dimensional elastohydrodynamics of a thin plate oscillating above a wall

We consider deflections of a thin rectangular elastic plate that is submerged within a Newtonian fluid. The plate is clamped along one edge and supported horizontally over a plane horizontal wall. We consider both external driving, where the clamped edge is vibrated vertically at high frequencies, and thermal driving, where the plate fluctuates under Brownian motion. In both cases, the amplitude of oscillation is assumed sufficiently small that the resulting flow has little convective inertia, although the oscillation frequency is sufficiently high to generate substantial unsteady inertia in the flow, a common scenario in many nano- and microdevices. We exploit the plate's thinness to develop an integral-equation representation for the three-dimensional flow (a so-called thin-plate theory) which offers considerable computational savings over a full boundary-integral formulation. Limiting cases of high oscillation frequencies and small wall-plate separation distances are studied separately, leading to further simplified descriptions for the hydrodynamics. We validate these reduced integral representations against full boundary-integral computations, and identify the parameter ranges over which these simplified formulations are valid. Addressing the full flow-structure interaction, we also examine the limits of simpler two-dimensional hydrodynamic models. We compare the responses of a narrow plate under two- and three-dimensional hydrodynamic loading, and report differences in the frequency response curves that occur when the plate operates in water, in contrast to the excellent agreement observed in air.