Vibration of plates in contact with viscous fluid: Extension of Lambʹs model

The natural frequency of a plate decreases when in contact with a fluid. An analytical calculation using the Navier–Stokes equations for Newtonian fluids is performed to find the dependency of the frequency decrease on the fluid viscosity. The analysis follows the rationale of Lambʹs model, and gives the viscosity contribution to the added virtual mass. The magnitude of the viscous effect is found to be inversely related to the thickness of the plate. Therefore, while the viscous effect is negligible for macroscopic plates, it is significant for thin plates incorporated in micro-sensors fabricated by silicon technology. The analytical results are shown to be well correlated with measured natural frequencies taken from published data of a micro-resonator in contact with viscous water–glycol mixtures.